lars/FYS-STK4155
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
Files
05a43432a732e438285bbcaab800237fe43d1215
FYS-STK4155/doc/LectureNotes/_build/html/chapter13.html
T
Morten Hjorth-Jensen 05a43432a7 updated schedule with video
2022-09-10 21:33:46 +02:00

2355 lines
206 KiB
HTML
Raw Blame History

This file contains ambiguous Unicode characters
This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>17. Recurrent neural networks: Overarching view &#8212; Applied Data Analysis and Machine Learning</title>
<link href="_static/css/theme.css" rel="stylesheet" />
<link href="_static/css/index.c5995385ac14fb8791e8eb36b4908be2.css" rel="stylesheet" />
<link rel="stylesheet"
href="_static/vendor/fontawesome/5.13.0/css/all.min.css">
<link rel="preload" as="font" type="font/woff2" crossorigin
href="_static/vendor/fontawesome/5.13.0/webfonts/fa-solid-900.woff2">
<link rel="preload" as="font" type="font/woff2" crossorigin
href="_static/vendor/fontawesome/5.13.0/webfonts/fa-brands-400.woff2">
<link rel="stylesheet" type="text/css" href="_static/pygments.css" />
<link rel="stylesheet" type="text/css" href="_static/sphinx-book-theme.css?digest=c3fdc42140077d1ad13ad2f1588a4309" />
<link rel="stylesheet" type="text/css" href="_static/togglebutton.css" />
<link rel="stylesheet" type="text/css" href="_static/copybutton.css" />
<link rel="stylesheet" type="text/css" href="_static/mystnb.css" />
<link rel="stylesheet" type="text/css" href="_static/sphinx-thebe.css" />
<link rel="stylesheet" type="text/css" href="_static/panels-main.c949a650a448cc0ae9fd3441c0e17fb0.css" />
<link rel="stylesheet" type="text/css" href="_static/panels-variables.06eb56fa6e07937060861dad626602ad.css" />
<link rel="preload" as="script" href="_static/js/index.1c5a1a01449ed65a7b51.js">
<script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script>
<script src="_static/jquery.js"></script>
<script src="_static/underscore.js"></script>
<script src="_static/doctools.js"></script>
<script src="_static/togglebutton.js"></script>
<script src="_static/clipboard.min.js"></script>
<script src="_static/copybutton.js"></script>
<script>var togglebuttonSelector = '.toggle, .admonition.dropdown, .tag_hide_input div.cell_input, .tag_hide-input div.cell_input, .tag_hide_output div.cell_output, .tag_hide-output div.cell_output, .tag_hide_cell.cell, .tag_hide-cell.cell';</script>
<script src="_static/sphinx-book-theme.12a9622fbb08dcb3a2a40b2c02b83a57.js"></script>
<script>const THEBE_JS_URL = "https://unpkg.com/thebe@0.8.2/lib/index.js"
const thebe_selector = ".thebe,.cell"
const thebe_selector_input = "pre"
const thebe_selector_output = ".output, .cell_output"
</script>
<script async="async" src="_static/sphinx-thebe.js"></script>
<script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<script>window.MathJax = {"options": {"processHtmlClass": "tex2jax_process|mathjax_process|math|output_area"}}</script>
<link rel="index" title="Index" href="genindex.html" />
<link rel="search" title="Search" href="search.html" />
<link rel="prev" title="16. Convolutional Neural Networks" href="chapter12.html" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta name="docsearch:language" content="en" />
</head>
<body data-spy="scroll" data-target="#bd-toc-nav" data-offset="80">
<div class="container-fluid" id="banner"></div>
<div class="container-xl">
<div class="row">
<div class="col-12 col-md-3 bd-sidebar site-navigation show" id="site-navigation">
<div class="navbar-brand-box">
<a class="navbar-brand text-wrap" href="index.html">
<!-- `logo` is deprecated in Sphinx 4.0, so remove this when we stop supporting 3 -->
<img src="_static/logo.png" class="logo" alt="logo">
<h1 class="site-logo" id="site-title">Applied Data Analysis and Machine Learning</h1>
</a>
</div><form class="bd-search d-flex align-items-center" action="search.html" method="get">
<i class="icon fas fa-search"></i>
<input type="search" class="form-control" name="q" id="search-input" placeholder="Search this book..." aria-label="Search this book..." autocomplete="off" >
</form><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
<div class="bd-toc-item active">
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="intro.html">
Applied Data Analysis and Machine Learning
</a>
</li>
</ul>
<p class="caption" role="heading">
<span class="caption-text">
About the course
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="schedule.html">
Teaching schedule with links to material
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="teachers.html">
Teachers and Grading
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="textbooks.html">
Textbooks
</a>
</li>
</ul>
<p class="caption" role="heading">
<span class="caption-text">
Review of Statistics with Resampling Techniques and Linear Algebra
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="statistics.html">
1. Elements of Probability Theory and Statistical Data Analysis
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="linalg.html">
