From 39e3ccd36e7eefbe4c93829aca5498081c3aef09 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Fri, 19 Oct 2018 05:52:15 +0200 Subject: [PATCH] correcting typos --- doc/pub/NeuralNet/html/._NeuralNet-bs000.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs001.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs002.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs003.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs004.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs005.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs006.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs007.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs008.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs009.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs010.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs011.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs012.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs013.html | 53 +- doc/pub/NeuralNet/html/._NeuralNet-bs014.html | 53 +- 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doc/pub/NeuralNet/html/._NeuralNet-bs087.html | 64 +- doc/pub/NeuralNet/html/NeuralNet-bs.html | 53 +- doc/pub/NeuralNet/html/NeuralNet-reveal.html | 868 +++++++++++++++ .../NeuralNet/html/NeuralNet-solarized.html | 848 ++++++++++++++- doc/pub/NeuralNet/html/NeuralNet.html | 848 ++++++++++++++- doc/pub/NeuralNet/ipynb/NeuralNet.ipynb | 989 +++++++++++++++++- .../ipynb/ipynb-NeuralNet-src.tar.gz | Bin 88051 -> 88051 bytes doc/pub/NeuralNet/pdf/NeuralNet-minted.pdf | Bin 591416 -> 664646 bytes doc/src/NeuralNet/NeuralNet.do.txt | 743 +++++++++++++ 96 files changed, 8888 insertions(+), 172 deletions(-) diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs000.html b/doc/pub/NeuralNet/html/._NeuralNet-bs000.html index 77def0bb5..ddfcae71f 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs000.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs000.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -357,7 +406,7 @@ MathJax.Hub.Config({
  • 9
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs001.html b/doc/pub/NeuralNet/html/._NeuralNet-bs001.html index 5bfe1a146..de81d371e 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs001.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs001.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -342,7 +391,7 @@ a weight variable.
  • 10
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs002.html b/doc/pub/NeuralNet/html/._NeuralNet-bs002.html index 111f8f033..e73bb72be 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs002.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs002.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -392,7 +441,7 @@ humanities to life science and medicine.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs003.html b/doc/pub/NeuralNet/html/._NeuralNet-bs003.html index 867ad951a..dcf0a0514 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs003.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs003.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -357,7 +406,7 @@ methods we discussed earlier.
  • 12
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  • ...
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs004.html b/doc/pub/NeuralNet/html/._NeuralNet-bs004.html index e70bdb090..9c745763f 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs004.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs004.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -349,7 +398,7 @@ to all nodes in the subsequent layer, making this a so-called
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs005.html b/doc/pub/NeuralNet/html/._NeuralNet-bs005.html index 675da577e..a0dfb186f 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs005.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs005.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -358,7 +407,7 @@ recognition.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs006.html b/doc/pub/NeuralNet/html/._NeuralNet-bs006.html index 3de479f0f..73eb8cfc7 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs006.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs006.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -350,7 +399,7 @@ especially well-suited for handwriting and speech recognition.
  • 15
  • 16
  • ...
    • -
  • 88
    • +
  • 109
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs007.html b/doc/pub/NeuralNet/html/._NeuralNet-bs007.html index 714defc0a..9d5562fd5 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs007.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs007.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -351,7 +400,7 @@ type of NN due the unusual activation functions.
  • 16
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs008.html b/doc/pub/NeuralNet/html/._NeuralNet-bs008.html index be95c9ebf..1b186864d 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs008.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs008.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -349,7 +398,7 @@ Such networks are often called multilayer perceptrons (MLPs).
  • 17
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs009.html b/doc/pub/NeuralNet/html/._NeuralNet-bs009.html index 24b560514..3ea03c32f 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs009.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs009.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -354,7 +403,7 @@ as to not restrict the range of output values.
  • 18
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  • ...
    • -
  • 88
    • +
  • 109
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs010.html b/doc/pub/NeuralNet/html/._NeuralNet-bs010.html index 7b0a3bf44..7cdbec57d 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs010.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs010.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -355,7 +404,7 @@ of the outputs of all neurons in the previous layer.
  • 19
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs011.html b/doc/pub/NeuralNet/html/._NeuralNet-bs011.html index 5bab11737..16d32e3f4 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs011.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs011.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -384,7 +433,7 @@ is obtained.
  • 20
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs012.html b/doc/pub/NeuralNet/html/._NeuralNet-bs012.html index 727a7ad2f..f26003662 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs012.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs012.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -366,7 +415,7 @@ $$
  • 21
  • 22
  • ...
    • -
  • 88
    • +
  • 109
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs013.html b/doc/pub/NeuralNet/html/._NeuralNet-bs013.html index 5efc8c4b8..b0b7038d1 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs013.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs013.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -357,7 +406,7 @@ variables are the input values \( x_n \).
  • 22
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  • ...
    • -
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    • +
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs014.html b/doc/pub/NeuralNet/html/._NeuralNet-bs014.html index 3f34aae22..eb4f94e71 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs014.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs014.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -366,7 +415,7 @@ flexibility of a neural network.
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    • -
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    • +
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs015.html b/doc/pub/NeuralNet/html/._NeuralNet-bs015.html index 3f6978e5c..00c8edb01 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs015.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs015.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -376,7 +425,7 @@ $$
  • 24
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    • -
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    • +
  • 109
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs016.html b/doc/pub/NeuralNet/html/._NeuralNet-bs016.html index 9f46a1d47..11a02a8f9 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs016.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs016.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -360,7 +409,7 @@ used as input to the activation functions. For each operation
  • 25
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs017.html b/doc/pub/NeuralNet/html/._NeuralNet-bs017.html index ab7e92de8..aef848854 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs017.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs017.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -354,7 +403,7 @@ for a FFNN to fulfill the universal approximation theorem
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs018.html b/doc/pub/NeuralNet/html/._NeuralNet-bs018.html index e0f264d7b..dc8c5a10f 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs018.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs018.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -362,7 +411,7 @@ $$
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    • +
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs019.html b/doc/pub/NeuralNet/html/._NeuralNet-bs019.html index bb6dc52f4..10d9e9386 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs019.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs019.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -424,7 +473,7 @@ plt.show()
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs020.html b/doc/pub/NeuralNet/html/._NeuralNet-bs020.html index cfc20678c..056ecf225 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs020.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs020.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -375,7 +424,7 @@ like logistic regression or linear regression and their modifications on the oth
  • 29
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs021.html b/doc/pub/NeuralNet/html/._NeuralNet-bs021.html index c61d8cbdf..ebd93e9a6 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs021.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs021.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -365,7 +414,7 @@ the potential of being universal approximators.
  • 30
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs022.html b/doc/pub/NeuralNet/html/._NeuralNet-bs022.html index 2d14d2d6d..e676a8b6e 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs022.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs022.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -371,7 +420,7 @@ classes.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs023.html b/doc/pub/NeuralNet/html/._NeuralNet-bs023.html index 410974113..6b78ec10c 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs023.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs023.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -375,7 +424,7 @@ $$
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs024.html b/doc/pub/NeuralNet/html/._NeuralNet-bs024.html index 8e3fca047..53be8dcc2 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs024.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs024.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -359,7 +408,7 @@ $$
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs025.html b/doc/pub/NeuralNet/html/._NeuralNet-bs025.html index 782f6db27..29491da62 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs025.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs025.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -362,7 +411,7 @@ $$
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs026.html b/doc/pub/NeuralNet/html/._NeuralNet-bs026.html index 0d8640b34..6ae0ebf5e 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs026.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs026.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -387,7 +436,7 @@ $$
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs027.html b/doc/pub/NeuralNet/html/._NeuralNet-bs027.html index c6be3044f..ef088d578 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs027.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs027.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -354,7 +403,7 @@ That is, the error \( \delta_j^L \) is exactly equal to the rate of change of th
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs028.html b/doc/pub/NeuralNet/html/._NeuralNet-bs028.html index 1c30f30f3..e40e694fe 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs028.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs028.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -399,7 +448,7 @@ one \( L-1 \) in terms of the errors in the final output layer.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs029.html b/doc/pub/NeuralNet/html/._NeuralNet-bs029.html index f581bfea2..39af94d22 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs029.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs029.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -369,7 +418,7 @@ We are now ready to set up the algorithm for back propagation and learning the w
  • 38
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs030.html b/doc/pub/NeuralNet/html/._NeuralNet-bs030.html index d69aa8c4e..1c51545ef 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs030.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs030.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -413,7 +462,7 @@ Here it is convenient to use stochastic gradient descent (see the examples below
  • 39
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs031.html b/doc/pub/NeuralNet/html/._NeuralNet-bs031.html index 93532e220..473fedee8 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs031.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs031.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -375,7 +424,7 @@ of our network.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs032.html b/doc/pub/NeuralNet/html/._NeuralNet-bs032.html index e0dd470ad..f28a6f4bc 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs032.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs032.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -397,7 +446,7 @@ The back propagation equations need now only a small change, namely the definiti
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs033.html b/doc/pub/NeuralNet/html/._NeuralNet-bs033.html index 2142367cc..64c11e975 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs033.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs033.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -385,7 +434,7 @@ In case we use another activation function than the logistic one, we need to eva
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs034.html b/doc/pub/NeuralNet/html/._NeuralNet-bs034.html index 8134c9260..b7eb14900 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs034.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs034.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -359,7 +408,7 @@ which in case of the simply binary model reduces to having \( i=j \).
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs035.html b/doc/pub/NeuralNet/html/._NeuralNet-bs035.html index 4b30168ff..71e9befe1 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs035.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs035.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -353,7 +402,7 @@ One can identify a set of key steps when using neural networks to solve supervis
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs036.html b/doc/pub/NeuralNet/html/._NeuralNet-bs036.html index 28ba61266..f33b5a40b 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs036.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs036.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -432,7 +481,7 @@ plt.show()
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs037.html b/doc/pub/NeuralNet/html/._NeuralNet-bs037.html index 59860c02b..c7ab2aeaf 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs037.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs037.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -386,7 +435,7 @@ X_train, X_test, Y_train, Y_test = train_tes
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs038.html b/doc/pub/NeuralNet/html/._NeuralNet-bs038.html index 46732053d..f2d8151c6 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs038.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs038.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -383,7 +432,7 @@ which is inspired by probability theory (see logistic regression) and was most c
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs039.html b/doc/pub/NeuralNet/html/._NeuralNet-bs039.html index 53ca3b191..38295173c 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs039.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs039.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -382,7 +431,7 @@ weights to the output layer.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs040.html b/doc/pub/NeuralNet/html/._NeuralNet-bs040.html index 357afc4f0..1de83339a 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs040.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs040.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -373,7 +422,7 @@ output_bias = np49
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs041.html b/doc/pub/NeuralNet/html/._NeuralNet-bs041.html index 3183c8684..b133d453e 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs041.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs041.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -364,7 +413,7 @@ $$ a_{j}^{L} = \frac{\exp{(z_j^{L})}}
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs042.html b/doc/pub/NeuralNet/html/._NeuralNet-bs042.html index b353fba5c..c2a2f4587 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs042.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs042.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -412,7 +461,7 @@ predictions = predict(X_train)
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs043.html b/doc/pub/NeuralNet/html/._NeuralNet-bs043.html index 0ea651383..b43f51687 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs043.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs043.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -368,7 +417,7 @@ you got the correct label. The probability of category \( c \) is given by the s
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs044.html b/doc/pub/NeuralNet/html/._NeuralNet-bs044.html index 5af3febfe..fa785a6c1 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs044.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs044.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -377,7 +426,7 @@ The various optmization methods, with codes and algorithms, are discussed in o
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs045.html b/doc/pub/NeuralNet/html/._NeuralNet-bs045.html index 7b00e6528..cb34f2101 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs045.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs045.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -370,7 +419,7 @@ calculate the gradient efficently.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs046.html b/doc/pub/NeuralNet/html/._NeuralNet-bs046.html index 723f10fd1..f7b6c6af9 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs046.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs046.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -451,7 +500,7 @@ lmbd = 0.0155
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs047.html b/doc/pub/NeuralNet/html/._NeuralNet-bs047.html index 0e2132b9f..353cd30f8 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs047.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs047.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -357,7 +406,7 @@ Andrew Ng goes through some of these considerations in this 56
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs048.html b/doc/pub/NeuralNet/html/._NeuralNet-bs048.html index f9c7cad52..d66d4c2b3 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs048.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs048.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -450,7 +499,7 @@ being realizations of this object with different hyperparameters. An implementat
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs049.html b/doc/pub/NeuralNet/html/._NeuralNet-bs049.html index bc6c9400e..e32de07a8 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs049.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs049.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -372,7 +421,7 @@ test_predict = dnn58
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs050.html b/doc/pub/NeuralNet/html/._NeuralNet-bs050.html index 83f6785c3..e1e9eaded 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs050.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs050.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -370,7 +419,7 @@ DNN_numpy = np.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs051.html b/doc/pub/NeuralNet/html/._NeuralNet-bs051.html index 2c92749e6..799cf8a88 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs051.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs051.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -379,7 +428,7 @@ plt.show()
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs052.html b/doc/pub/NeuralNet/html/._NeuralNet-bs052.html index 4ff3e2de7..bf2b566f0 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs052.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs052.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -376,7 +425,7 @@ DNN_scikit = np
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs053.html b/doc/pub/NeuralNet/html/._NeuralNet-bs053.html index 0b4856ce2..0010c5a4f 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs053.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs053.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -379,7 +428,7 @@ plt.show()
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs054.html b/doc/pub/NeuralNet/html/._NeuralNet-bs054.html index 697bd76e4..7470337de 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs054.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs054.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -352,7 +401,7 @@ NumPy arrays.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs055.html b/doc/pub/NeuralNet/html/._NeuralNet-bs055.html index 3c198c16b..5c345ef38 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs055.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs055.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -382,7 +431,7 @@ and/or if you use anaconda, just write (or install from the graphical use
  • 64
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs056.html b/doc/pub/NeuralNet/html/._NeuralNet-bs056.html index c47fa4b14..2b57e12cd 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs056.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs056.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -403,7 +452,7 @@ X_train, X_test, Y_train, Y_test = train_tes
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs057.html b/doc/pub/NeuralNet/html/._NeuralNet-bs057.html index 5494018a7..27f07c6ed 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs057.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs057.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -482,7 +531,7 @@ MathJax.Hub.Config({
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs058.html b/doc/pub/NeuralNet/html/._NeuralNet-bs058.html index 5a6bdba2a..0ea92c54e 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs058.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs058.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -415,7 +464,7 @@ writer.add_graph(tf67
  • 68
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs059.html b/doc/pub/NeuralNet/html/._NeuralNet-bs059.html index ccb0885e0..f16661bde 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs059.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs059.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -436,7 +485,7 @@ plt.show()
  • 68
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs060.html b/doc/pub/NeuralNet/html/._NeuralNet-bs060.html index 879ad5e94..4d2c5638f 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs060.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs060.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -366,7 +415,7 @@ learn at widely different speeds
  • 69
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs061.html b/doc/pub/NeuralNet/html/._NeuralNet-bs061.html index 9c490338c..e4c2f4f32 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs061.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs061.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -367,7 +416,7 @@ better than the logistic function in deep networks).
  • 70
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs062.html b/doc/pub/NeuralNet/html/._NeuralNet-bs062.html index cf07f6aae..fa2d57b53 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs062.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs062.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -373,7 +422,7 @@ fast to compute).
  • 71
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs063.html b/doc/pub/NeuralNet/html/._NeuralNet-bs063.html index 4b19e9839..f8aefc426 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs063.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs063.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -364,7 +413,7 @@ $$
  • 72
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    • +
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs064.html b/doc/pub/NeuralNet/html/._NeuralNet-bs064.html index 20b28a4f2..d20879a62 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs064.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs064.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -357,7 +406,7 @@ bootstrap to evaluate other activation functions.
  • 73
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs065.html b/doc/pub/NeuralNet/html/._NeuralNet-bs065.html index c94bbe5eb..dfb40a9c2 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs065.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs065.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -379,7 +428,7 @@ supervised learning.
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    • +
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs066.html b/doc/pub/NeuralNet/html/._NeuralNet-bs066.html index 79b256a05..1e69e0d38 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs066.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs066.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -363,7 +412,7 @@ Some of these remarks are particular to DNNs, others are shared by all supervise
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs067.html b/doc/pub/NeuralNet/html/._NeuralNet-bs067.html index 952bd8b66..3c7e90fa6 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs067.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs067.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -367,7 +416,7 @@ and the slides of 76
  • 77
  • ...
    • -
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    • +
  • 109
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs068.html b/doc/pub/NeuralNet/html/._NeuralNet-bs068.html index d0b0cf8a5..d7b435e61 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs068.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs068.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -365,7 +414,7 @@ would quickly lead to possible overfitting.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs069.html b/doc/pub/NeuralNet/html/._NeuralNet-bs069.html index fad9843d6..ededfecf4 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs069.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs069.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -377,7 +426,7 @@ dimension.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs070.html b/doc/pub/NeuralNet/html/._NeuralNet-bs070.html index ae325d61b..c97306768 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs070.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs070.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -360,7 +409,7 @@ A simple CNN for image classification could have the architecture:
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs071.html b/doc/pub/NeuralNet/html/._NeuralNet-bs071.html index 142441fe9..6f036636b 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs071.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs071.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -356,7 +405,7 @@ are consistent with the labels in the training set for each image.
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs072.html b/doc/pub/NeuralNet/html/._NeuralNet-bs072.html index a1ff786b4..2cc1fbb00 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs072.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs072.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -358,7 +407,7 @@ and the slides of 81
  • 82
  • ...
    • -
  • 88
    • +
  • 109
  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs073.html b/doc/pub/NeuralNet/html/._NeuralNet-bs073.html index ee414f64a..ffe6cc5c9 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs073.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs073.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -353,7 +402,7 @@ matrices, typically 1 for each color dimension (Red, Green, Blue).
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs074.html b/doc/pub/NeuralNet/html/._NeuralNet-bs074.html index 101ae23e5..b45367c01 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs074.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs074.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -349,7 +398,7 @@ $$
  • 83
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  • ...
    • -
  • 88
    • +
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  • »
    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs075.html b/doc/pub/NeuralNet/html/._NeuralNet-bs075.html index af828474c..72fda2090 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs075.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs075.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -355,7 +404,7 @@ single neuron in the first hidden layer.
  • 84
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs076.html b/doc/pub/NeuralNet/html/._NeuralNet-bs076.html index 23daa7f6b..dc0bcc4b5 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs076.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs076.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -353,7 +402,7 @@ fixed, and known as a 85
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs077.html b/doc/pub/NeuralNet/html/._NeuralNet-bs077.html index c610ec456..b1edc81d0 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs077.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs077.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -359,7 +408,7 @@ layer.
  • 86
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs078.html b/doc/pub/NeuralNet/html/._NeuralNet-bs078.html index b3d6cd552..f455a85a1 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs078.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs078.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -351,6 +400,8 @@ classification.
  • 86
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs079.html b/doc/pub/NeuralNet/html/._NeuralNet-bs079.html index 5ccfc29e2..cc3e2fe20 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs079.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs079.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -383,6 +432,9 @@ plt.show()
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs080.html b/doc/pub/NeuralNet/html/._NeuralNet-bs080.html index 91e2e7667..813ed2ac9 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs080.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs080.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -354,6 +403,10 @@ X_train, X_test, Y_train, Y_test = train_tes
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs081.html b/doc/pub/NeuralNet/html/._NeuralNet-bs081.html index e8a28e958..2fe02e70a 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs081.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs081.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -482,6 +531,11 @@ class ConvolutionalNeuralNetworkTensorflow:
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    • diff --git a/doc/pub/NeuralNet/html/._NeuralNet-bs082.html b/doc/pub/NeuralNet/html/._NeuralNet-bs082.html index 7b237939b..7566455ed 100644 --- a/doc/pub/NeuralNet/html/._NeuralNet-bs082.html +++ b/doc/pub/NeuralNet/html/._NeuralNet-bs082.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
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  • Using a deep neural network
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  • Wrapping it up
    • @@ -367,6 +416,12 @@ CNN_tf = np.86
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  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
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  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
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  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -370,6 +419,13 @@ plt.show()
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  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
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  • Setting up the code, feed forward part
    • +
  • Backpropagation
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  • Gradient Descent
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  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -373,6 +422,14 @@ lmbd_vals = np.
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  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -354,6 +403,15 @@ MathJax.Hub.Config({
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  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -367,6 +416,16 @@ plt.show()
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  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -321,7 +370,6 @@ MathJax.Hub.Config({
  • Abstract art using convolutional neural networks
    • -

