rnn update

doc/pub/Recurrent/html/Recurrent.html

@@ -92,7 +92,7 @@ MathJax.Hub.Config({
 <center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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-<center><h4>Dec 16, 2019</h4></center> <!-- date -->
+<center><h4>Dec 17, 2019</h4></center> <!-- date -->
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@@ -100,10 +100,9 @@ MathJax.Hub.Config({
 <h2 id="___sec0">Recurrent neural networks: Overarching view </h2>
 
 <p>
-Till now our focus has been, including convolutional neural networks as well, 
-on feedforward neural networks. The output or the
-activations flow only in one direction, from the input layer to the
-output layer.
+Till now our focus has been, including convolutional neural networks
+as well, on feedforward neural networks. The output or the activations
+flow only in one direction, from the input layer to the output layer.
 
 <p>
 A recurrent neural network (RNN) looks very much like a feedforward
@@ -121,7 +120,7 @@ input, making them extremely useful for natural language processing
 systems such as automatic translation and speech-to-text.
 
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-<b>The text here is under development</b>. Planned finished Jan 2020.
+<b>The text here is under development</b>. Planned finished mid Jan 2020.
 
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@@ -136,11 +135,10 @@ to the inputs \( x_t \), the layer at a time \( t \) receives also as input
 the output from the previous layer \( t-1 \), that is \( y_{t1} \).
 
 <p>
-This means also that we need to have weights that link both the inputs \( x_t \) to the outputs \( y_t \) as well as weights that link
-the output from the previous time \( y_{t-1} \) and \( y_t \).   The figure here shows an example of a simple RNN.
-
-<p>
-Add figures and bring up equations.
+This means also that we need to have weights that link both the inputs
+\( x_t \) to the outputs \( y_t \) as well as weights that link the output
+from the previous time \( y_{t-1} \) and \( y_t \).  The figure here shows an
+example of a simple RNN.
 
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@@ -148,7 +146,15 @@ Add figures and bring up equations.
 <h2 id="___sec2">Solving differential equations and eigenvalue problems with RNNs </h2>
 
 <p>
-Have example with eigenvalues solvers as well.
+In our discussions of ordinary differential equations and partial
+differential equations using neural networks. Here we will discuss how
+we can solve say ordinary differential equations and eigenvalue
+problems using RNNs. Eigenvalue problems can be solved using RNNs by
+rewriting such a problems as a non-linear differential equation.
+
+<p>
+Instead of starting with a well-known ordinary differential equation,
+we start directly with an eigenvaule problem.
 
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