2. Linear Algebra, Handling of Arrays and more Python Features
</a>
</li>
</ul>
<p class="caption" role="heading">
<span class="caption-text">
From Regression to Support Vector Machines
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter1.html">
3. Linear Regression
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter2.html">
4. Ridge and Lasso Regression
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter3.html">
5. Resampling Methods
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter4.html">
6. Logistic Regression
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapteroptimization.html">
7. Optimization, the central part of any Machine Learning algortithm
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter5.html">
8. Support Vector Machines, overarching aims
</a>
</li>
</ul>
<p class="caption" role="heading">
<span class="caption-text">
Decision Trees, Ensemble Methods and Boosting
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter6.html">
9. Decision trees, overarching aims
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter7.html">
10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
</a>
</li>
</ul>
<p class="caption" role="heading">
<span class="caption-text">
Dimensionality Reduction
</span>
</p>
<ul class="nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter8.html">
11. Basic ideas of the Principal Component Analysis (PCA)
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="clustering.html">
12. Clustering and Unsupervised Learning
</a>
</li>
</ul>
<p class="caption" role="heading">
<span class="caption-text">
Deep Learning Methods
</span>
</p>
<ul class="current nav bd-sidenav">
<li class="toctree-l1">
<a class="reference internal" href="chapter9.html">
13. Neural networks
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter10.html">
14. Building a Feed Forward Neural Network
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter11.html">
15. Solving Differential Equations with Deep Learning
</a>
</li>
<li class="toctree-l1">
<a class="reference internal" href="chapter12.html">
16. Convolutional Neural Networks
</a>
</li>
<li class="toctree-l1 current active">
<a class="current reference internal" href="#">
17. Recurrent neural networks: Overarching view
</a>
</li>
</ul>
</div>
</nav> <!-- To handle the deprecated key -->
<div class="navbar_extra_footer">
Powered by <a href="https://jupyterbook.org">Jupyter Book</a>
</div>
</div>
<main class="col py-md-3 pl-md-4 bd-content overflow-auto" role="main">
<div class="topbar container-xl fixed-top">
<div class="topbar-contents row">
<div class="col-12 col-md-3 bd-topbar-whitespace site-navigation show"></div>
<div class="col pl-md-4 topbar-main">
<button id="navbar-toggler" class="navbar-toggler ml-0" type="button" data-toggle="collapse"
data-toggle="tooltip" data-placement="bottom" data-target=".site-navigation" aria-controls="navbar-menu"
aria-expanded="true" aria-label="Toggle navigation" aria-controls="site-navigation"
title="Toggle navigation" data-toggle="tooltip" data-placement="left">
<i class="fas fa-bars"></i>
<i class="fas fa-arrow-left"></i>
<i class="fas fa-arrow-up"></i>
</button>
<div class="dropdown-buttons-trigger">
<button id="dropdown-buttons-trigger" class="btn btn-secondary topbarbtn" aria-label="Download this page"><i
class="fas fa-download"></i></button>
<div class="dropdown-buttons">
<!-- ipynb file if we had a myst markdown file -->
<!-- Download raw file -->
<a class="dropdown-buttons" href="_sources/chapter13.ipynb"><button type="button"
class="btn btn-secondary topbarbtn" title="Download source file" data-toggle="tooltip"
data-placement="left">.ipynb</button></a>
<!-- Download PDF via print -->
<button type="button" id="download-print" class="btn btn-secondary topbarbtn" title="Print to PDF"
onClick="window.print()" data-toggle="tooltip" data-placement="left">.pdf</button>
</div>
</div>
<!-- Source interaction buttons -->
<!-- Full screen (wrap in <a> to have style consistency -->
<a class="full-screen-button"><button type="button" class="btn btn-secondary topbarbtn" data-toggle="tooltip"
data-placement="bottom" onclick="toggleFullScreen()" aria-label="Fullscreen mode"
title="Fullscreen mode"><i
class="fas fa-expand"></i></button></a>
<!-- Launch buttons -->
</div>
<!-- Table of contents -->
<div class="d-none d-md-block col-md-2 bd-toc show">
<div class="tocsection onthispage pt-5 pb-3">
<i class="fas fa-list"></i> Contents
</div>
<nav id="bd-toc-nav" aria-label="Page">
<ul class="visible nav section-nav flex-column">
<li class="toc-h1 nav-item toc-entry">
<a class="reference internal nav-link" href="#">
17. Recurrent neural networks: Overarching view
</a>
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#a-simple-example">
17.1. A simple example
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#an-extrapolation-example">
17.2. An extrapolation example