    diff --git a/doc/pub/NeuralNet/html/NeuralNet-bs.html b/doc/pub/NeuralNet/html/NeuralNet-bs.html index 77def0bb5..ddfcae71f 100644 --- a/doc/pub/NeuralNet/html/NeuralNet-bs.html +++ b/doc/pub/NeuralNet/html/NeuralNet-bs.html @@ -174,7 +174,35 @@ Automatically generated HTML file from DocOnce source ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -299,6 +327,27 @@ MathJax.Hub.Config({
  • Final part
  • Final visualization
  • Fun links
    • +
  • Applications: solving ordinary differential equations with Neural Networks
    • +
  • Trial solution
    • +
  • More details
    • +
  • Reformulating the problem
    • +
  • Estimating errors
    • +
  • Creating a simple Deep Neural Net
    • +
  • Feedforward
    • +
  • Result after weighting
    • +
  • Output
    • +
  • Setting up the code, feed forward part
    • +
  • Backpropagation
    • +
  • Gradient Descent
    • +
  • More on GD and cost function
    • +
  • An implementation of a Deep Neural Network
    • +
  • Feed forward again
    • +
  • The final parts of the code
    • +
  • And adding Back propagation
    • +
  • Solving the ODE
    • +
  • Using neural network
    • +
  • Using a deep neural network
    • +
  • Wrapping it up
    • @@ -357,7 +406,7 @@ MathJax.Hub.Config({
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    • diff --git a/doc/pub/NeuralNet/html/NeuralNet-reveal.html b/doc/pub/NeuralNet/html/NeuralNet-reveal.html index e9961348d..6e4a8b736 100644 --- a/doc/pub/NeuralNet/html/NeuralNet-reveal.html +++ b/doc/pub/NeuralNet/html/NeuralNet-reveal.html @@ -3770,6 +3770,874 @@ plt.show() +
    +

    Applications: solving ordinary differential equations with Neural Networks

    + +

    +We end our discussion on neural networks with a discussion on how to solve differential equations. Here we focus +first on the classical exponential decay in one dimension. Thereafter we switch to the Poisson equation in one dimension. + +

    +The aim is to see if we can use a neural network to solve + +

     
    +$$ +\begin{equation} +\tag{16} +g'(x) = -\gamma g(x) +\end{equation} +$$ +

     
    + +

    +where \( g(0) = g_0 \) with \( \gamma \) and \( g_0 \) being some chosen +values. This equation is an ordinary differential equation since the +function we have to solve for, \( g(x) \), is of one variable. + +

    +Here we set \( \gamma = 2 \) and \( g_0 = 10 \) but feel free to change +them and see how the neural network performs. +

    + + +
    +

    Trial solution

    + +

    +To begin with, a trial solution \( g_t(t) \) must be chosen. A general +trial solution for ordinary differential equations could be + +

     
    +$$ +g_t(x, P) = h_1(x) + h_2(x, N(x, P)), +$$ +

     
    + +

    +with \( h_1(x) \) ensuring that \( g_t(x) \) satisfies some conditions and +\( h_2(x,N(x, P)) \) an expression involving \( x \) and the output from the +neural network \( N(x,P) \) with \( P \) being the collection of the weights +and biases for each layer. + +

    +It is assumed that there are no weights and +bias at the input layer, so \( P = \{ P_{\text{hidden}}, +P_{\text{output}} \} \). If there are \( N_{\text{hidden} } \) neurons in +the hidden layer, then \( P_{\text{hidden}} \) is an \( N_{\text{hidden} } +\times 2 \) matrix. + +

    +The first column in \( P_{\text{hidden} } \) represents +the bias for each neuron in the hidden layer and the second column +represents the weigths for each neuron. If there are \( N_{\text{output} +} \) neurons in the output layer, then \( P_{\text{output}} \) is a +\( N_{\text{output} } \times (1 + N_{\text{hidden} }) \) matrix. Its first +column represents the bias of each neuron and the remaining columns +represents the weights to each neuron. +

    + + +
    +

    More details

    + +

    +We have \( g(0) = g_0 \). The trial solution must fulfill this +condition to be a proper solution of (16). + +

    +A possible way to +ensure that \( g_t(0, P) = g_0 \), is to let \( F(N(x,P)) = x\cdot N(x,P) \) +and \( A(x) = g_0 \). This gives the following trial solution: + +

     
    +$$ +\begin{equation} +g_t(x, P) = g_0 + x \cdot N(x, P). +\tag{17} +\end{equation} +$$ +

     
    +

    + + +
    +

    Reformulating the problem

    + +

    +Often, the role of a neural network is to minimize its parameters with +respect to some given error criteria. This criteria, the cost or loss +function, is a measure of how much error the output of the network has +compared to some given known answers. A reformulation of +(16) must therefore be done, such that it describes the +problem a neural network can solve. + +

    +The neural network must find the set of weigths and biases \( P \) such +that the trial solution in satisfies +(16). The trial solution has been chosen such that it +already solves the condition \( g(0) = g_0 \). What remains, is to find +\( P \) such that + +

     
    +$$ +\begin{equation} +g_t'(x, P) = - \gamma g_t(x, P) +\tag{18} +\end{equation} +$$ +

     
    + +

    +is fulfilled as best as possible. +

    + + +
    +

    Estimating errors

    + +

    +Having two sides of an equation as equal as +possible, means that the absolute or squared difference between the +sides must be as close to zero as small. In this case, the difference +squared is an appropiate measurement of how errorneous the trial +solution is with respect to \( P \) of the neural network. Therefore, the +problem our network must solve, is + +

     
    +$$ +\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\} +$$ +

     
    + +

    +or, in terms of weights and biases for each layer: + +

     
    +$$ +\min_{P_{\text{hidden} }, \ P_{\text{output} }}\Big\{ \big(g_t'(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) - ( -\gamma g_t(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) \big)^2 \Big\} +$$ +

     
    + +

    +for an input value \( x \). +If the neural network evaluates \( g_t(x, P) \) at more avalues for \( x \), say \( N \) values \( x_i \) for \( i = 1, \dots, N \), then the total error to minimize is + +

     
    +$$ +\begin{equation} +\tag{19} +\min_{P}\Big\{\sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \Big\} +\end{equation} +$$ +

     
    + +

    +Letting \( c(x, P) = \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) +\big)^2 \) denote the cost function, the minimization problem of which +our network must solve, is + +

     
    +$$ +\min_{P} c(x, P) +$$ +

     
    + +

    +or in terms of \( P_{\text{hidden} } \) and \( P_{\text{output} } \) + +

     
    +$$ +\min_{P_{\text{hidden} }, \ P_{\text{output} }} c(x, \{P_{\text{hidden} }, P_{\text{output} }\}) +$$ +

     
    +

    + + +
    +

    Creating a simple Deep Neural Net

    + +

    +The next step is to decide how the neural net \( N(x, P) \) +should be. In this case, the neural network is made +from scratch to understand better how a neural network works, gain +more control over its architecture, and see how Autograd can be used +to simplify the implementation. + +

    +Since a deep neural network (DNN) is a neural network with more than +one hidden layer, we can first look on how to implement a neural +network. Having an implementation of a neural network at hand, an +extension of it into a deep neural network would (hopefully) be +painless. + +

    +For simplicity, we assume that the input is an array \( \vec x = +(x_1, \dots, x_N) \) with \( N \) elements. It is at these points the neural +network should find \( P \) such that it fulfills (19). +

    + + +
    +

    Feedforward

    + +

    +First, a feedforward of the inputs must be done. This means that \( \vec + x \) must be passed through an input layer, a hidden layer and a output + layer. The input layer in this case, does not need to process the + data any further. The input layer will consist of \( N_{\text{input} } \) + neurons, passing its element to each neuron in the hidden layer. The + number of neurons in the hidden layer will be \( N_{\text{hidden} } \). + +

    +For the \( i \)-th in the hidden layer with weight \( w_i^{\text{hidden} } \) +and bias \( b_i^{\text{hidden} } \), the weighting from the \( j \)-th neuron +at the input layer is: + +

     
    +$$ +\begin{aligned} +z_{i,j}^{\text{hidden}} &= b_i^{\text{hidden}} + w_i^{\text{hidden}}x_j \\ +&= +\begin{pmatrix} +b_i^{\text{hidden}} & w_i^{\text{hidden}} +\end{pmatrix} +\begin{pmatrix} +1 \\ +x_j +\end{pmatrix} +\end{aligned} +$$ +

     
    +

    + + +
    +

    Result after weighting

    + +

    +The result after weighting the input at the \( i \)-th hidden neuron can be written as a vector: +

     
    +$$ +\begin{aligned} +\vec{z}_{i}^{\text{hidden}} &= \Big( b_i^{\text{hidden}} + w_i^{\text{hidden}}x_1 , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_2, \ \dots \, , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_N\Big) \\ +&= +\begin{pmatrix} + b_i^{\text{hidden}} & w_i^{\text{hidden}} +\end{pmatrix} +\begin{pmatrix} +1 & 1 & \dots & 1 \\ +x_1 & x_2 & \dots & x_N +\end{pmatrix} \\ +&= \vec{p}_{i, \text{hidden}}^T X +\end{aligned} +$$ +

     
    + +

    +It is the vector \( \vec{p}_{i, \text{hidden}}^T \) that defines each row +in \( P_{\text{hidden} } \), which contains the weights for the neural +network to minimize according to (19). + +

    +After having found \( \vec{z}_{i}^{\text{hidden}} \) for every neuron \( i \) +in the hidden layer, the vector will be sent to an activation function +\( a_i(\vec{z}) \). In this example, the sigmoid function has been used: + +

     
    +$$ +f(z) = \frac{1}{1 + \exp{(-z)}}. +$$ +

     
    +

    + + +
    +

    Output

    + +

    +The output $\vec{x}_i^{\text{hidden} }$from each \( i \)-th hidden neuron is: + +

     
    +$$ +\vec{x}_i^{\text{hidden} } = f\big( \vec{z}_{i}^{\text{hidden}} \big). +$$ +

     
    + +

    +The outputs \( \vec{x}_i^{\text{hidden} } \) are then sent to the output layer. + +

    +The output layer consist of one neuron in this case, and combines the +output from each of the neurons in the hidden layers. The output layer +combines the results from the hidden layer using some weights \( +w_i^{\text{output}} \) and biases \( b_i^{\text{output}} \). In this case, +it is assumes that the number of neurons in the output layer is one. + +

    +The procedure of weigthing the output neuron \( j \) in the hidden layer +to the \( i \)-th neuron in the output layer is similar as for the hidden +layer described previously. + +

     
    +$$ +\begin{aligned} +z_{1,j}^{\text{output}} & = +\begin{pmatrix} +b_1^{\text{output}} & \vec{w}_1^{\text{output}} +\end{pmatrix} +\begin{pmatrix} +1 \\ +\vec{x}_j^{\text{hidden}} +\end{pmatrix} +\end{aligned} +$$ +

