Morten Hjorth-Jensen
621 lines
39 KiB
HTML
621 lines
39 KiB
HTML
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<a class="navbar-brand" href="week41-bs.html">Week 41 Neural networks and constructing a neural network code</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week41-bs001.html#plan-for-week-41-october-7-11" style="font-size: 80%;"><b>Plan for week 41, October 7-11</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs002.html#material-for-the-lecture-on-monday-october-7-2024" style="font-size: 80%;"><b>Material for the lecture on Monday October 7, 2024</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs003.html#material-for-the-active-learning-sessions-on-tuesday-and-wednesday" style="font-size: 80%;"><b>Material for the active learning sessions on Tuesday and Wednesday</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs004.html#lecture-monday-october-7" style="font-size: 80%;"><b>Lecture Monday October 7</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs005.html#introduction-to-neural-networks" style="font-size: 80%;"><b>Introduction to Neural networks</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="#artificial-neurons" style="font-size: 80%;"><b>Artificial neurons</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs007.html#neural-network-types" style="font-size: 80%;"><b>Neural network types</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs008.html#feed-forward-neural-networks" style="font-size: 80%;"><b>Feed-forward neural networks</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs009.html#convolutional-neural-network" style="font-size: 80%;"><b>Convolutional Neural Network</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs010.html#recurrent-neural-networks" style="font-size: 80%;"><b>Recurrent neural networks</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs011.html#other-types-of-networks" style="font-size: 80%;"><b>Other types of networks</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs012.html#multilayer-perceptrons" style="font-size: 80%;"><b>Multilayer perceptrons</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs013.html#why-multilayer-perceptrons" style="font-size: 80%;"><b>Why multilayer perceptrons?</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs014.html#illustration-of-a-single-perceptron-model-and-a-multi-perceptron-model" style="font-size: 80%;"><b>Illustration of a single perceptron model and a multi-perceptron model</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs015.html#examples-of-xor-or-and-and-gates" style="font-size: 80%;"><b>Examples of XOR, OR and AND gates</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs016.html#does-logistic-regression-do-a-better-job" style="font-size: 80%;"><b>Does Logistic Regression do a better Job?</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs017.html#adding-neural-networks" style="font-size: 80%;"><b>Adding Neural Networks</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs018.html#mathematics-of-deep-learning" style="font-size: 80%;"><b>Mathematics of deep learning</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs019.html#reminder-on-books-with-hands-on-material-and-codes" style="font-size: 80%;"><b>Reminder on books with hands-on material and codes</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs020.html#reading-recommendations" style="font-size: 80%;"><b>Reading recommendations</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs021.html#mathematics-of-deep-learning-and-neural-networks" style="font-size: 80%;"><b>Mathematics of deep learning and neural networks</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs022.html#basics-of-an-nn" style="font-size: 80%;"><b>Basics of an NN</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._week41-bs023.html#overarching-view-of-a-neural-network" style="font-size: 80%;"><b>Overarching view of a neural network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs024.html#the-optimization-problem" style="font-size: 80%;"><b>The optimization problem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs025.html#parameters-of-neural-networks" style="font-size: 80%;"><b>Parameters of neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs026.html#other-ingredients-of-a-neural-network" style="font-size: 80%;"><b>Other ingredients of a neural network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs027.html#other-parameters" style="font-size: 80%;"><b>Other parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs028.html#universal-approximation-theorem" style="font-size: 80%;"><b>Universal approximation theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs029.html#some-parallels-from-real-analysis" style="font-size: 80%;"><b>Some parallels from real analysis</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs030.html#the-approximation-theorem-in-words" style="font-size: 80%;"><b>The approximation theorem in words</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs031.html#more-on-the-general-approximation-theorem" style="font-size: 80%;"><b>More on the general approximation theorem</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs032.html#class-of-functions-we-can-approximate" style="font-size: 80%;"><b>Class of functions we can approximate</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs033.html#setting-up-the-equations-for-a-neural-network" style="font-size: 80%;"><b>Setting up the equations for a neural network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs034.html#layout-of-a-neural-network-with-three-hidden-layers" style="font-size: 80%;"><b>Layout of a neural network with three hidden layers</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs035.html#definitions" style="font-size: 80%;"><b>Definitions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs036.html#inputs-to-the-activation-function" style="font-size: 80%;"><b>Inputs to the activation function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs037.html#derivatives-and-the-chain-rule" style="font-size: 80%;"><b>Derivatives and the chain rule</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs038.html#derivative-of-the-cost-function" style="font-size: 80%;"><b>Derivative of the cost function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs039.html#simpler-examples-first-and-automatic-differentiation" style="font-size: 80%;"><b>Simpler examples first, and automatic differentiation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs040.html#reminder-on-the-chain-rule-and-gradients" style="font-size: 80%;"><b>Reminder on the chain rule and gradients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs041.html#multivariable-functions" style="font-size: 80%;"><b>Multivariable functions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs042.html#automatic-differentiation-through-examples" style="font-size: 80%;"><b>Automatic differentiation through examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs043.html#simple-example" style="font-size: 80%;"><b>Simple example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs044.html#smarter-way-of-evaluating-the-above-function" style="font-size: 80%;"><b>Smarter way of evaluating the above function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs045.html#reducing-the-number-of-operations" style="font-size: 80%;"><b>Reducing the number of operations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs046.html#chain-rule-forward-and-reverse-modes" style="font-size: 