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#predicting-new-points-with-a-trained-recurrent-neural-network">
17.3. Predicting New Points With A Trained Recurrent Neural Network
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#other-types-of-recurrent-neural-networks">
17.4. Other Types of Recurrent Neural Networks
</a>
</li>
</ul>
</li>
<li class="toc-h1 nav-item toc-entry">
<a class="reference internal nav-link" href="#generative-models">
18. Generative Models
</a>
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#generative-adversarial-networks">
18.1. Generative Adversarial Networks
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#writing-our-first-generative-adversarial-network">
18.2. Writing Our First Generative Adversarial Network
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#mnist-and-gans">
18.2.1. MNIST and GANs
</a>
</li>
</ul>
</li>
</ul>
</li>
</ul>
</nav>
</div>
</div>
</div>
<div id="main-content" class="row">
<div class="col-12 col-md-9 pl-md-3 pr-md-0">
<div>
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
doconce format html chapter13.do.txt --><div class="tex2jax_ignore mathjax_ignore section" id="recurrent-neural-networks-overarching-view">
<h1><span class="section-number">17. </span>Recurrent neural networks: Overarching view<a class="headerlink" href="#recurrent-neural-networks-overarching-view" title="Permalink to this headline"></a></h1>
<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.</p>
<p>A recurrent neural network (RNN) looks very much like a feedforward
neural network, except that it also has connections pointing
backward.</p>
<p>RNNs are used to analyze time series data such as stock prices, and
tell you when to buy or sell. In autonomous driving systems, they can
anticipate car trajectories and help avoid accidents. More generally,
they can work on sequences of arbitrary lengths, rather than on
fixed-sized inputs like all the nets we have discussed so far. For
example, they can take sentences, documents, or audio samples as
input, making them extremely useful for natural language processing
systems such as automatic translation and speech-to-text.</p>
<p>More to text to be added</p>
<div class="section" id="a-simple-example">
<h2><span class="section-number">17.1. </span>A simple example<a class="headerlink" href="#a-simple-example" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="c1"># Start importing packages</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">datasets</span><span class="p">,</span> <span class="n">layers</span><span class="p">,</span> <span class="n">models</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Input</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.models</span> <span class="kn">import</span> <span class="n">Model</span><span class="p">,</span> <span class="n">Sequential</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Dense</span><span class="p">,</span> <span class="n">SimpleRNN</span><span class="p">,</span> <span class="n">LSTM</span><span class="p">,</span> <span class="n">GRU</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">optimizers</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">regularizers</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.utils</span> <span class="kn">import</span> <span class="n">to_categorical</span>
<span class="c1"># convert into dataset matrix</span>
<span class="k">def</span> <span class="nf">convertToMatrix</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">step</span><span class="p">):</span>
<span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span><span class="p">[],</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">data</span><span class="p">)</span><span class="o">-</span><span class="n">step</span><span class="p">):</span>
<span class="n">d</span><span class="o">=</span><span class="n">i</span><span class="o">+</span><span class="n">step</span>
<span class="n">X</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">d</span><span class="p">,])</span>
<span class="n">Y</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">d</span><span class="p">,])</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
<span class="n">step</span> <span class="o">=</span> <span class="mi">4</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">Tp</span> <span class="o">=</span> <span class="mi">800</span>
<span class="n">t</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">N</span><span class="p">)</span>
<span class="n">x</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mf">0.02</span><span class="o">*</span><span class="n">t</span><span class="p">)</span><span class="o">+</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">N</span><span class="p">)</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">df</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">df</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="n">values</span><span class="o">=</span><span class="n">df</span><span class="o">.</span><span class="n">values</span>