     
    + +

    +Expressing \( z_{1,j}^{\text{output}} \) as a vector gives the following procedure of weighting the inputs from the hidden layer: + +

     
    +$$ +\vec{z}_{1}^{\text{output}} = +\begin{pmatrix} +b_1^{\text{output}} & \vec{w}_1^{\text{output}} +\end{pmatrix} +\begin{pmatrix} +1 & 1 & \dots & 1 \\ +\vec{x}_1^{\text{hidden}} & \vec{x}_2^{\text{hidden}} & \dots & \vec{x}_N^{\text{hidden}} +\end{pmatrix} +$$ +

     
    + +

    +In this case we seek a continous range of values since we are +approximating a function. This means that after computing +\( \vec{z}_{1}^{\text{output}} \) the neural network has finished its +feedforward step, and \( \vec{z}_{1}^{\text{output}} \) is the final +output of the network. +

    + + +
    +

    Setting up the code, feed forward part

    +

    + + +

    # Note that we use the  numpy wrapper for Autograd (see the gradient descent  slides)
+import autograd.numpy as np
+from autograd import grad, elementwise_grad
+import autograd.numpy.random as npr
+from matplotlib import pyplot as plt
+
+def sigmoid(z):
+    return 1/(1 + np.exp(-z))
+
+def neural_network(params, x):
+    
+    # Find the weights (including and biases) for the hidden and output layer.
+    # Assume that params is a list of parameters for each layer. 
+    # The biases are the first element for each array in params, 
+    # and the weights are the remaning elements in each array in params.   
+    
+    w_hidden = params[0]
+    w_output = params[1]
+
+    # Assumes input x being an one-dimensional array
+    num_values = np.size(x)
+    x = x.reshape(-1, num_values)
+    
+    # Assume that the input layer does nothing to the input x
+    x_input = x
+
+    ## Hidden layer:
+    
+    # Add a row of ones to include bias
+    x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)
+    
+    z_hidden = np.matmul(w_hidden, x_input)
+    x_hidden = sigmoid(z_hidden)
+
+    ## Output layer:
+    
+    # Include bias:
+    x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)
+
+    z_output = np.matmul(w_output, x_hidden)
+    x_output = z_output
+
+    return x_output
+
    +
    + + +
    +

    Backpropagation

    + +

    +Now that feedforward can be done, the next step is to decide how the +parameters should change such that they minimize the cost function. + +

    +Recall that the chosen cost function for this problem is + +

     
    +$$ +c(x, P) = \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 +$$ +

     
    + +

    +In order to minimize it, an optimalization method must be chosen. + +

    +Here, gradient descent with a constant step size has been chosen. + +

    +Before looking at the gradient descent method, let us set up the cost +function along with the right ride of the ODE and trial solution. + +

    + + +

    # The trial solution using the deep neural network:
+def g_trial(x,params, g0 = 10):
+    return g0 + x*neural_network(params,x)
+
+# The right side of the ODE:
+def g(x, g_trial, gamma = 2):
+    return -gamma*g_trial
+
+# The cost function:
+def cost_function(P, x):
+    
+    # Evaluate the trial function with the current parameters P
+    g_t = g_trial(x,P)
+    
+    # Find the derivative w.r.t x of the neural network
+    d_net_out = elementwise_grad(neural_network,1)(P,x) 
+    
+    # Find the derivative w.r.t x of the trial function
+    d_g_t = elementwise_grad(g_trial,0)(x,P)  
+    
+    # The right side of the ODE 
+    func = g(x, g_t)
+
+    err_sqr = (d_g_t - func)**2
+    cost_sum = np.sum(err_sqr)
+    
+    return cost_sum
+
    +
    + + +
    +

    Gradient Descent

    + +

    +The idea of the gradient descent algorithm is to update parameters in +direction where the cost function decreases goes to a minimum. + +

    +In general, the update of some parameters \( \vec \omega \) given a cost +function defined by some weights \( \vec \omega \), \( c(x, \vec \omega) \), +goes as follows: + +

     
    +$$ +\vec \omega_{\text{new} } = \vec \omega - \lambda \nabla_{\vec \omega} c(x, \vec \omega), +$$ +

     
    + +

    +for a number of iterations or until $ \big|\big| \vec +\omega_{\text{new} } - \vec \omega \big|\big|$ is smaller than some +given tolerance. + +

    +The value of \( \lambda \) decides how large steps the algorithm must take +in the direction of $ \nabla_{\vec \omega} c(x, \vec \omega)$. The +notatation \( \nabla_{\vec \omega} \) denotes the gradient with respect to +the elements in \( \vec \omega \). +

    + + +
    +

    More on GD and cost function

    + +

    +In our case, we have to minimize the cost function \( c(x, P) \) with +respect to the two sets of weights and bisases, that is for the hidden +layer \( P_{\text{hidden} } \) and for the ouput layer \( P_{\text{output} +} \) . + +

    +This means that \( P_{\text{hidden} } \) and \( P_{\text{output} } \) is +updated by + +

     
    +$$ +\begin{aligned} +P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} c(x, P) \\ +P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} c(x, P) +\end{aligned} +$$ +

     
    + +

    +This might look like a cumberstone to set up the correct expression +for finding the gradients. Luckily, Autograd comes to the rescue. + +

    + + +

    def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):
+    ## Set up initial weigths and biases 
+    
+    # For the hidden layer
+    p0 = npr.randn(num_neurons_hidden, 2 ) 
+
+    # For the output layer
+    p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included
+
+    P = [p0, p1]
+
+    print('Initial cost: %g'%cost_function(P, x))
+    
+    ## Start finding the optimal weigths using gradient descent
+    
+    # Find the Python function that represents the gradient of the cost function
+    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
+    cost_function_grad = grad(cost_function,0)
+    
+    # Let the update be done num_iter times
+    for i in range(num_iter):
+        # Evaluate the gradient at the current weights and biases in P. 
+        # The cost_grad consist now of two arrays; 
+        # one for the gradient w.r.t P_hidden and 
+        # one for the gradient w.r.t P_output
+        cost_grad =  cost_function_grad(P, x)
+    
+        P[0] = P[0] - lmb * cost_grad[0]
+        P[1] = P[1] - lmb * cost_grad[1]
+
+    print('Final cost: %g'%cost_function(P, x))
+    
+    return P
+
    +
    + + +
    +

    An implementation of a Deep Neural Network

    + +

    +As previously stated, a Deep Neural Network (DNN) follows the same +concept of a neural network, but having more than one hidden +layer. Suppose that the network has \( N_{\text{hidden}} \) hidden layers +where the \( l \)-th layer has \( N_{\text{hidden}}^{(l)} \) neurons. The +input is still assumed to be an array of size \( 1 \times N \). The +network must now try to optimalize its output with respect to the +collection of weigths and biases \( P = \big\{P_{\text{input} }, \ +P_{\text{hidden} }^{(1)}, \ P_{\text{hidden} }^{(2)}, \ \dots , \ +P_{\text{hidden} }^{(N_{\text{hidden}})}, \ P_{\text{output} }\big\} \). +

    + + +
    +

    Feed forward again

    + +

    +The feedforward step is similar to as for the neural netowork, but now considering more than one hidden layer. + +

    +The \( i \)-th neuron at layer \( l \) recieves the result +\( \vec{x}_j^{(l-1),\text{hidden} } \) from the \( j \)-th neuron at layer +\( l-1 \). The \( i \)-th neuron at layer \( l \) weights all of the elements in +\( \vec{x}_j^{(l-1),\text{hidden} } \) with a weight vector \( \vec +w_{i,j}^{(l), \ \text{hidden} } \) with as many weigths as there are +elements in$\vec{x}_j^{(l-1),\text{hidden} }$, and adds a bias +\( b_i^{(l), \ \text{hidden} } \): + +

     
    +$$ +\begin{aligned} +z_{i,j}^{(l),\ \text{hidden}} &= b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_j^{(l-1),\text{hidden} } \\ +&= +\begin{pmatrix} +b_i^{(l), \ \text{hidden}} & \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T +\end{pmatrix} +\begin{pmatrix} +1 \\ +\vec{x}_j^{(l-1),\text{hidden} } +\end{pmatrix} +\end{aligned} +$$ +

     
    + +

    +The output from the \( i \)-th neuron at the hidden layer \( l \) becomes a vector \( \vec{z}_{i}^{(l),\ \text{hidden}} \): + +

     
    +$$ +\begin{aligned} +\vec{z}_{i}^{(l),\ \text{hidden}} &= \Big( b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_1^{(l-1),\text{hidden} }, \ \dots \ , \ b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_{N_{hidden}^{(l-1)}}^{(l-1),\text{hidden} } \Big) \\ +&= +\begin{pmatrix} +b_i^{(l), \ \text{hidden}} & \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T +\end{pmatrix} +\begin{pmatrix} +1 & 1 & \dots & 1 \\ +\vec{x}_{1}^{(l-1),\text{hidden} } & \vec{x}_{2}^{(l-1),\text{hidden} } & \dots & \vec{x}_{N_{hidden}^{(l-1)}}^{(l-1),\text{hidden} } +\end{pmatrix} +\end{aligned} +$$ +

     
    +

    + + +
    +

    The final parts of the code

    +

    + + +

    def deep_neural_network(deep_params, x):
+    # N_hidden is the number of hidden layers  
+    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
+        
+    # Assumes input x being an one-dimensional array
+    num_values = np.size(x)
+    x = x.reshape(-1, num_values)
+    
+    # Assume that the input layer does nothing to the input x
+    x_input = x
+    
+    # Due to multiple hidden layers, define a variable referencing to the
+    # output of the previous layer:
+    x_prev = x_input 
+    
+    ## Hidden layers:
+    
+    for l in range(N_hidden):
+        # From the list of parameters P; find the correct weigths and bias for this layer
+        w_hidden = deep_params[l]
+        
+        # Add a row of ones to include bias
+        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
+
+        z_hidden = np.matmul(w_hidden, x_prev)
+        x_hidden = sigmoid(z_hidden)
+
+        # Update x_prev such that next layer can use the output from this layer
+        x_prev = x_hidden 
+
+    ## Output layer:
+    
+    # Get the weights and bias for this layer
+    w_output = deep_params[-1]
+    
+    # Include bias:
+    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
+
+    z_output = np.matmul(w_output, x_prev)
+    x_output = z_output
+
+    return x_output
+
    +
    + + +
    +

    And adding Back propagation

    + +

    +This step is very similar for the neural network. The idea in this +step is the same as for the neural network, but with more parameters +to update for. Again there is no need for computing the gradients +analytically since Autograd does the work for us. + +

    + + +

    # The trial solution using the deep neural network:
+def g_trial_deep(x,params, g0 = 10):
+    return g0 + x*deep_neural_network(params,x)
+
+# The same cost function as for the neural network, but calls deep_neural_network instead.
+def cost_function_deep(P, x):
+    
+    # Evaluate the trial function with the current parameters P
+    g_t = g_trial_deep(x,P)
+    
+    # Find the derivative w.r.t x of the neural network
+    d_net_out = elementwise_grad(deep_neural_network,1)(P,x) 
+    
+    # Find the derivative w.r.t x of the trial function
+    d_g_t = elementwise_grad(g_trial_deep,0)(x,P)  
+    
+    # The right side of the ODE 
+    func = g(x, g_t)
+
+    err_sqr = (d_g_t - func)**2
+    cost_sum = np.sum(err_sqr)
+    
+    return cost_sum
+
+def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
+    # num_hidden_neurons is now a list of number of neurons within each hidden layer
+
+    # Find the number of hidden layers:
+    N_hidden = np.size(num_neurons)
+    
+    ## Set up initial weigths and biases 
+    
+    # Initialize the list of parameters:
+    P = [None]*(N_hidden + 1) # + 1 to include the output layer
+
+    P[0] = npr.randn(num_neurons[0], 2 ) 
+    for l in range(1,N_hidden):
+        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias 
+    
+    # For the output layer
+    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
+
+    print('Initial cost: %g'%cost_function_deep(P, x))
+    
+    ## Start finding the optimal weigths using gradient descent
+    
+    # Find the Python function that represents the gradient of the cost function
+    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
+    cost_function_deep_grad = grad(cost_function_deep,0)
+    
+    # Let the update be done num_iter times
+    for i in range(num_iter):
+        # Evaluate the gradient at the current weights and biases in P. 
+        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
+        # in the hidden layers and output layers evaluated at x.
+        cost_deep_grad =  cost_function_deep_grad(P, x)
+        
+        for l in range(N_hidden+1):
+            P[l] = P[l] - lmb * cost_deep_grad[l]
+
+    print('Final cost: %g'%cost_function_deep(P, x))
+    
+    return P
+
    +
    + + +
    +

    Solving the ODE

    + +

    +Finally, having set up the networks we are ready to use them to solve the ODE problem. +We add the analytical solution + +

    + + +

    def g_analytic(x, gamma = 2, g0 = 10):
+    return g0*np.exp(-gamma*x)
+
    +
    + + +
    +

    Using neural network

    + +

    +The code below solves the ODE using a neural network. The number of +values for the input \( \vec x \) is 10, number of hidden neurons in the +hidden layer being 10 and th step size used in gradien descent +\( \lambda = 0.001 \). The program updates the weights and biases in the +network num_iter times. Finally, it plots the results from using the +neural network along with the analytical solution. + +

    + + +

    npr.seed(15)
+
+## Decide the vales of arguments to the function to solve
+N = 10
+x = np.linspace(0, 1, N)
+
+## Set up the initial parameters
+num_hidden_neurons = 10
+num_iter = 10000
+lmb = 0.001
+
+P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)
+
+res = g_trial(x,P) 
+res_analytical = g_analytic(x)
+
+print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))
+
+plt.figure(figsize=(10,10))
+
+plt.title('Performance of neural network solving an ODE compared to the analytical solution')
+plt.plot(x, res_analytical)
+plt.plot(x, res[0,:])
+plt.legend(['analytical','nn'])
+plt.xlabel('x')
+plt.ylabel('g(x)')
+plt.show()
+
    +
    + + +
    +

    Using a deep neural network

    + +

    + + +

    npr.seed(15)
+
+## Decide the vales of arguments to the function to solve
+N = 10
+x = np.linspace(0, 1, N)
+
+## Set up the initial parameters
+num_hidden_neurons = np.array([10,10])
+num_iter = 10000
+lmb = 0.001
+
+P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
+
+res = g_trial_deep(x,P) 
+res_analytical = g_analytic(x)
+
+plt.figure(figsize=(10,10))
+
+plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')
+plt.plot(x, res_analytical)
+plt.plot(x, res[0,:])
+plt.legend(['analytical','dnn'])
+plt.ylabel('g(x)')
+plt.show()
+
    +
    + + +
    +

    Wrapping it up

    + +

    +By rewriting the ODE as a minimization problem, it was possible to +solve equation using either a neural network (one hidden layer) or a +deep neural network (more than one hidden layers). How well the +network performed is measured by a specified cost function, which is +the function the network tries to minimize. Using a trial solution +which satisfies the additional condition and being defined by using +the output from the network in some way, the minimization problem +could be explicitly defined for out network to solve. The proposed +solution from the network is then the trial solution with parameters, +that is weights and biases within each layer in the network, such that +the solution minimizes the cost function. +

    + + diff --git a/doc/pub/NeuralNet/html/NeuralNet-solarized.html b/doc/pub/NeuralNet/html/NeuralNet-solarized.html index 7dd813788..ca1df75cf 100644 --- a/doc/pub/NeuralNet/html/NeuralNet-solarized.html +++ b/doc/pub/NeuralNet/html/NeuralNet-solarized.html @@ -194,7 +194,35 @@ div { text-align: justify; text-justify: inter-word; } ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -3627,6 +3655,824 @@ plt.show()
  • Abstract art using convolutional neural networks
    • +