80%;"><b>Chain rule, forward and reverse modes</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs047.html#forward-and-reverse-modes" style="font-size: 80%;"><b>Forward and reverse modes</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs048.html#more-complicated-function" style="font-size: 80%;"><b>More complicated function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs049.html#counting-the-number-of-floating-point-operations" style="font-size: 80%;"><b>Counting the number of floating point operations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs050.html#defining-intermediate-operations" style="font-size: 80%;"><b>Defining intermediate operations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs051.html#new-expression-for-the-derivative" style="font-size: 80%;"><b>New expression for the derivative</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs052.html#final-derivatives" style="font-size: 80%;"><b>Final derivatives</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs053.html#in-general-not-this-simple" style="font-size: 80%;"><b>In general not this simple</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs054.html#automatic-differentiation" style="font-size: 80%;"><b>Automatic differentiation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs055.html#chain-rule" style="font-size: 80%;"><b>Chain rule</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs056.html#first-network-example-simple-percepetron-with-one-input" style="font-size: 80%;"><b>First network example, simple percepetron with one input</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs057.html#layout-of-a-simple-neural-network-with-no-hidden-layer" style="font-size: 80%;"><b>Layout of a simple neural network with no hidden layer</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs058.html#optimizing-the-parameters" style="font-size: 80%;"><b>Optimizing the parameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs059.html#adding-a-hidden-layer" style="font-size: 80%;"><b>Adding a hidden layer</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs060.html#layout-of-a-simple-neural-network-with-one-hidden-layer" style="font-size: 80%;"><b>Layout of a simple neural network with one hidden layer</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs061.html#the-derivatives" style="font-size: 80%;"><b>The derivatives</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs062.html#important-observations" style="font-size: 80%;"><b>Important observations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs063.html#the-training" style="font-size: 80%;"><b>The training</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs064.html#code-example" style="font-size: 80%;"><b>Code example</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs065.html#exercise-1-including-more-data" style="font-size: 80%;"><b>Exercise 1: Including more data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs066.html#simple-neural-network-and-the-back-propagation-equations" style="font-size: 80%;"><b>Simple neural network and the back propagation equations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs067.html#layout-of-a-simple-neural-network-with-two-input-nodes-one-hidden-layer-and-one-output-node" style="font-size: 80%;"><b>Layout of a simple neural network with two input nodes, one hidden layer and one output node</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs068.html#the-ouput-layer" style="font-size: 80%;"><b>The ouput layer</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs069.html#compact-expressions" style="font-size: 80%;"><b>Compact expressions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs070.html#output-layer" style="font-size: 80%;"><b>Output layer</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs071.html#explicit-derivatives" style="font-size: 80%;"><b>Explicit derivatives</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs072.html#derivatives-of-the-hidden-layer" style="font-size: 80%;"><b>Derivatives of the hidden layer</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs073.html#final-expression" style="font-size: 80%;"><b>Final expression</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs074.html#completing-the-list" style="font-size: 80%;"><b>Completing the list</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs075.html#final-expressions-for-the-biases-of-the-hidden-layer" style="font-size: 80%;"><b>Final expressions for the biases of the hidden layer</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs076.html#gradient-expressions" style="font-size: 80%;"><b>Gradient expressions</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs077.html#exercise-2-extended-program" style="font-size: 80%;"><b>Exercise 2: Extended program</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs078.html#getting-serious-the-back-propagation-equations-for-a-neural-network" style="font-size: 80%;"><b>Getting serious, the back propagation equations for a neural network</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs079.html#analyzing-the-last-results" style="font-size: 80%;"><b>Analyzing the last results</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs080.html#more-considerations" style="font-size: 80%;"><b>More considerations</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs081.html#derivatives-in-terms-of-z-j-l" style="font-size: 80%;"><b>Derivatives in terms of \( z_j^L \)</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs082.html#bringing-it-together" style="font-size: 80%;"><b>Bringing it together</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs083.html#final-back-propagating-equation" style="font-size: 80%;"><b>Final back propagating equation</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs084.html#using-the-chain-rule-and-summing-over-all-k-entries" style="font-size: 80%;"><b>Using the chain rule and summing over all \( k \) entries</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs085.html#setting-up-the-back-propagation-algorithm" style="font-size: 80%;"><b>Setting up the back propagation algorithm</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs086.html#setting-up-the-back-propagation-algorithm-part-2" style="font-size: 80%;"><b>Setting up the back propagation algorithm, part 2</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs087.html#setting-up-the-back-propagation-algorithm-part-3" style="font-size: 80%;"><b>Setting up the Back propagation algorithm, part 3</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs088.html#updating-the-gradients" style="font-size: 80%;"><b>Updating the gradients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs089.html#activation-functions" style="font-size: 80%;"> Activation functions</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs090.html#activation-functions-logistic-and-hyperbolic-ones" style="font-size: 80%;"> Activation functions, Logistic and Hyperbolic ones</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs091.html#relevance" style="font-size: 80%;"> Relevance</a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs092.html#fine-tuning-neural-network-hyperparameters" style="font-size: 80%;"><b>Fine-tuning neural network hyperparameters</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs093.html#hidden-layers" style="font-size: 80%;"><b>Hidden layers</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs094.html#vanishing-gradients" style="font-size: 80%;"><b>Vanishing gradients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs095.html#exploding-gradients" style="font-size: 80%;"><b>Exploding gradients</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs096.html#is-the-logistic-activation-function-sigmoid-our-choice" style="font-size: 80%;"><b>Is the Logistic activation function (Sigmoid) our choice?