<span class="n">train</span><span class="p">,</span><span class="n">test</span> <span class="o">=</span> <span class="n">values</span><span class="p">[</span><span class="mi">0</span><span class="p">:</span><span class="n">Tp</span><span class="p">,:],</span> <span class="n">values</span><span class="p">[</span><span class="n">Tp</span><span class="p">:</span><span class="n">N</span><span class="p">,:]</span>
<span class="c1"># add step elements into train and test</span>
<span class="n">test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">test</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">test</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,],</span><span class="n">step</span><span class="p">))</span>
<span class="n">train</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">train</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">repeat</span><span class="p">(</span><span class="n">train</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,],</span><span class="n">step</span><span class="p">))</span>
<span class="n">trainX</span><span class="p">,</span><span class="n">trainY</span> <span class="o">=</span><span class="n">convertToMatrix</span><span class="p">(</span><span class="n">train</span><span class="p">,</span><span class="n">step</span><span class="p">)</span>
<span class="n">testX</span><span class="p">,</span><span class="n">testY</span> <span class="o">=</span><span class="n">convertToMatrix</span><span class="p">(</span><span class="n">test</span><span class="p">,</span><span class="n">step</span><span class="p">)</span>
<span class="n">trainX</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">trainX</span><span class="p">,</span> <span class="p">(</span><span class="n">trainX</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">trainX</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
<span class="n">testX</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">testX</span><span class="p">,</span> <span class="p">(</span><span class="n">testX</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mi">1</span><span class="p">,</span> <span class="n">testX</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Sequential</span><span class="p">()</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">SimpleRNN</span><span class="p">(</span><span class="n">units</span><span class="o">=</span><span class="mi">32</span><span class="p">,</span> <span class="n">input_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">step</span><span class="p">),</span> <span class="n">activation</span><span class="o">=</span><span class="s2">&quot;relu&quot;</span><span class="p">))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s2">&quot;relu&quot;</span><span class="p">))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Dense</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span>
<span class="n">model</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s1">&#39;mean_squared_error&#39;</span><span class="p">,</span> <span class="n">optimizer</span><span class="o">=</span><span class="s1">&#39;rmsprop&#39;</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">summary</span><span class="p">()</span>
<span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">trainX</span><span class="p">,</span><span class="n">trainY</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="mi">16</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">trainPredict</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">trainX</span><span class="p">)</span>
<span class="n">testPredict</span><span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">testX</span><span class="p">)</span>
<span class="n">predicted</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">trainPredict</span><span class="p">,</span><span class="n">testPredict</span><span class="p">),</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">trainScore</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="n">trainX</span><span class="p">,</span> <span class="n">trainY</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">trainScore</span><span class="p">)</span>
<span class="n">index</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">index</span><span class="o">.</span><span class="n">values</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">index</span><span class="p">,</span><span class="n">df</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">index</span><span class="p">,</span><span class="n">predicted</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">index</span><span class="p">[</span><span class="n">Tp</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;r&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/jax/_src/lib/__init__.py:33: UserWarning: JAX on Mac ARM machines is experimental and minimally tested. Please see https://github.com/google/jax/issues/5501 in the event of problems.