    + +

    Applications: solving ordinary differential equations with Neural Networks

    + +

    +We end our discussion on neural networks with a discussion on how to solve differential equations. Here we focus +first on the classical exponential decay in one dimension. Thereafter we switch to the Poisson equation in one dimension. + +

    +The aim is to see if we can use a neural network to solve + +$$ +\begin{equation} +\label{eq:ode} +g'(x) = -\gamma g(x) +\end{equation} +$$ + +

    +where \( g(0) = g_0 \) with \( \gamma \) and \( g_0 \) being some chosen +values. This equation is an ordinary differential equation since the +function we have to solve for, \( g(x) \), is of one variable. + +

    +Here we set \( \gamma = 2 \) and \( g_0 = 10 \) but feel free to change +them and see how the neural network performs. + +

    +









    + +

    Trial solution

    + +

    +To begin with, a trial solution \( g_t(t) \) must be chosen. A general +trial solution for ordinary differential equations could be + +$$ +g_t(x, P) = h_1(x) + h_2(x, N(x, P)), +$$ + +

    +with \( h_1(x) \) ensuring that \( g_t(x) \) satisfies some conditions and +\( h_2(x,N(x, P)) \) an expression involving \( x \) and the output from the +neural network \( N(x,P) \) with \( P \) being the collection of the weights +and biases for each layer. + +

    +It is assumed that there are no weights and +bias at the input layer, so \( P = \{ P_{\text{hidden}}, +P_{\text{output}} \} \). If there are \( N_{\text{hidden} } \) neurons in +the hidden layer, then \( P_{\text{hidden}} \) is an \( N_{\text{hidden} } +\times 2 \) matrix. + +

    +The first column in \( P_{\text{hidden} } \) represents +the bias for each neuron in the hidden layer and the second column +represents the weigths for each neuron. If there are \( N_{\text{output} +} \) neurons in the output layer, then \( P_{\text{output}} \) is a +\( N_{\text{output} } \times (1 + N_{\text{hidden} }) \) matrix. Its first +column represents the bias of each neuron and the remaining columns +represents the weights to each neuron. + +

    +









    + +

    More details

    + +

    +We have \( g(0) = g_0 \). The trial solution must fulfill this +condition to be a proper solution of \eqref{eq:ode}. + +

    +A possible way to +ensure that \( g_t(0, P) = g_0 \), is to let \( F(N(x,P)) = x\cdot N(x,P) \) +and \( A(x) = g_0 \). This gives the following trial solution: + +$$ +\begin{equation} +g_t(x, P) = g_0 + x \cdot N(x, P). +\label{_auto11} +\end{equation} +$$ + +

    +









    + +

    Reformulating the problem

    + +

    +Often, the role of a neural network is to minimize its parameters with +respect to some given error criteria. This criteria, the cost or loss +function, is a measure of how much error the output of the network has +compared to some given known answers. A reformulation of +\eqref{eq:ode} must therefore be done, such that it describes the +problem a neural network can solve. + +

    +The neural network must find the set of weigths and biases \( P \) such +that the trial solution in satisfies +\eqref{eq:ode}. The trial solution has been chosen such that it +already solves the condition \( g(0) = g_0 \). What remains, is to find +\( P \) such that + +$$ +\begin{equation} +g_t'(x, P) = - \gamma g_t(x, P) +\label{_auto12} +\end{equation} +$$ + +

    +is fulfilled as best as possible. + +

    +









    + +

    Estimating errors

    + +

    +Having two sides of an equation as equal as +possible, means that the absolute or squared difference between the +sides must be as close to zero as small. In this case, the difference +squared is an appropiate measurement of how errorneous the trial +solution is with respect to \( P \) of the neural network. Therefore, the +problem our network must solve, is + +$$ +\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\} +$$ + +

    +or, in terms of weights and biases for each layer: + +$$ +\min_{P_{\text{hidden} }, \ P_{\text{output} }}\Big\{ \big(g_t'(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) - ( -\gamma g_t(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) \big)^2 \Big\} +$$ + +

    +for an input value \( x \). +If the neural network evaluates \( g_t(x, P) \) at more avalues for \( x \), say \( N \) values \( x_i \) for \( i = 1, \dots, N \), then the total error to minimize is + +$$ +\begin{equation} +\label{eq:min} +\min_{P}\Big\{\sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \Big\} +\end{equation} +$$ + +

    +Letting \( c(x, P) = \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) +\big)^2 \) denote the cost function, the minimization problem of which +our network must solve, is + +$$ +\min_{P} c(x, P) +$$ + +

    +or in terms of \( P_{\text{hidden} } \) and \( P_{\text{output} } \) + +$$ +\min_{P_{\text{hidden} }, \ P_{\text{output} }} c(x, \{P_{\text{hidden} }, P_{\text{output} }\}) +$$ + +

    +









    + +

    Creating a simple Deep Neural Net

    + +

    +The next step is to decide how the neural net \( N(x, P) \) +should be. In this case, the neural network is made +from scratch to understand better how a neural network works, gain +more control over its architecture, and see how Autograd can be used +to simplify the implementation. + +

    +Since a deep neural network (DNN) is a neural network with more than +one hidden layer, we can first look on how to implement a neural +network. Having an implementation of a neural network at hand, an +extension of it into a deep neural network would (hopefully) be +painless. + +

    +For simplicity, we assume that the input is an array \( \vec x = +(x_1, \dots, x_N) \) with \( N \) elements. It is at these points the neural +network should find \( P \) such that it fulfills \eqref{eq:min}. + +

    +









    + +

    Feedforward

    + +

    +First, a feedforward of the inputs must be done. This means that \( \vec + x \) must be passed through an input layer, a hidden layer and a output + layer. The input layer in this case, does not need to process the + data any further. The input layer will consist of \( N_{\text{input} } \) + neurons, passing its element to each neuron in the hidden layer. The + number of neurons in the hidden layer will be \( N_{\text{hidden} } \). + +

    +For the \( i \)-th in the hidden layer with weight \( w_i^{\text{hidden} } \) +and bias \( b_i^{\text{hidden} } \), the weighting from the \( j \)-th neuron +at the input layer is: + +$$ +\begin{aligned} +z_{i,j}^{\text{hidden}} &= b_i^{\text{hidden}} + w_i^{\text{hidden}}x_j \\ +&= +\begin{pmatrix} +b_i^{\text{hidden}} & w_i^{\text{hidden}} +\end{pmatrix} +\begin{pmatrix} +1 \\ +x_j +\end{pmatrix} +\end{aligned} +$$ + +

    + + +

    Result after weighting

    + +

    +The result after weighting the input at the \( i \)-th hidden neuron can be written as a vector: +$$ +\begin{aligned} +\vec{z}_{i}^{\text{hidden}} &= \Big( b_i^{\text{hidden}} + w_i^{\text{hidden}}x_1 , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_2, \ \dots \, , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_N\Big) \\ +&= +\begin{pmatrix} + b_i^{\text{hidden}} & w_i^{\text{hidden}} +\end{pmatrix} +\begin{pmatrix} +1 & 1 & \dots & 1 \\ +x_1 & x_2 & \dots & x_N +\end{pmatrix} \\ +&= \vec{p}_{i, \text{hidden}}^T X +\end{aligned} +$$ + +

    +It is the vector \( \vec{p}_{i, \text{hidden}}^T \) that defines each row +in \( P_{\text{hidden} } \), which contains the weights for the neural +network to minimize according to \eqref{eq:min}. + +

    +After having found \( \vec{z}_{i}^{\text{hidden}} \) for every neuron \( i \) +in the hidden layer, the vector will be sent to an activation function +\( a_i(\vec{z}) \). In this example, the sigmoid function has been used: + +$$ +f(z) = \frac{1}{1 + \exp{(-z)}}. +$$ + +

    +









    + +

    Output

    + +

    +The output $\vec{x}_i^{\text{hidden} }$from each \( i \)-th hidden neuron is: + +$$ +\vec{x}_i^{\text{hidden} } = f\big( \vec{z}_{i}^{\text{hidden}} \big). +$$ + +

    +The outputs \( \vec{x}_i^{\text{hidden} } \) are then sent to the output layer. + +

    +The output layer consist of one neuron in this case, and combines the +output from each of the neurons in the hidden layers. The output layer +combines the results from the hidden layer using some weights \( +w_i^{\text{output}} \) and biases \( b_i^{\text{output}} \). In this case, +it is assumes that the number of neurons in the output layer is one. + +

    +The procedure of weigthing the output neuron \( j \) in the hidden layer +to the \( i \)-th neuron in the output layer is similar as for the hidden +layer described previously. + +$$ +\begin{aligned} +z_{1,j}^{\text{output}} & = +\begin{pmatrix} +b_1^{\text{output}} & \vec{w}_1^{\text{output}} +\end{pmatrix} +\begin{pmatrix} +1 \\ +\vec{x}_j^{\text{hidden}} +\end{pmatrix} +\end{aligned} +$$ + +

    +Expressing \( z_{1,j}^{\text{output}} \) as a vector gives the following procedure of weighting the inputs from the hidden layer: + +$$ +\vec{z}_{1}^{\text{output}} = +\begin{pmatrix} +b_1^{\text{output}} & \vec{w}_1^{\text{output}} +\end{pmatrix} +\begin{pmatrix} +1 & 1 & \dots & 1 \\ +\vec{x}_1^{\text{hidden}} & \vec{x}_2^{\text{hidden}} & \dots & \vec{x}_N^{\text{hidden}} +\end{pmatrix} +$$ + +

    +In this case we seek a continous range of values since we are +approximating a function. This means that after computing +\( \vec{z}_{1}^{\text{output}} \) the neural network has finished its +feedforward step, and \( \vec{z}_{1}^{\text{output}} \) is the final +output of the network. + +

    +









    + +

    Setting up the code, feed forward part

    +

    + + +

    # Note that we use the  numpy wrapper for Autograd (see the gradient descent  slides)
+import autograd.numpy as np
+from autograd import grad, elementwise_grad
+import autograd.numpy.random as npr
+from matplotlib import pyplot as plt
+
+def sigmoid(z):
+    return 1/(1 + np.exp(-z))
+
+def neural_network(params, x):
+    
+    # Find the weights (including and biases) for the hidden and output layer.
+    # Assume that params is a list of parameters for each layer. 
+    # The biases are the first element for each array in params, 
+    # and the weights are the remaning elements in each array in params.   
+    
+    w_hidden = params[0]
+    w_output = params[1]
+
+    # Assumes input x being an one-dimensional array
+    num_values = np.size(x)
+    x = x.reshape(-1, num_values)
+    
+    # Assume that the input layer does nothing to the input x
+    x_input = x
+
+    ## Hidden layer:
+    
+    # Add a row of ones to include bias
+    x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)
+    
+    z_hidden = np.matmul(w_hidden, x_input)
+    x_hidden = sigmoid(z_hidden)
+
+    ## Output layer:
+    
+    # Include bias:
+    x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)
+
+    z_output = np.matmul(w_output, x_hidden)
+    x_output = z_output
+
+    return x_output
+
    +

    +









    + +

    Backpropagation

    + +

    +Now that feedforward can be done, the next step is to decide how the +parameters should change such that they minimize the cost function. + +

    +Recall that the chosen cost function for this problem is + +$$ +c(x, P) = \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 +$$ + +

    +In order to minimize it, an optimalization method must be chosen. + +

    +Here, gradient descent with a constant step size has been chosen. + +

    +Before looking at the gradient descent method, let us set up the cost +function along with the right ride of the ODE and trial solution. + +

    + + +

    # The trial solution using the deep neural network:
+def g_trial(x,params, g0 = 10):
+    return g0 + x*neural_network(params,x)
+
+# The right side of the ODE:
+def g(x, g_trial, gamma = 2):
+    return -gamma*g_trial
+
+# The cost function:
+def cost_function(P, x):
+    
+    # Evaluate the trial function with the current parameters P
+    g_t = g_trial(x,P)
+    
+    # Find the derivative w.r.t x of the neural network
+    d_net_out = elementwise_grad(neural_network,1)(P,x) 
+    
+    # Find the derivative w.r.t x of the trial function
+    d_g_t = elementwise_grad(g_trial,0)(x,P)  
+    
+    # The right side of the ODE 
+    func = g(x, g_t)
+
+    err_sqr = (d_g_t - func)**2
+    cost_sum = np.sum(err_sqr)
+    
+    return cost_sum
+
    +

    +









    + +

    Gradient Descent

    + +

    +The idea of the gradient descent algorithm is to update parameters in +direction where the cost function decreases goes to a minimum. + +

    +In general, the update of some parameters \( \vec \omega \) given a cost +function defined by some weights \( \vec \omega \), \( c(x, \vec \omega) \), +goes as follows: + +$$ +\vec \omega_{\text{new} } = \vec \omega - \lambda \nabla_{\vec \omega} c(x, \vec \omega), +$$ + +

    +for a number of iterations or until $ \big|\big| \vec +\omega_{\text{new} } - \vec \omega \big|\big|$ is smaller than some +given tolerance. + +

    +The value of \( \lambda \) decides how large steps the algorithm must take +in the direction of $ \nabla_{\vec \omega} c(x, \vec \omega)$. The +notatation \( \nabla_{\vec \omega} \) denotes the gradient with respect to +the elements in \( \vec \omega \). + +

    +









    + +

    More on GD and cost function

    + +

    +In our case, we have to minimize the cost function \( c(x, P) \) with +respect to the two sets of weights and bisases, that is for the hidden +layer \( P_{\text{hidden} } \) and for the ouput layer \( P_{\text{output} +} \) . + +

    +This means that \( P_{\text{hidden} } \) and \( P_{\text{output} } \) is +updated by + +$$ +\begin{aligned} +P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} c(x, P) \\ +P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} c(x, P) +\end{aligned} +$$ + +

    +This might look like a cumberstone to set up the correct expression +for finding the gradients. Luckily, Autograd comes to the rescue. + +

    + + +

    def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):
+    ## Set up initial weigths and biases 
+    
+    # For the hidden layer
+    p0 = npr.randn(num_neurons_hidden, 2 ) 
+
+    # For the output layer
+    p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included
+
+    P = [p0, p1]
+
+    print('Initial cost: %g'%cost_function(P, x))
+    
+    ## Start finding the optimal weigths using gradient descent
+    
+    # Find the Python function that represents the gradient of the cost function
+    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
+    cost_function_grad = grad(cost_function,0)
+    
+    # Let the update be done num_iter times
+    for i in range(num_iter):
+        # Evaluate the gradient at the current weights and biases in P. 
+        # The cost_grad consist now of two arrays; 
+        # one for the gradient w.r.t P_hidden and 
+        # one for the gradient w.r.t P_output
+        cost_grad =  cost_function_grad(P, x)
+    
+        P[0] = P[0] - lmb * cost_grad[0]
+        P[1] = P[1] - lmb * cost_grad[1]
+
+    print('Final cost: %g'%cost_function(P, x))
+    
+    return P
+
    +

    +









    + +

    An implementation of a Deep Neural Network

    + +

    +As previously stated, a Deep Neural Network (DNN) follows the same +concept of a neural network, but having more than one hidden +layer. Suppose that the network has \( N_{\text{hidden}} \) hidden layers +where the \( l \)-th layer has \( N_{\text{hidden}}^{(l)} \) neurons. The +input is still assumed to be an array of size \( 1 \times N \). The +network must now try to optimalize its output with respect to the +collection of weigths and biases \( P = \big\{P_{\text{input} }, \ +P_{\text{hidden} }^{(1)}, \ P_{\text{hidden} }^{(2)}, \ \dots , \ +P_{\text{hidden} }^{(N_{\text{hidden}})}, \ P_{\text{output} }\big\} \). + +

    +









    + +

    Feed forward again

    + +

    +The feedforward step is similar to as for the neural netowork, but now considering more than one hidden layer. + +