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs097.html#logistic-function-as-the-root-of-problems" style="font-size: 80%;"><b>Logistic function as the root of problems</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs098.html#the-derivative-of-the-logistic-funtion" style="font-size: 80%;"><b>The derivative of the Logistic funtion</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs099.html#insights-from-the-paper-by-glorot-and-bengio" style="font-size: 80%;"><b>Insights from the paper by Glorot and Bengio</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs100.html#the-relu-function-family" style="font-size: 80%;"><b>The RELU function family</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs101.html#elu-function" style="font-size: 80%;"><b>ELU function</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs102.html#which-activation-function-should-we-use" style="font-size: 80%;"><b>Which activation function should we use?</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs103.html#more-on-activation-functions-output-layers" style="font-size: 80%;"><b>More on activation functions, output layers</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs104.html#batch-normalization" style="font-size: 80%;"><b>Batch Normalization</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs105.html#dropout" style="font-size: 80%;"><b>Dropout</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs106.html#gradient-clipping" style="font-size: 80%;"><b>Gradient Clipping</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs107.html#a-top-down-perspective-on-neural-networks" style="font-size: 80%;"><b>A top-down perspective on Neural networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs108.html#more-top-down-perspectives" style="font-size: 80%;"><b>More top-down perspectives</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs109.html#limitations-of-supervised-learning-with-deep-networks" style="font-size: 80%;"><b>Limitations of supervised learning with deep networks</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs110.html#limitations-of-nns" style="font-size: 80%;"><b>Limitations of NNs</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs111.html#homogeneous-data" style="font-size: 80%;"><b>Homogeneous data</b></a></li>
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<!-- navigation toc: --> <li><a href="._week41-bs112.html#more-limitations" style="font-size: 80%;"><b>More limitations</b></a></li>
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<div class="container">
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0006"></a>
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<!-- !split -->
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<h2 id="artificial-neurons" class="anchor">Artificial neurons </h2>
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<p>The field of artificial neural networks has a long history of
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development, and is closely connected with the advancement of computer
|
|
science and computers in general. A model of artificial neurons was
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first developed by McCulloch and Pitts in 1943 to study signal
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processing in the brain and has later been refined by others. The
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general idea is to mimic neural networks in the human brain, which is
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composed of billions of neurons that communicate with each other by
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sending electrical signals. Each neuron accumulates its incoming
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signals, which must exceed an activation threshold to yield an
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output. If the threshold is not overcome, the neuron remains inactive,
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i.e. has zero output.
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</p>
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<p>This behaviour has inspired a simple mathematical model for an artificial neuron.</p>
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|
|
$$
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\begin{equation}
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y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u)
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\tag{1}
|
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\end{equation}
|
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$$
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<p>Here, the output \( y \) of the neuron is the value of its activation function, which have as input
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|
a weighted sum of signals \( x_i, \dots ,x_n \) received by \( n \) other neurons.
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</p>
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|
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<p>Conceptually, it is helpful to divide neural networks into four
|
|
categories:
|
|
</p>
|
|
<ol>
|
|
<li> general purpose neural networks for supervised learning,</li>
|
|
<li> neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),</li>
|
|
<li> neural networks for sequential data such as Recurrent Neural Networks (RNNs), and</li>
|
|
<li> neural networks for unsupervised learning such as Deep Boltzmann Machines.</li>
|
|
</ol>
|
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<p>In natural science, DNNs and CNNs have already found numerous
|
|
applications. In statistical physics, they have been applied to detect
|
|
phase transitions in 2D Ising and Potts models, lattice gauge
|
|
theories, and different phases of polymers, or solving the
|
|
Navier-Stokes equation in weather forecasting. Deep learning has also
|
|
found interesting applications in quantum physics. Various quantum
|
|
phase transitions can be detected and studied using DNNs and CNNs,
|
|
topological phases, and even non-equilibrium many-body
|
|
localization. Representing quantum states as DNNs quantum state
|
|
tomography are among some of the impressive achievements to reveal the
|
|
potential of DNNs to facilitate the study of quantum systems.
|
|
</p>
|
|
|
|
<p>In quantum information theory, it has been shown that one can perform
|
|
gate decompositions with the help of neural.
|
|
</p>
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|
|
|
<p>The applications are not limited to the natural sciences. There is a
|
|
plethora of applications in essentially all disciplines, from the
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humanities to life science and medicine.
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</p>
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<p>
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