warnings.warn(&quot;JAX on Mac ARM machines is experimental and minimally tested. &quot;
</pre></div>
</div>
<img alt="_images/chapter13_3_1.png" src="_images/chapter13_3_1.png" />
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Model: &quot;sequential&quot;
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>_________________________________________________________________
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> Layer (type) Output Shape Param #
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>=================================================================
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> simple_rnn (SimpleRNN) (None, 32) 1184
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> dense (Dense) (None, 8) 264
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> dense_1 (Dense) (None, 1) 9
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>=================================================================
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Total params: 1,457
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Trainable params: 1,457
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Non-trainable params: 0
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>_________________________________________________________________
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 1/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 2s - loss: 1.1810 - 2s/epoch - 33ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 2/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4703 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 3/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4379 - 303ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 4/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4275 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 5/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4224 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 6/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4224 - 308ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 7/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4212 - 308ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 8/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4198 - 308ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 9/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4198 - 304ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 10/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4177 - 304ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 11/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4186 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 12/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4176 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 13/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4167 - 303ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 14/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4143 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 15/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4128 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 16/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4144 - 324ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 17/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4146 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 18/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4131 - 303ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 19/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4117 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 20/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4115 - 319ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 21/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4100 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 22/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4088 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 23/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4089 - 303ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 24/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4082 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 25/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4084 - 304ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 26/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4068 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 27/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4075 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 28/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4055 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 29/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4069 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 30/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4048 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 31/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4038 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 32/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4059 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 33/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4039 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 34/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4045 - 320ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 35/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4030 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 36/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4012 - 304ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 37/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3994 - 304ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 38/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4027 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 39/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4011 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 40/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4017 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 41/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3996 - 303ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 42/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3996 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 43/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.4005 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 44/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3994 - 303ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 45/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3979 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 46/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3983 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 47/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3979 - 304ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 48/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3955 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 49/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3970 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 50/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3962 - 304ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 51/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3943 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 52/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3975 - 304ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 53/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3944 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 54/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3950 - 304ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 55/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3960 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 56/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3922 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 57/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3922 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 58/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3928 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 59/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3915 - 317ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 60/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3916 - 304ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 61/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3889 - 308ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 62/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3917 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 63/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3891 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 64/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3906 - 308ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 65/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3914 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 66/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3889 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 67/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3872 - 302ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 68/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3881 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 69/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3869 