    +The \( i \)-th neuron at layer \( l \) recieves the result +\( \vec{x}_j^{(l-1),\text{hidden} } \) from the \( j \)-th neuron at layer +\( l-1 \). The \( i \)-th neuron at layer \( l \) weights all of the elements in +\( \vec{x}_j^{(l-1),\text{hidden} } \) with a weight vector \( \vec +w_{i,j}^{(l), \ \text{hidden} } \) with as many weigths as there are +elements in$\vec{x}_j^{(l-1),\text{hidden} }$, and adds a bias +\( b_i^{(l), \ \text{hidden} } \): + +$$ +\begin{aligned} +z_{i,j}^{(l),\ \text{hidden}} &= b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_j^{(l-1),\text{hidden} } \\ +&= +\begin{pmatrix} +b_i^{(l), \ \text{hidden}} & \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T +\end{pmatrix} +\begin{pmatrix} +1 \\ +\vec{x}_j^{(l-1),\text{hidden} } +\end{pmatrix} +\end{aligned} +$$ + +

    +The output from the \( i \)-th neuron at the hidden layer \( l \) becomes a vector \( \vec{z}_{i}^{(l),\ \text{hidden}} \): + +$$ +\begin{aligned} +\vec{z}_{i}^{(l),\ \text{hidden}} &= \Big( b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_1^{(l-1),\text{hidden} }, \ \dots \ , \ b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_{N_{hidden}^{(l-1)}}^{(l-1),\text{hidden} } \Big) \\ +&= +\begin{pmatrix} +b_i^{(l), \ \text{hidden}} & \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T +\end{pmatrix} +\begin{pmatrix} +1 & 1 & \dots & 1 \\ +\vec{x}_{1}^{(l-1),\text{hidden} } & \vec{x}_{2}^{(l-1),\text{hidden} } & \dots & \vec{x}_{N_{hidden}^{(l-1)}}^{(l-1),\text{hidden} } +\end{pmatrix} +\end{aligned} +$$ + +

    +









    + +

    The final parts of the code

    +

    + + +

    def deep_neural_network(deep_params, x):
+    # N_hidden is the number of hidden layers  
+    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
+        
+    # Assumes input x being an one-dimensional array
+    num_values = np.size(x)
+    x = x.reshape(-1, num_values)
+    
+    # Assume that the input layer does nothing to the input x
+    x_input = x
+    
+    # Due to multiple hidden layers, define a variable referencing to the
+    # output of the previous layer:
+    x_prev = x_input 
+    
+    ## Hidden layers:
+    
+    for l in range(N_hidden):
+        # From the list of parameters P; find the correct weigths and bias for this layer
+        w_hidden = deep_params[l]
+        
+        # Add a row of ones to include bias
+        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
+
+        z_hidden = np.matmul(w_hidden, x_prev)
+        x_hidden = sigmoid(z_hidden)
+
+        # Update x_prev such that next layer can use the output from this layer
+        x_prev = x_hidden 
+
+    ## Output layer:
+    
+    # Get the weights and bias for this layer
+    w_output = deep_params[-1]
+    
+    # Include bias:
+    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
+
+    z_output = np.matmul(w_output, x_prev)
+    x_output = z_output
+
+    return x_output
+
    +

    +









    + +

    And adding Back propagation

    + +

    +This step is very similar for the neural network. The idea in this +step is the same as for the neural network, but with more parameters +to update for. Again there is no need for computing the gradients +analytically since Autograd does the work for us. + +

    + + +

    # The trial solution using the deep neural network:
+def g_trial_deep(x,params, g0 = 10):
+    return g0 + x*deep_neural_network(params,x)
+
+# The same cost function as for the neural network, but calls deep_neural_network instead.
+def cost_function_deep(P, x):
+    
+    # Evaluate the trial function with the current parameters P
+    g_t = g_trial_deep(x,P)
+    
+    # Find the derivative w.r.t x of the neural network
+    d_net_out = elementwise_grad(deep_neural_network,1)(P,x) 
+    
+    # Find the derivative w.r.t x of the trial function
+    d_g_t = elementwise_grad(g_trial_deep,0)(x,P)  
+    
+    # The right side of the ODE 
+    func = g(x, g_t)
+
+    err_sqr = (d_g_t - func)**2
+    cost_sum = np.sum(err_sqr)
+    
+    return cost_sum
+
+def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
+    # num_hidden_neurons is now a list of number of neurons within each hidden layer
+
+    # Find the number of hidden layers:
+    N_hidden = np.size(num_neurons)
+    
+    ## Set up initial weigths and biases 
+    
+    # Initialize the list of parameters:
+    P = [None]*(N_hidden + 1) # + 1 to include the output layer
+
+    P[0] = npr.randn(num_neurons[0], 2 ) 
+    for l in range(1,N_hidden):
+        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias 
+    
+    # For the output layer
+    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
+
+    print('Initial cost: %g'%cost_function_deep(P, x))
+    
+    ## Start finding the optimal weigths using gradient descent
+    
+    # Find the Python function that represents the gradient of the cost function
+    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
+    cost_function_deep_grad = grad(cost_function_deep,0)
+    
+    # Let the update be done num_iter times
+    for i in range(num_iter):
+        # Evaluate the gradient at the current weights and biases in P. 
+        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
+        # in the hidden layers and output layers evaluated at x.
+        cost_deep_grad =  cost_function_deep_grad(P, x)
+        
+        for l in range(N_hidden+1):
+            P[l] = P[l] - lmb * cost_deep_grad[l]
+
+    print('Final cost: %g'%cost_function_deep(P, x))
+    
+    return P
+
    +

    +









    + +

    Solving the ODE

    + +

    +Finally, having set up the networks we are ready to use them to solve the ODE problem. +We add the analytical solution + +

    + + +

    def g_analytic(x, gamma = 2, g0 = 10):
+    return g0*np.exp(-gamma*x)
+
    +

    +









    + +

    Using neural network

    + +

    +The code below solves the ODE using a neural network. The number of +values for the input \( \vec x \) is 10, number of hidden neurons in the +hidden layer being 10 and th step size used in gradien descent +\( \lambda = 0.001 \). The program updates the weights and biases in the +network num_iter times. Finally, it plots the results from using the +neural network along with the analytical solution. + +

    + + +

    npr.seed(15)
+
+## Decide the vales of arguments to the function to solve
+N = 10
+x = np.linspace(0, 1, N)
+
+## Set up the initial parameters
+num_hidden_neurons = 10
+num_iter = 10000
+lmb = 0.001
+
+P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)
+
+res = g_trial(x,P) 
+res_analytical = g_analytic(x)
+
+print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))
+
+plt.figure(figsize=(10,10))
+
+plt.title('Performance of neural network solving an ODE compared to the analytical solution')
+plt.plot(x, res_analytical)
+plt.plot(x, res[0,:])
+plt.legend(['analytical','nn'])
+plt.xlabel('x')
+plt.ylabel('g(x)')
+plt.show()
+
    +

    + + +

    Using a deep neural network

    + +

    + + +

    npr.seed(15)
+
+## Decide the vales of arguments to the function to solve
+N = 10
+x = np.linspace(0, 1, N)
+
+## Set up the initial parameters
+num_hidden_neurons = np.array([10,10])
+num_iter = 10000
+lmb = 0.001
+
+P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
+
+res = g_trial_deep(x,P) 
+res_analytical = g_analytic(x)
+
+plt.figure(figsize=(10,10))
+
+plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')
+plt.plot(x, res_analytical)
+plt.plot(x, res[0,:])
+plt.legend(['analytical','dnn'])
+plt.ylabel('g(x)')
+plt.show()
+
    +

    +









    + +

    Wrapping it up

    + +

    +By rewriting the ODE as a minimization problem, it was possible to +solve equation using either a neural network (one hidden layer) or a +deep neural network (more than one hidden layers). How well the +network performed is measured by a specified cost function, which is +the function the network tries to minimize. Using a trial solution +which satisfies the additional condition and being defined by using +the output from the network in some way, the minimization problem +could be explicitly defined for out network to solve. The proposed +solution from the network is then the trial solution with parameters, +that is weights and biases within each layer in the network, such that +the solution minimizes the cost function. diff --git a/doc/pub/NeuralNet/html/NeuralNet.html b/doc/pub/NeuralNet/html/NeuralNet.html index afffc73d6..680938c01 100644 --- a/doc/pub/NeuralNet/html/NeuralNet.html +++ b/doc/pub/NeuralNet/html/NeuralNet.html @@ -199,7 +199,35 @@ div { text-align: justify; text-justify: inter-word; } ('Running with Keras', 2, None, '___sec83'), ('Final part', 2, None, '___sec84'), ('Final visualization', 2, None, '___sec85'), - ('Fun links', 2, None, '___sec86')]} + ('Fun links', 2, None, '___sec86'), + ('Applications: solving ordinary differential equations with ' + 'Neural Networks', + 2, + None, + '___sec87'), + ('Trial solution', 2, None, '___sec88'), + ('More details', 2, None, '___sec89'), + ('Reformulating the problem', 2, None, '___sec90'), + ('Estimating errors', 2, None, '___sec91'), + ('Creating a simple Deep Neural Net', 2, None, '___sec92'), + ('Feedforward', 2, None, '___sec93'), + ('Result after weighting', 2, None, '___sec94'), + ('Output', 2, None, '___sec95'), + ('Setting up the code, feed forward part', 2, None, '___sec96'), + ('Backpropagation', 2, None, '___sec97'), + ('Gradient Descent', 2, None, '___sec98'), + ('More on GD and cost function', 2, None, '___sec99'), + ('An implementation of a Deep Neural Network', + 2, + None, + '___sec100'), + ('Feed forward again', 2, None, '___sec101'), + ('The final parts of the code', 2, None, '___sec102'), + ('And adding Back propagation', 2, None, '___sec103'), + ('Solving the ODE', 2, None, '___sec104'), + ('Using neural network', 2, None, '___sec105'), + ('Using a deep neural network', 2, None, '___sec106'), + ('Wrapping it up', 2, None, '___sec107')]} end of tocinfo --> @@ -3632,6 +3660,824 @@ plt.show()

  • Abstract art using convolutional neural networks
    • +









    + +

    Applications: solving ordinary differential equations with Neural Networks

    + +

    +We end our discussion on neural networks with a discussion on how to solve differential equations. Here we focus +first on the classical exponential decay in one dimension. Thereafter we switch to the Poisson equation in one dimension. + +

    +The aim is to see if we can use a neural network to solve + +$$ +\begin{equation} +\label{eq:ode} +g'(x) = -\gamma g(x) +\end{equation} +$$ + +

    +where \( g(0) = g_0 \) with \( \gamma \) and \( g_0 \) being some chosen +values. This equation is an ordinary differential equation since the +function we have to solve for, \( g(x) \), is of one variable. + +

    +Here we set \( \gamma = 2 \) and \( g_0 = 10 \) but feel free to change +them and see how the neural network performs. + +

    +









    + +

    Trial solution

    + +

    +To begin with, a trial solution \( g_t(t) \) must be chosen. A general +trial solution for ordinary differential equations could be + +$$ +g_t(x, P) = h_1(x) + h_2(x, N(x, P)), +$$ + +

    +with \( h_1(x) \) ensuring that \( g_t(x) \) satisfies some conditions and +\( h_2(x,N(x, P)) \) an expression involving \( x \) and the output from the +neural network \( N(x,P) \) with \( P \) being the collection of the weights +and biases for each layer. + +

    +It is assumed that there are no weights and +bias at the input layer, so \( P = \{ P_{\text{hidden}}, +P_{\text{output}} \} \). If there are \( N_{\text{hidden} } \) neurons in +the hidden layer, then \( P_{\text{hidden}} \) is an \( N_{\text{hidden} } +\times 2 \) matrix. + +

    +The first column in \( P_{\text{hidden} } \) represents +the bias for each neuron in the hidden layer and the second column +represents the weigths for each neuron. If there are \( N_{\text{output} +} \) neurons in the output layer, then \( P_{\text{output}} \) is a +\( N_{\text{output} } \times (1 + N_{\text{hidden} }) \) matrix. Its first +column represents the bias of each neuron and the remaining columns +represents the weights to each neuron. + +

    +









    + +

    More details

    + +

    +We have \( g(0) = g_0 \). The trial solution must fulfill this +condition to be a proper solution of \eqref{eq:ode}. + +

    +A possible way to +ensure that \( g_t(0, P) = g_0 \), is to let \( F(N(x,P)) = x\cdot N(x,P) \) +and \( A(x) = g_0 \). This gives the following trial solution: + +$$ +\begin{equation} +g_t(x, P) = g_0 + x \cdot N(x, P). +\label{_auto11} +\end{equation} +$$ + +

    +









    + +

    Reformulating the problem

    + +

    +Often, the role of a neural network is to minimize its parameters with +respect to some given error criteria. This criteria, the cost or loss +function, is a measure of how much error the output of the network has +compared to some given known answers. A reformulation of +\eqref{eq:ode} must therefore be done, such that it describes the +problem a neural network can solve. + +

    +The neural network must find the set of weigths and biases \( P \) such +that the trial solution in satisfies +\eqref{eq:ode}. The trial solution has been chosen such that it +already solves the condition \( g(0) = g_0 \). What remains, is to find +\( P \) such that + +$$ +\begin{equation} +g_t'(x, P) = - \gamma g_t(x, P) +\label{_auto12} +\end{equation} +$$ + +

    +is fulfilled as best as possible. + +

    +









    + +

    Estimating errors

    + +

    +Having two sides of an equation as equal as +possible, means that the absolute or squared difference between the +sides must be as close to zero as small. In this case, the difference +squared is an appropiate measurement of how errorneous the trial +solution is with respect to \( P \) of the neural network. Therefore, the +problem our network must solve, is + +$$ +\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\} +$$ + +

    +or, in terms of weights and biases for each layer: + +$$ +\min_{P_{\text{hidden} }, \ P_{\text{output} }}\Big\{ \big(g_t'(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) - ( -\gamma g_t(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) \big)^2 \Big\} +$$ + +

    +for an input value \( x \). +If the neural network evaluates \( g_t(x, P) \) at more avalues for \( x \), say \( N \) values \( x_i \) for \( i = 1, \dots, N \), then the total error to minimize is + +$$ +\begin{equation} +\label{eq:min} +\min_{P}\Big\{\sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \Big\} +\end{equation} +$$ + +

    +Letting \( c(x, P) = \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) +\big)^2 \) denote the cost function, the minimization problem of which +our network must solve, is + +$$ +\min_{P} c(x, P) +$$ + +

    +or in terms of \( P_{\text{hidden} } \) and \( P_{\text{output} } \) + +$$ +\min_{P_{\text{hidden} }, \ P_{\text{output} }} c(x, \{P_{\text{hidden} }, P_{\text{output} }\}) +$$ + +

    +









    + +

    Creating a simple Deep Neural Net

    + +

    +The next step is to decide how the neural net \( N(x, P) \) +should be. In this case, the neural network is made +from scratch to understand better how a neural network works, gain +more control over its architecture, and see how Autograd can be used +to simplify the implementation. + +

    +Since a deep neural network (DNN) is a neural network with more than +one hidden layer, we can first look on how to implement a neural +network. Having an implementation of a neural network at hand, an +extension of it into a deep neural network would (hopefully) be +painless. + +

    +For simplicity, we assume that the input is an array \( \vec x = +(x_1, \dots, x_N) \) with \( N \) elements. It is at these points the neural +network should find \( P \) such that it fulfills \eqref{eq:min}. + +

    +









    + +

    Feedforward

    + +

    +First, a feedforward of the inputs must be done. This means that \( \vec + x \) must be passed through an input layer, a hidden layer and a output + layer. The input layer in this case, does not need to process the + data any further. The input layer will consist of \( N_{\text{input} } \) + neurons, passing its element to each neuron in the hidden layer. The + number of neurons in the hidden layer will be \( N_{\text{hidden} } \). + +

    +For the \( i \)-th in the hidden layer with weight \( w_i^{\text{hidden} } \) +and bias \( b_i^{\text{hidden} } \), the weighting from the \( j \)-th neuron +at the input layer is: + +$$ +\begin{aligned} +z_{i,j}^{\text{hidden}} &= b_i^{\text{hidden}} + w_i^{\text{hidden}}x_j \\ +&= +\begin{pmatrix} +b_i^{\text{hidden}} & w_i^{\text{hidden}} +\end{pmatrix} +\begin{pmatrix} +1 \\ +x_j +\end{pmatrix} +\end{aligned} +$$ + +

    + + +

    Result after weighting

    + +

    +The result after weighting the input at the \( i \)-th hidden neuron can be written as a vector: +$$ +\begin{aligned} +\vec{z}_{i}^{\text{hidden}} &= \Big( b_i^{\text{hidden}} + w_i^{\text{hidden}}x_1 , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_2, \ \dots \, , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_N\Big) \\ +&= +\begin{pmatrix} + b_i^{\text{hidden}} & w_i^{\text{hidden}} +\end{pmatrix} +\begin{pmatrix} +1 & 1 & \dots & 1 \\ +x_1 & x_2 & \dots & x_N +\end{pmatrix} \\ +&= \vec{p}_{i, \text{hidden}}^T X +\end{aligned} +$$ + +

    +It is the vector \( \vec{p}_{i, \text{hidden}}^T \) that defines each row +in \( P_{\text{hidden} } \), which contains the weights for the neural +network to minimize according to \eqref{eq:min}. + +

    +After having found \( \vec{z}_{i}^{\text{hidden}} \) for every neuron \( i \) +in the hidden layer, the vector will be sent to an activation function +\( a_i(\vec{z}) \). In this example, the sigmoid function has been used: + +$$ +f(z) = \frac{1}{1 + \exp{(-z)}}. +$$ + +

    +









    + +

    Output

    + +

    +The output $\vec{x}_i^{\text{hidden} }$from each \( i \)-th hidden neuron is: + +$$ +\vec{x}_i^{\text{hidden} } = f\big( \vec{z}_{i}^{\text{hidden}} \big). +$$ + +

    +The outputs \( \vec{x}_i^{\text{hidden} } \) are then sent to the output layer. + +

    +The output layer consist of one neuron in this case, and combines the +output from each of the neurons in the hidden layers. The output layer +combines the results from the hidden layer using some weights \( +w_i^{\text{output}} \) and biases \( b_i^{\text{output}} \). In this case, +it is assumes that the number of neurons in the output layer is one. + +

    +The procedure of weigthing the output neuron \( j \) in the hidden layer +to the \( i \)-th neuron in the output layer is similar as for the hidden +layer described previously. + +$$ +\begin{aligned} +z_{1,j}^{\text{output}} & = +\begin{pmatrix} +b_1^{\text{output}} & \vec{w}_1^{\text{output}} +\end{pmatrix} +\begin{pmatrix} +1 \\ +\vec{x}_j^{\text{hidden}} +\end{pmatrix} +\end{aligned} +$$ + +

    +Expressing \( z_{1,j}^{\text{output}} \) as a vector gives the following procedure of weighting the inputs from the hidden layer: + +$$ +\vec{z}_{1}^{\text{output}} = +\begin{pmatrix} +b_1^{\text{output}} & \vec{w}_1^{\text{output}} +\end{pmatrix} +\begin{pmatrix} +1 & 1 & \dots & 1 \\ +\vec{x}_1^{\text{hidden}} & \vec{x}_2^{\text{hidden}} & \dots & \vec{x}_N^{\text{hidden}} +\end{pmatrix} +$$ + +

    +In this case we seek a continous range of values since we are +approximating a function. This means that after computing +\( \vec{z}_{1}^{\text{output}} \) the neural network has finished its +feedforward step, and \( \vec{z}_{1}^{\text{output}} \) is the final +output of the network. + +

    +









    + +

    Setting up the code, feed forward part

    +

    + + +

    # Note that we use the  numpy wrapper for Autograd (see the gradient descent  slides)
+import autograd.numpy as np
+from autograd import grad, elementwise_grad
+import autograd.numpy.random as npr
+from matplotlib import pyplot as plt
+
+def sigmoid(z):
+    return 1/(1 + np.exp(-z))
+
+def neural_network(params, x):
+    
+    # Find the weights (including and biases) for the hidden and output layer.
+    # Assume that params is a list of parameters for each layer. 
+    # The biases are the first element for each array in params, 
+    # and the weights are the remaning elements in each array in params.   
+    
+    w_hidden = params[0]
+    w_output = params[1]
+
+    # Assumes input x being an one-dimensional array
+    num_values = np.size(x)
+    x = x.reshape(-1, num_values)
+    
+    # Assume that the input layer does nothing to the input x
+    x_input = x
+
+    ## Hidden layer:
+    
+    # Add a row of ones to include bias
+    x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)
+    
+    z_hidden = np.matmul(w_hidden, x_input)
+    x_hidden = sigmoid(z_hidden)
+
+    ## Output layer:
+    
+    # Include bias:
+    x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)
+
+    z_output = np.matmul(w_output, x_hidden)
+    x_output = z_output
+
+    return x_output
+
    +

    +









    + +

    Backpropagation

    + +

    +Now that feedforward can be done, the next step is to decide how the +parameters should change such that they minimize the cost function. + +

    +Recall that the chosen cost function for this problem is + +$$ +c(x, P) = \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 +$$ + +

    +In order to minimize it, an optimalization method must be chosen. + +

    +Here, gradient descent with a constant step size has been chosen. + +

    +Before looking at the gradient descent method, let us set up the cost +function along with the right ride of the ODE and trial solution. + +

    + + +

    # The trial solution using the deep neural network:
+def g_trial(x,params, g0 = 10):
+    return g0 + x*neural_network(params,x)
+
+# The right side of the ODE:
+def g(x, g_trial, gamma = 2):
+    return -gamma*g_trial
+
+# The cost function:
+def cost_function(P, x):
+    
+    # Evaluate the trial function with the current parameters P
+    g_t = g_trial(x,P)
+    
+    # Find the derivative w.r.t x of the neural network
+    d_net_out = elementwise_grad(neural_network,1)(P,x) 
+    
+    # Find the derivative w.r.t x of the trial function
+    d_g_t = elementwise_grad(g_trial,0)(x,P)  
+    
+    # The right side of the ODE 
+    func = g(x, g_t)
+
+    err_sqr = (d_g_t - func)**2
+    cost_sum = np.sum(err_sqr)
+    
+    return cost_sum
+
    +

    +









    + +

    Gradient Descent

    + +

    +The idea of the gradient descent algorithm is to update parameters in +direction where the cost function decreases goes to a minimum. + +

    +In general, the update of some parameters \( \vec \omega \) given a cost +function defined by some weights \( \vec \omega \), \( c(x, \vec \omega) \), +goes as follows: + +$$ +\vec \omega_{\text{new} } = \vec \omega - \lambda \nabla_{\vec \omega} c(x, \vec \omega), +$$ + +

    +for a number of iterations or until $ \big|\big| \vec +\omega_{\text{new} } - \vec \omega \big|\big|$ is smaller than some +given tolerance. + +

    +The value of \( \lambda \) decides how large steps the algorithm must take +in the direction of $ \nabla_{\vec \omega} c(x, \vec \omega)$. The +notatation \( \nabla_{\vec \omega} \) denotes the gradient with respect to +the elements in \( \vec \omega \). + +

    +









    + +

    More on GD and cost function

    + +

    +In our case, we have to minimize the cost function \( c(x, P) \) with +respect to the two sets of weights and bisases, that is for the hidden +layer \( P_{\text{hidden} } \) and for the ouput layer \( P_{\text{output} +} \) . + +

    +This means that \( P_{\text{hidden} } \) and \( P_{\text{output} } \) is +updated by + +$$ +\begin{aligned} +P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} c(x, P) \\ +P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} c(x, P) +\end{aligned} +$$ + +

    +This might look like a cumberstone to set up the correct expression +for finding the gradients. Luckily, Autograd comes to the rescue. + +

    + + +

    def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):
+    ## Set up initial weigths and biases 
+    
+    # For the hidden layer
+    p0 = npr.randn(num_neurons_hidden, 2 ) 
+
+    # For the output layer
+    p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included
+
+    P = [p0, p1]
+
+    print('Initial cost: %g'%cost_function(P, x))
+    
+    ## Start finding the optimal weigths using gradient descent
+    
+    # Find the Python function that represents the gradient of the cost function
+    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
+    cost_function_grad = grad(cost_function,0)
+    
+    # Let the update be done num_iter times
+    for i in range(num_iter):
+        # Evaluate the gradient at the current weights and biases in P. 
+        # The cost_grad consist now of two arrays; 
+        # one for the gradient w.r.t P_hidden and 
+        # one for the gradient w.r.t P_output
+        cost_grad =  cost_function_grad(P, x)
+    
+        P[0] = P[0] - lmb * cost_grad[0]
+        P[1] = P[1] - lmb * cost_grad[1]
+
+    print('Final cost: %g'%cost_function(P, x))
+    
+    return P
+
    +

    +









    + +

    An implementation of a Deep Neural Network

    + +

    +As previously stated, a Deep Neural Network (DNN) follows the same +concept of a neural network, but having more than one hidden +layer. Suppose that the network has \( N_{\text{hidden}} \) hidden layers +where the \( l \)-th layer has \( N_{\text{hidden}}^{(l)} \) neurons. The +input is still assumed to be an array of size \( 1 \times N \). The +network must now try to optimalize its output with respect to the +collection of weigths and biases \( P = \big\{P_{\text{input} }, \ +P_{\text{hidden} }^{(1)}, \ P_{\text{hidden} }^{(2)}, \ \dots , \ +P_{\text{hidden} }^{(N_{\text{hidden}})}, \ P_{\text{output} }\big\} \). + +

    +









    + +

    Feed forward again

    + +

    +The feedforward step is similar to as for the neural netowork, but now considering more than one hidden layer. + +

    +The \( i \)-th neuron at layer \( l \) recieves the result +\( \vec{x}_j^{(l-1),\text{hidden} } \) from the \( j \)-th neuron at layer +\( l-1 \). The \( i \)-th neuron at layer \( l \) weights all of the elements in +\( \vec{x}_j^{(l-1),\text{hidden} } \) with a weight vector \( \vec +w_{i,j}^{(l), \ \text{hidden} } \) with as many weigths as there are +elements in$\vec{x}_j^{(l-1),\text{hidden} }$, and adds a bias +\( b_i^{(l), \ \text{hidden} } \): + +$$ +\begin{aligned} +z_{i,j}^{(l),\ \text{hidden}} &= b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_j^{(l-1),\text{hidden} } \\ +&= +\begin{pmatrix} +b_i^{(l), \ \text{hidden}} & \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T +\end{pmatrix} +\begin{pmatrix} +1 \\ +\vec{x}_j^{(l-1),\text{hidden} } +\end{pmatrix} +\end{aligned} +$$ + +

    +The output from the \( i \)-th neuron at the hidden layer \( l \) becomes a vector \( \vec{z}_{i}^{(l),\ \text{hidden}} \): + +$$ +\begin{aligned} +\vec{z}_{i}^{(l),\ \text{hidden}} &= \Big( b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_1^{(l-1),\text{hidden} }, \ \dots \ , \ b_i^{(l), \ \text{hidden}} + \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T\vec{x}_{N_{hidden}^{(l-1)}}^{(l-1),\text{hidden} } \Big) \\ +&= +\begin{pmatrix} +b_i^{(l), \ \text{hidden}} & \big(\vec{w}_{i}^{(l), \ \text{hidden}}\big)^T +\end{pmatrix} +\begin{pmatrix} +1 & 1 & \dots & 1 \\ +\vec{x}_{1}^{(l-1),\text{hidden} } & \vec{x}_{2}^{(l-1),\text{hidden} } & \dots & \vec{x}_{N_{hidden}^{(l-1)}}^{(l-1),\text{hidden} } +\end{pmatrix} +\end{aligned} +$$ + +

    +









    + +

    The final parts of the code

    +

    + + +

    def deep_neural_network(deep_params, x):
+    # N_hidden is the number of hidden layers  
+    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
+        
+    # Assumes input x being an one-dimensional array
+    num_values = np.size(x)
+    x = x.reshape(-1, num_values)
+    
+    # Assume that the input layer does nothing to the input x
+    x_input = x
+    
+    # Due to multiple hidden layers, define a variable referencing to the
+    # output of the previous layer:
+    x_prev = x_input 
+    
+    ## Hidden layers:
+    
+    for l in range(N_hidden):
+        # From the list of parameters P; find the correct weigths and bias for this layer
+        w_hidden = deep_params[l]
+        
+        # Add a row of ones to include bias
+        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
+
+        z_hidden = np.matmul(w_hidden, x_prev)
+        x_hidden = sigmoid(z_hidden)
+
+        # Update x_prev such that next layer can use the output from this layer
+        x_prev = x_hidden 
+
+    ## Output layer:
+    
+    # Get the weights and bias for this layer
+    w_output = deep_params[-1]
+    
+    # Include bias:
+    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
+
+    z_output = np.matmul(w_output, x_prev)
+    x_output = z_output
+
+    return x_output
+
    +

    +









    + +

    And adding Back propagation

    + +

    +This step is very similar for the neural network. The idea in this +step is the same as for the neural network, but with more parameters +to update for. Again there is no need for computing the gradients +analytically since Autograd does the work for us. + +

    + + +

    # The trial solution using the deep neural network:
+def g_trial_deep(x,params, g0 = 10):
+    return g0 + x*deep_neural_network(params,x)
+
+# The same cost function as for the neural network, but calls deep_neural_network instead.
+def cost_function_deep(P, x):
+    
+    # Evaluate the trial function with the current parameters P
+    g_t = g_trial_deep(x,P)
+    
+    # Find the derivative w.r.t x of the neural network
+    d_net_out = elementwise_grad(deep_neural_network,1)(P,x) 
+    
+    # Find the derivative w.r.t x of the trial function
+    d_g_t = elementwise_grad(g_trial_deep,0)(x,P)  
+    
+    # The right side of the ODE 
+    func = g(x, g_t)
+
+    err_sqr = (d_g_t - func)**2
+    cost_sum = np.sum(err_sqr)
+    
+    return cost_sum
+
+def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
+    # num_hidden_neurons is now a list of number of neurons within each hidden layer
+
+    # Find the number of hidden layers:
+    N_hidden = np.size(num_neurons)
+    
+    ## Set up initial weigths and biases 
+    
+    # Initialize the list of parameters:
+    P = [None]*(N_hidden + 1) # + 1 to include the output layer
+
+    P[0] = npr.randn(num_neurons[0], 2 ) 
+    for l in range(1,N_hidden):
+        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias 
+    
+    # For the output layer
+    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
+
+    print('Initial cost: %g'%cost_function_deep(P, x))
+    
+    ## Start finding the optimal weigths using gradient descent
+    
+    # Find the Python function that represents the gradient of the cost function
+    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
+    cost_function_deep_grad = grad(cost_function_deep,0)
+    
+    # Let the update be done num_iter times
+    for i in range(num_iter):
+        # Evaluate the gradient at the current weights and biases in P. 
+        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
+        # in the hidden layers and output layers evaluated at x.
+        cost_deep_grad =  cost_function_deep_grad(P, x)
+        
+        for l in range(N_hidden+1):
+            P[l] = P[l] - lmb * cost_deep_grad[l]
+
+    print('Final cost: %g'%cost_function_deep(P, x))
+    
+    return P
+
    +

    +









    + +

    Solving the ODE

    + +

    +Finally, having set up the networks we are ready to use them to solve the ODE problem. +We add the analytical solution + +

    + + +

    def g_analytic(x, gamma = 2, g0 = 10):
+    return g0*np.exp(-gamma*x)
+
    +

    +









    + +

    Using neural network

    + +

    +The code below solves the ODE using a neural network. The number of +values for the input \( \vec x \) is 10, number of hidden neurons in the +hidden layer being 10 and th step size used in gradien descent +\( \lambda = 0.001 \). The program updates the weights and biases in the +network num_iter times. Finally, it plots the results from using the +neural network along with the analytical solution. + +

    + + +

    npr.seed(15)
+
+## Decide the vales of arguments to the function to solve
+N = 10
+x = np.linspace(0, 1, N)
+
+## Set up the initial parameters
+num_hidden_neurons = 10
+num_iter = 10000
+lmb = 0.001
+
+P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)
+
+res = g_trial(x,P) 
+res_analytical = g_analytic(x)
+
+print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))
+
+plt.figure(figsize=(10,10))
+
+plt.title('Performance of neural network solving an ODE compared to the analytical solution')
+plt.plot(x, res_analytical)
+plt.plot(x, res[0,:])
+plt.legend(['analytical','nn'])
+plt.xlabel('x')
+plt.ylabel('g(x)')
+plt.show()
+
    +

    + + +

    Using a deep neural network

    + +

    + + +

    npr.seed(15)
+
+## Decide the vales of arguments to the function to solve
+N = 10
+x = np.linspace(0, 1, N)
+
+## Set up the initial parameters
+num_hidden_neurons = np.array([10,10])
+num_iter = 10000
+lmb = 0.001
+
+P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
+
+res = g_trial_deep(x,P) 
+res_analytical = g_analytic(x)
+
+plt.figure(figsize=(10,10))
+
+plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')
+plt.plot(x, res_analytical)
+plt.plot(x, res[0,:])
+plt.legend(['analytical','dnn'])
+plt.ylabel('g(x)')
+plt.show()
+
    +

    +









    + +

    Wrapping it up

    + +

    +By rewriting the ODE as a minimization problem, it was possible to +solve equation using either a neural network (one hidden layer) or a +deep neural network (more than one hidden layers). How well the +network performed is measured by a specified cost function, which is +the function the network tries to minimize. Using a trial solution +which satisfies the additional condition and being defined by using +the output from the network in some way, the minimization problem +could be explicitly defined for out network to solve. The proposed +solution from the network is then the trial solution with parameters, +that is weights and biases within each layer in the network, such that +the solution minimizes the cost function. diff --git a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb index 011e92f5f..61d8d5658 100644 --- a/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb +++ b/doc/pub/NeuralNet/ipynb/NeuralNet.ipynb @@ -4010,7 +4010,994 @@ "\n", "1. [Self-Driving cars using a convolutional neural network](https://arxiv.org/abs/1604.07316)\n", "\n", - "2. [Abstract art using convolutional neural networks](https://deepdreamgenerator.com/)" + "2. [Abstract art using convolutional neural networks](https://deepdreamgenerator.com/)\n", + "\n", + "## Applications: solving ordinary differential equations with Neural Networks\n", + "\n", + "We end our discussion on neural networks with a discussion on how to solve differential equations. Here we focus \n", + "first on the classical exponential decay in one dimension. Thereafter we switch to the Poisson equation in one dimension.\n", + "\n", + "The aim is to see if we can use a neural network to solve" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "

    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\label{eq:ode} \\tag{16}\n", + "g'(x) = -\\gamma g(x) \n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $g(0) = g_0$ with $\\gamma$ and $g_0$ being some chosen\n", + "values. This equation is an ordinary differential equation since the\n", + "function we have to solve for, $g(x)$, is of one variable.\n", + "\n", + "Here we set $\\gamma = 2$ and $g_0 = 10$ but feel free to change\n", + "them and see how the neural network performs.\n", + "\n", + "## Trial solution\n", + "\n", + "To begin with, a trial solution $g_t(t)$ must be chosen. A general\n", + "trial solution for ordinary differential equations could be" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "g_t(x, P) = h_1(x) + h_2(x, N(x, P)),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $h_1(x)$ ensuring that $g_t(x)$ satisfies some conditions and\n", + "$h_2(x,N(x, P))$ an expression involving $x$ and the output from the\n", + "neural network $N(x,P)$ with $P $ being the collection of the weights\n", + "and biases for each layer. \n", + "\n", + "It is assumed that there are no weights and\n", + "bias at the input layer, so $P = \\{ P_{\\text{hidden}},\n", + "P_{\\text{output}} \\}$. If there are $N_{\\text{hidden} }$ neurons in\n", + "the hidden layer, then $P_{\\text{hidden}}$ is an $N_{\\text{hidden} }\n", + "\\times 2$ matrix. \n", + "\n", + "The first column in $P_{\\text{hidden} }$ represents\n", + "the bias for each neuron in the hidden layer and the second column\n", + "represents the weigths for each neuron. If there are $N_{\\text{output}\n", + "}$ neurons in the output layer, then $P_{\\text{output}} $ is a\n", + "$N_{\\text{output} } \\times (1 + N_{\\text{hidden} })$ matrix. Its first\n", + "column represents the bias of each neuron and the remaining columns\n", + "represents the weights to each neuron.\n", + "\n", + "\n", + "## More details\n", + "\n", + "We have $g(0) = g_0$. The trial solution must fulfill this\n", + "condition to be a proper solution of ([eq:ode](#eq:ode)). \n", + "\n", + "A possible way to\n", + "ensure that $g_t(0, P) = g_0$, is to let $F(N(x,P)) = x\\cdot N(x,P)$\n", + "and $A(x) = g_0$. This gives the following trial solution:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "g_t(x, P) = g_0 + x \\cdot N(x, P).\n", + "\\label{_auto11} \\tag{17}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Reformulating the problem\n", + "\n", + "Often, the role of a neural network is to minimize its parameters with\n", + "respect to some given error criteria. This criteria, the cost or loss\n", + "function, is a measure of how much error the output of the network has\n", + "compared to some given known answers. A reformulation of\n", + "([eq:ode](#eq:ode)) must therefore be done, such that it describes the\n", + "problem a neural network can solve.\n", + "\n", + "The neural network must find the set of weigths and biases $P$ such\n", + "that the trial solution in satisfies\n", + "([eq:ode](#eq:ode)). The trial solution has been chosen such that it\n", + "already solves the condition $g(0) = g_0$. What remains, is to find\n", + "$P$ such that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "g_t'(x, P) = - \\gamma g_t(x, P) \n", + "\\label{_auto12} \\tag{18}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "is fulfilled as *best as possible*. \n", + "\n", + "\n", + "## Estimating errors\n", + "\n", + "Having two sides of an equation as equal as\n", + "possible, means that the absolute or squared difference between the\n", + "sides must be as close to zero as small. In this case, the difference\n", + "squared is an appropiate measurement of how errorneous the trial\n", + "solution is with respect to $P$ of the neural network. Therefore, the\n", + "problem our network must solve, is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\min_{P}\\Big\\{ \\big(g_t'(x, P) - ( -\\gamma g_t(x, P) \\big)^2 \\Big\\}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or, in terms of weights and biases for each layer:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }}\\Big\\{ \\big(g_t'(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) - ( -\\gamma g_t(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) \\big)^2 \\Big\\}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "for an input value $x$. \n", + "If the neural network evaluates $g_t(x, P)$ at more avalues for $x$, say $N$ values $x_i$ for $i = 1, \\dots, N$, then the *total* error to minimize is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
    \n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\label{eq:min} \\tag{19}\n", + "\\min_{P}\\Big\\{\\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2 \\Big\\} \n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Letting $c(x, P) = \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P)\n", + "\\big)^2$ denote the cost function, the minimization problem of which\n", + "our network must solve, is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\min_{P} c(x, P)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or in terms of $P_{\\text{hidden} }$ and $P_{\\text{output} }$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }} c(x, \\{P_{\\text{hidden} }, P_{\\text{output} }\\})\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Creating a simple Deep Neural Net\n", + "\n", + "The next step is to decide how the neural net $N(x, P)$ \n", + "should be. In this case, the neural network is made\n", + "from scratch to understand better how a neural network works, gain\n", + "more control over its architecture, and see how Autograd can be used\n", + "to simplify the implementation.\n", + "\n", + "\n", + "Since a deep neural network (DNN) is a neural network with more than\n", + "one hidden layer, we can first look on how to implement a neural\n", + "network. Having an implementation of a neural network at hand, an\n", + "extension of it into a deep neural network would (hopefully) be\n", + "painless.\n", + "\n", + "For simplicity, we assume that the input is an array $\\vec x =\n", + "(x_1, \\dots, x_N)$ with $N$ elements. It is at these points the neural\n", + "network should find $P$ such that it fulfills ([eq:min](#eq:min)).\n", + "\n", + "## Feedforward\n", + "\n", + "First, a feedforward of the inputs must be done. This means that $\\vec\n", + " x$ must be passed through an input layer, a hidden layer and a output\n", + " layer. The input layer in this case, does not need to process the\n", + " data any further. The input layer will consist of $N_{\\text{input} }$\n", + " neurons, passing its element to each neuron in the hidden layer. The\n", + " number of neurons in the hidden layer will be $N_{\\text{hidden} }$.\n", + "\n", + "For the $i$-th in the hidden layer with weight $w_i^{\\text{hidden} }$\n", + "and bias $b_i^{\\text{hidden} }$, the weighting from the $j$-th neuron\n", + "at the input layer is:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "z_{i,j}^{\\text{hidden}} &= b_i^{\\text{hidden}} + w_i^{\\text{hidden}}x_j \\\\\n", + "&= \n", + "\\begin{pmatrix}\n", + "b_i^{\\text{hidden}} & w_i^{\\text{hidden}}\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "1 \\\\\n", + "x_j\n", + "\\end{pmatrix} \n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Result after weighting\n", + "\n", + "The result after weighting the input at the $i$-th hidden neuron can be written as a vector:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "\\vec{z}_{i}^{\\text{hidden}} &= \\Big( b_i^{\\text{hidden}} + w_i^{\\text{hidden}}x_1 , \\ b_i^{\\text{hidden}} + w_i^{\\text{hidden}} x_2, \\ \\dots \\, , \\ b_i^{\\text{hidden}} + w_i^{\\text{hidden}} x_N\\Big) \\\\\n", + "&= \n", + "\\begin{pmatrix}\n", + " b_i^{\\text{hidden}} & w_i^{\\text{hidden}}\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "1 & 1 & \\dots & 1 \\\\\n", + "x_1 & x_2 & \\dots & x_N\n", + "\\end{pmatrix} \\\\\n", + "&= \\vec{p}_{i, \\text{hidden}}^T X\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is the vector $\\vec{p}_{i, \\text{hidden}}^T$ that defines each row\n", + "in $P_{\\text{hidden} }$, which contains the weights for the neural\n", + "network to minimize according to ([eq:min](#eq:min)).\n", + "\n", + "After having found $\\vec{z}_{i}^{\\text{hidden}} $ for every neuron $i$\n", + "in the hidden layer, the vector will be sent to an activation function\n", + "$a_i(\\vec{z})$. In this example, the sigmoid function has been used:\n", + "\n", + "$$\n", + "f(z) = \\frac{1}{1 + \\exp{(-z)}}.\n", + "$$\n", + "\n", + "\n", + "## Output\n", + "\n", + "The output $\\vec{x}_i^{\\text{hidden} }$from each $i$-th hidden neuron is:\n", + "\n", + "$$\n", + "\\vec{x}_i^{\\text{hidden} } = f\\big( \\vec{z}_{i}^{\\text{hidden}} \\big).\n", + "$$\n", + "\n", + "The outputs $\\vec{x}_i^{\\text{hidden} } $ are then sent to the output layer. \n", + "\n", + "The output layer consist of one neuron in this case, and combines the\n", + "output from each of the neurons in the hidden layers. The output layer\n", + "combines the results from the hidden layer using some weights $\n", + "w_i^{\\text{output}}$ and biases $b_i^{\\text{output}}$. In this case,\n", + "it is assumes that the number of neurons in the output layer is one.\n", + "\n", + "The procedure of weigthing the output neuron $j$ in the hidden layer\n", + "to the $i$-th neuron in the output layer is similar as for the hidden\n", + "layer described previously." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "z_{1,j}^{\\text{output}} & = \n", + "\\begin{pmatrix}\n", + "b_1^{\\text{output}} & \\vec{w}_1^{\\text{output}}\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "1 \\\\\n", + "\\vec{x}_j^{\\text{hidden}}\n", + "\\end{pmatrix}\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Expressing $z_{1,j}^{\\text{output}}$ as a vector gives the following procedure of weighting the inputs from the hidden layer:\n", + "\n", + "$$\n", + "\\vec{z}_{1}^{\\text{output}} = \n", + "\\begin{pmatrix}\n", + "b_1^{\\text{output}} & \\vec{w}_1^{\\text{output}}\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "1 & 1 & \\dots & 1 \\\\\n", + "\\vec{x}_1^{\\text{hidden}} & \\vec{x}_2^{\\text{hidden}} & \\dots & \\vec{x}_N^{\\text{hidden}}\n", + "\\end{pmatrix}\n", + "$$\n", + "\n", + "In this case we seek a continous range of values since we are\n", + "approximating a function. This means that after computing\n", + "$\\vec{z}_{1}^{\\text{output}}$ the neural network has finished its\n", + "feedforward step, and $\\vec{z}_{1}^{\\text{output}}$ is the final\n", + "output of the network.\n", + "\n", + "\n", + "## Setting up the code, feed forward part" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Note that we use the numpy wrapper for Autograd (see the gradient descent slides)\n", + "import autograd.numpy as np\n", + "from autograd import grad, elementwise_grad\n", + "import autograd.numpy.random as npr\n", + "from matplotlib import pyplot as plt\n", + "\n", + "def sigmoid(z):\n", + " return 1/(1 + np.exp(-z))\n", + "\n", + "def neural_network(params, x):\n", + " \n", + " # Find the weights (including and biases) for the hidden and output layer.\n", + " # Assume that params is a list of parameters for each layer. \n", + " # The biases are the first element for each array in params, \n", + " # and the weights are the remaning elements in each array in params. \n", + " \n", + " w_hidden = params[0]\n", + " w_output = params[1]\n", + "\n", + " # Assumes input x being an one-dimensional array\n", + " num_values = np.size(x)\n", + " x = x.reshape(-1, num_values)\n", + " \n", + " # Assume that the input layer does nothing to the input x\n", + " x_input = x\n", + "\n", + " ## Hidden layer:\n", + " \n", + " # Add a row of ones to include bias\n", + " x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)\n", + " \n", + " z_hidden = np.matmul(w_hidden, x_input)\n", + " x_hidden = sigmoid(z_hidden)\n", + "\n", + " ## Output layer:\n", + " \n", + " # Include bias:\n", + " x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)\n", + "\n", + " z_output = np.matmul(w_output, x_hidden)\n", + " x_output = z_output\n", + "\n", + " return x_output" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Backpropagation\n", + "\n", + "Now that feedforward can be done, the next step is to decide how the\n", + "parameters should change such that they minimize the cost function.\n", + "\n", + "Recall that the chosen cost function for this problem is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "c(x, P) = \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In order to minimize it, an optimalization method must be chosen. \n", + "\n", + "Here, gradient descent with a constant step size has been chosen. \n", + "\n", + "Before looking at the gradient descent method, let us set up the cost\n", + "function along with the right ride of the ODE and trial solution." + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# The trial solution using the deep neural network:\n", + "def g_trial(x,params, g0 = 10):\n", + " return g0 + x*neural_network(params,x)\n", + "\n", + "# The right side of the ODE:\n", + "def g(x, g_trial, gamma = 2):\n", + " return -gamma*g_trial\n", + "\n", + "# The cost function:\n", + "def cost_function(P, x):\n", + " \n", + " # Evaluate the trial function with the current parameters P\n", + " g_t = g_trial(x,P)\n", + " \n", + " # Find the derivative w.r.t x of the neural network\n", + " d_net_out = elementwise_grad(neural_network,1)(P,x) \n", + " \n", + " # Find the derivative w.r.t x of the trial function\n", + " d_g_t = elementwise_grad(g_trial,0)(x,P) \n", + " \n", + " # The right side of the ODE \n", + " func = g(x, g_t)\n", + "\n", + " err_sqr = (d_g_t - func)**2\n", + " cost_sum = np.sum(err_sqr)\n", + " \n", + " return cost_sum" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Gradient Descent\n", + "\n", + "The idea of the gradient descent algorithm is to update parameters in\n", + "direction where the cost function decreases goes to a minimum.\n", + "\n", + "In general, the update of some parameters $\\vec \\omega$ given a cost\n", + "function defined by some weights $\\vec \\omega$, $c(x, \\vec \\omega)$,\n", + "goes as follows:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\vec \\omega_{\\text{new} } = \\vec \\omega - \\lambda \\nabla_{\\vec \\omega} c(x, \\vec \\omega),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "for a number of iterations or until $ \\big|\\big| \\vec\n", + "\\omega_{\\text{new} } - \\vec \\omega \\big|\\big|$ is smaller than some\n", + "given tolerance.\n", + "\n", + "The value of $\\lambda$ decides how large steps the algorithm must take\n", + "in the direction of $ \\nabla_{\\vec \\omega} c(x, \\vec \\omega)$. The\n", + "notatation $\\nabla_{\\vec \\omega}$ denotes the gradient with respect to\n", + "the elements in $\\vec \\omega$.\n", + "\n", + "\n", + "## More on GD and cost function\n", + "\n", + "In our case, we have to minimize the cost function $c(x, P)$ with\n", + "respect to the two sets of weights and bisases, that is for the hidden\n", + "layer $P_{\\text{hidden} }$ and for the ouput layer $P_{\\text{output}\n", + "}$ .\n", + "\n", + "This means that $P_{\\text{hidden} }$ and $P_{\\text{output} }$ is\n", + "updated by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "P_{\\text{hidden},\\text{new}} &= P_{\\text{hidden}} - \\lambda \\nabla_{P_{\\text{hidden}}} c(x, P) \\\\\n", + "P_{\\text{output},\\text{new}} &= P_{\\text{output}} - \\lambda \\nabla_{P_{\\text{output}}} c(x, P) \n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This might look like a cumberstone to set up the correct expression\n", + "for finding the gradients. Luckily, Autograd comes to the rescue." + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):\n", + " ## Set up initial weigths and biases \n", + " \n", + " # For the hidden layer\n", + " p0 = npr.randn(num_neurons_hidden, 2 ) \n", + "\n", + " # For the output layer\n", + " p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included\n", + "\n", + " P = [p0, p1]\n", + "\n", + " print('Initial cost: %g'%cost_function(P, x))\n", + " \n", + " ## Start finding the optimal weigths using gradient descent\n", + " \n", + " # Find the Python function that represents the gradient of the cost function\n", + " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", + " cost_function_grad = grad(cost_function,0)\n", + " \n", + " # Let the update be done num_iter times\n", + " for i in range(num_iter):\n", + " # Evaluate the gradient at the current weights and biases in P. \n", + " # The cost_grad consist now of two arrays; \n", + " # one for the gradient w.r.t P_hidden and \n", + " # one for the gradient w.r.t P_output\n", + " cost_grad = cost_function_grad(P, x)\n", + " \n", + " P[0] = P[0] - lmb * cost_grad[0]\n", + " P[1] = P[1] - lmb * cost_grad[1]\n", + "\n", + " print('Final cost: %g'%cost_function(P, x))\n", + " \n", + " return P" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## An implementation of a Deep Neural Network\n", + "\n", + "As previously stated, a Deep Neural Network (DNN) follows the same\n", + "concept of a neural network, but having more than one hidden\n", + "layer. Suppose that the network has $N_{\\text{hidden}}$ hidden layers\n", + "where the $l$-th layer has $N_{\\text{hidden}}^{(l)}$ neurons. The\n", + "input is still assumed to be an array of size $1 \\times N$. The\n", + "network must now try to optimalize its output with respect to the\n", + "collection of weigths and biases $P = \\big\\{P_{\\text{input} }, \\\n", + "P_{\\text{hidden} }^{(1)}, \\ P_{\\text{hidden} }^{(2)}, \\ \\dots , \\\n", + "P_{\\text{hidden} }^{(N_{\\text{hidden}})}, \\ P_{\\text{output} }\\big\\}$.\n", + "\n", + "\n", + "## Feed forward again\n", + "\n", + "The feedforward step is similar to as for the neural netowork, but now considering more than one hidden layer. \n", + "\n", + "The $i$-th neuron at layer $l$ recieves the result\n", + "$\\vec{x}_j^{(l-1),\\text{hidden} }$ from the $j$-th neuron at layer\n", + "$l-1$. The $i$-th neuron at layer $l$ weights all of the elements in\n", + "$\\vec{x}_j^{(l-1),\\text{hidden} }$ with a weight vector $\\vec\n", + "w_{i,j}^{(l), \\ \\text{hidden} }$ with as many weigths as there are\n", + "elements in$\\vec{x}_j^{(l-1),\\text{hidden} }$, and adds a bias\n", + "$b_i^{(l), \\ \\text{hidden} }$:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "z_{i,j}^{(l),\\ \\text{hidden}} &= b_i^{(l), \\ \\text{hidden}} + \\big(\\vec{w}_{i}^{(l), \\ \\text{hidden}}\\big)^T\\vec{x}_j^{(l-1),\\text{hidden} } \\\\\n", + "&= \n", + "\\begin{pmatrix}\n", + "b_i^{(l), \\ \\text{hidden}} & \\big(\\vec{w}_{i}^{(l), \\ \\text{hidden}}\\big)^T\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "1 \\\\\n", + "\\vec{x}_j^{(l-1),\\text{hidden} }\n", + "\\end{pmatrix} \n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The output from the $i$-th neuron at the hidden layer $l$ becomes a vector $\\vec{z}_{i}^{(l),\\ \\text{hidden}}$:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\begin{aligned}\n", + "\\vec{z}_{i}^{(l),\\ \\text{hidden}} &= \\Big( b_i^{(l), \\ \\text{hidden}} + \\big(\\vec{w}_{i}^{(l), \\ \\text{hidden}}\\big)^T\\vec{x}_1^{(l-1),\\text{hidden} }, \\ \\dots \\ , \\ b_i^{(l), \\ \\text{hidden}} + \\big(\\vec{w}_{i}^{(l), \\ \\text{hidden}}\\big)^T\\vec{x}_{N_{hidden}^{(l-1)}}^{(l-1),\\text{hidden} } \\Big) \\\\\n", + "&= \n", + "\\begin{pmatrix}\n", + "b_i^{(l), \\ \\text{hidden}} & \\big(\\vec{w}_{i}^{(l), \\ \\text{hidden}}\\big)^T\n", + "\\end{pmatrix}\n", + "\\begin{pmatrix}\n", + "1 & 1 & \\dots & 1 \\\\\n", + "\\vec{x}_{1}^{(l-1),\\text{hidden} } & \\vec{x}_{2}^{(l-1),\\text{hidden} } & \\dots & \\vec{x}_{N_{hidden}^{(l-1)}}^{(l-1),\\text{hidden} }\n", + "\\end{pmatrix}\n", + "\\end{aligned}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The final parts of the code" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def deep_neural_network(deep_params, x):\n", + " # N_hidden is the number of hidden layers \n", + " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", + " \n", + " # Assumes input x being an one-dimensional array\n", + " num_values = np.size(x)\n", + " x = x.reshape(-1, num_values)\n", + " \n", + " # Assume that the input layer does nothing to the input x\n", + " x_input = x\n", + " \n", + " # Due to multiple hidden layers, define a variable referencing to the\n", + " # output of the previous layer:\n", + " x_prev = x_input \n", + " \n", + " ## Hidden layers:\n", + " \n", + " for l in range(N_hidden):\n", + " # From the list of parameters P; find the correct weigths and bias for this layer\n", + " w_hidden = deep_params[l]\n", + " \n", + " # Add a row of ones to include bias\n", + " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", + "\n", + " z_hidden = np.matmul(w_hidden, x_prev)\n", + " x_hidden = sigmoid(z_hidden)\n", + "\n", + " # Update x_prev such that next layer can use the output from this layer\n", + " x_prev = x_hidden \n", + "\n", + " ## Output layer:\n", + " \n", + " # Get the weights and bias for this layer\n", + " w_output = deep_params[-1]\n", + " \n", + " # Include bias:\n", + " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", + "\n", + " z_output = np.matmul(w_output, x_prev)\n", + " x_output = z_output\n", + "\n", + " return x_output" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## And adding Back propagation\n", + "\n", + "This step is very similar for the neural network. The idea in this\n", + "step is the same as for the neural network, but with more parameters\n", + "to update for. Again there is no need for computing the gradients\n", + "analytically since Autograd does the work for us." + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# The trial solution using the deep neural network:\n", + "def g_trial_deep(x,params, g0 = 10):\n", + " return g0 + x*deep_neural_network(params,x)\n", + "\n", + "# The same cost function as for the neural network, but calls deep_neural_network instead.\n", + "def cost_function_deep(P, x):\n", + " \n", + " # Evaluate the trial function with the current parameters P\n", + " g_t = g_trial_deep(x,P)\n", + " \n", + " # Find the derivative w.r.t x of the neural network\n", + " d_net_out = elementwise_grad(deep_neural_network,1)(P,x) \n", + " \n", + " # Find the derivative w.r.t x of the trial function\n", + " d_g_t = elementwise_grad(g_trial_deep,0)(x,P) \n", + " \n", + " # The right side of the ODE \n", + " func = g(x, g_t)\n", + "\n", + " err_sqr = (d_g_t - func)**2\n", + " cost_sum = np.sum(err_sqr)\n", + " \n", + " return cost_sum\n", + "\n", + "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", + " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", + "\n", + " # Find the number of hidden layers:\n", + " N_hidden = np.size(num_neurons)\n", + " \n", + " ## Set up initial weigths and biases \n", + " \n", + " # Initialize the list of parameters:\n", + " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", + "\n", + " P[0] = npr.randn(num_neurons[0], 2 ) \n", + " for l in range(1,N_hidden):\n", + " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias \n", + " \n", + " # For the output layer\n", + " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", + "\n", + " print('Initial cost: %g'%cost_function_deep(P, x))\n", + " \n", + " ## Start finding the optimal weigths using gradient descent\n", + " \n", + " # Find the Python function that represents the gradient of the cost function\n", + " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", + " cost_function_deep_grad = grad(cost_function_deep,0)\n", + " \n", + " # Let the update be done num_iter times\n", + " for i in range(num_iter):\n", + " # Evaluate the gradient at the current weights and biases in P. \n", + " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", + " # in the hidden layers and output layers evaluated at x.\n", + " cost_deep_grad = cost_function_deep_grad(P, x)\n", + " \n", + " for l in range(N_hidden+1):\n", + " P[l] = P[l] - lmb * cost_deep_grad[l]\n", + "\n", + " print('Final cost: %g'%cost_function_deep(P, x))\n", + " \n", + " return P" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Solving the ODE\n", + "\n", + "Finally, having set up the networks we are ready to use them to solve the ODE problem. \n", + "We add the analytical solution" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "def g_analytic(x, gamma = 2, g0 = 10):\n", + " return g0*np.exp(-gamma*x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using neural network\n", + "\n", + "The code below solves the ODE using a neural network. The number of\n", + "values for the input $\\vec x$ is 10, number of hidden neurons in the\n", + "hidden layer being 10 and th step size used in gradien descent\n", + "$\\lambda = 0.001$. The program updates the weights and biases in the\n", + "network *num_iter* times. Finally, it plots the results from using the\n", + "neural network along with the analytical solution." + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "npr.seed(15)\n", + "\n", + "## Decide the vales of arguments to the function to solve\n", + "N = 10\n", + "x = np.linspace(0, 1, N)\n", + "\n", + "## Set up the initial parameters\n", + "num_hidden_neurons = 10\n", + "num_iter = 10000\n", + "lmb = 0.001\n", + "\n", + "P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", + "\n", + "res = g_trial(x,P) \n", + "res_analytical = g_analytic(x)\n", + "\n", + "print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))\n", + "\n", + "plt.figure(figsize=(10,10))\n", + "\n", + "plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", + "plt.plot(x, res_analytical)\n", + "plt.plot(x, res[0,:])\n", + "plt.legend(['analytical','nn'])\n", + "plt.xlabel('x')\n", + "plt.ylabel('g(x)')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Using a deep neural network" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "npr.seed(15)\n", + "\n", + "## Decide the vales of arguments to the function to solve\n", + "N = 10\n", + "x = np.linspace(0, 1, N)\n", + "\n", + "## Set up the initial parameters\n", + "num_hidden_neurons = np.array([10,10])\n", + "num_iter = 10000\n", + "lmb = 0.001\n", + "\n", + "P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", + "\n", + "res = g_trial_deep(x,P) \n", + "res_analytical = g_analytic(x)\n", + "\n", + "plt.figure(figsize=(10,10))\n", + "\n", + "plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')\n", + "plt.plot(x, res_analytical)\n", + "plt.plot(x, res[0,:])\n", + "plt.legend(['analytical','dnn'])\n", + "plt.ylabel('g(x)')\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Wrapping it up\n", + "\n", + "By rewriting the ODE as a minimization problem, it was possible to\n", + "solve equation using either a neural network (one hidden layer) or a\n", + "deep neural network (more than one hidden layers). How well the\n", + "network performed is measured by a specified cost function, which is\n", + "the function the network tries to minimize. Using a trial solution\n", + "which satisfies the additional condition and being defined by using\n", + "the output from the network in some way, the minimization problem\n", + "could be explicitly defined for out network to solve. The proposed\n", + "solution from the network is then the trial solution with parameters,\n", + "that is weights and biases within each layer in the network, such that\n", + "the solution minimizes the cost function." ] } ], diff --git a/doc/pub/NeuralNet/ipynb/ipynb-NeuralNet-src.tar.gz b/doc/pub/NeuralNet/ipynb/ipynb-NeuralNet-src.tar.gz index b3dd04a5793087757b4c015525fa9f262861d696..a3461374c3f44d0e9d62191e423feca78ca7b4e1 100644 GIT binary patch delta 20 ccmeyop7ry3RyO%=4u)ePjci-l7~h5i09T#|r~m)} delta 20 ccmeyop7ry3RyO%=4u*dojci-l7~h5i09iH%)Bpeg diff --git a/doc/pub/NeuralNet/pdf/NeuralNet-minted.pdf b/doc/pub/NeuralNet/pdf/NeuralNet-minted.pdf index 96f40fd3edd4952c69fc4d67066d16cbd07e7cdc..e6e3cc0f6a457fed58b1cca7bc332dcb0d6b751f 100644 GIT binary patch delta 254957 zcmZs?^IxEi`}Cc?*|uGqZQJ%{TW9uW*Jj(cZM)62*}kvW=llHl{1eB_`VOc?5U?ije@0-eY-xMWptQiUwshR)7)H;#Cd>#v^^pGHhL$?c%O3Hq*|8CJWB~q1 ze*l9rg9&+qs7gz=VQwv^3Oxsf8(uI3x2sEK>ymEIvjL4hAN|5`TSD^EJ&8 zyu4g$$u_W+vGtH*(GrMzhrj 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