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 70/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3888 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 71/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3864 - 305ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 72/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3871 - 302ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 73/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3862 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 74/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3861 - 303ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 75/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3858 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 76/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3864 - 326ms/epoch - 7ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 77/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3851 - 331ms/epoch - 7ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 78/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3862 - 312ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 79/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3837 - 324ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 80/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3815 - 341ms/epoch - 7ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 81/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3838 - 306ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 82/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3834 - 307ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 83/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3830 - 311ms/epoch - 6ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 84/100
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>50/50 - 0s - loss: 0.3839 - 368ms/epoch - 7ms/step
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Epoch 85/100
</pre></div>
</div>
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">KeyboardInterrupt</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="o">&lt;</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">fdfcc778e1f8</span><span class="o">&gt;</span> <span class="ow">in</span> <span class="o">&lt;</span><span class="n">module</span><span class="o">&gt;</span>
<span class="g g-Whitespace"> </span><span class="mi">56</span> <span class="n">model</span><span class="o">.</span><span class="n">summary</span><span class="p">()</span>
<span class="g g-Whitespace"> </span><span class="mi">57</span>
<span class="ne">---&gt; </span><span class="mi">58</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">trainX</span><span class="p">,</span><span class="n">trainY</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="mi">16</span><span class="p">,</span> <span class="n">verbose</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">59</span> <span class="n">trainPredict</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">trainX</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">60</span> <span class="n">testPredict</span><span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">testX</span><span class="p">)</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/keras/utils/traceback_utils.py</span> in <span class="ni">error_handler</span><span class="nt">(*args, **kwargs)</span>
<span class="g g-Whitespace"> </span><span class="mi">62</span> <span class="n">filtered_tb</span> <span class="o">=</span> <span class="kc">None</span>
<span class="g g-Whitespace"> </span><span class="mi">63</span> <span class="k">try</span><span class="p">:</span>
<span class="ne">---&gt; </span><span class="mi">64</span> <span class="k">return</span> <span class="n">fn</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">65</span> <span class="k">except</span> <span class="ne">Exception</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span> <span class="c1"># pylint: disable=broad-except</span>
<span class="g g-Whitespace"> </span><span class="mi">66</span> <span class="n">filtered_tb</span> <span class="o">=</span> <span class="n">_process_traceback_frames</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">__traceback__</span><span class="p">)</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/keras/engine/training.py</span> in <span class="ni">fit</span><span class="nt">(self, x, y, batch_size, epochs, verbose, callbacks, validation_split, validation_data, shuffle, class_weight, sample_weight, initial_epoch, steps_per_epoch, validation_steps, validation_batch_size, validation_freq, max_queue_size, workers, use_multiprocessing)</span>
<span class="g g-Whitespace"> </span><span class="mi">1382</span> <span class="n">_r</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
<span class="g g-Whitespace"> </span><span class="mi">1383</span> <span class="n">callbacks</span><span class="o">.</span><span class="n">on_train_batch_begin</span><span class="p">(</span><span class="n">step</span><span class="p">)</span>
<span class="ne">-&gt; </span><span class="mi">1384</span> <span class="n">tmp_logs</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">train_function</span><span class="p">(</span><span class="n">iterator</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">1385</span> <span class="k">if</span> <span class="n">data_handler</span><span class="o">.</span><span class="n">should_sync</span><span class="p">:</span>
<span class="g g-Whitespace"> </span><span class="mi">1386</span> <span class="n">context</span><span class="o">.</span><span class="n">async_wait</span><span class="p">()</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/util/traceback_utils.py</span> in <span class="ni">error_handler</span><span class="nt">(*args, **kwargs)</span>
<span class="g g-Whitespace"> </span><span class="mi">148</span> <span class="n">filtered_tb</span> <span class="o">=</span> <span class="kc">None</span>
<span class="g g-Whitespace"> </span><span class="mi">149</span> <span class="k">try</span><span class="p">:</span>
<span class="ne">--&gt; </span><span class="mi">150</span> <span class="k">return</span> <span class="n">fn</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">151</span> <span class="k">except</span> <span class="ne">Exception</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
<span class="g g-Whitespace"> </span><span class="mi">152</span> <span class="n">filtered_tb</span> <span class="o">=</span> <span class="n">_process_traceback_frames</span><span class="p">(</span><span class="n">e</span><span class="o">.</span><span class="n">__traceback__</span><span class="p">)</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/def_function.py</span> in <span class="ni">__call__</span><span class="nt">(self, *args, **kwds)</span>
<span class="g g-Whitespace"> </span><span class="mi">913</span>
<span class="g g-Whitespace"> </span><span class="mi">914</span> <span class="k">with</span> <span class="n">OptionalXlaContext</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_jit_compile</span><span class="p">):</span>
<span class="ne">--&gt; </span><span class="mi">915</span> <span class="n">result</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_call</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwds</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">916</span>
<span class="g g-Whitespace"> </span><span class="mi">917</span> <span class="n">new_tracing_count</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">experimental_get_tracing_count</span><span class="p">()</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/def_function.py</span> in <span class="ni">_call</span><span class="nt">(self, *args, **kwds)</span>
<span class="g g-Whitespace"> </span><span class="mi">945</span> <span class="c1"># In this case we have created variables on the first call, so we run the</span>
<span class="g g-Whitespace"> </span><span class="mi">946</span> <span class="c1"># defunned version which is guaranteed to never create variables.</span>
<span class="ne">--&gt; </span><span class="mi">947</span> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_stateless_fn</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">**</span><span class="n">kwds</span><span class="p">)</span> <span class="c1"># pylint: disable=not-callable</span>
<span class="g g-Whitespace"> </span><span class="mi">948</span> <span class="k">elif</span> <span class="bp">self</span><span class="o">.</span><span class="n">_stateful_fn</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="g g-Whitespace"> </span><span class="mi">949</span> <span class="c1"># Release the lock early so that multiple threads can perform the call</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/function.py</span> in <span class="ni">__call__</span><span class="nt">(self, *args, **kwargs)</span>
<span class="g g-Whitespace"> </span><span class="mi">2954</span> <span class="p">(</span><span class="n">graph_function</span><span class="p">,</span>
<span class="g g-Whitespace"> </span><span class="mi">2955</span> <span class="n">filtered_flat_args</span><span class="p">)</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_maybe_define_function</span><span class="p">(</span><span class="n">args</span><span class="p">,</span> <span class="n">kwargs</span><span class="p">)</span>
<span class="ne">-&gt; </span><span class="mi">2956</span> <span class="k">return</span> <span class="n">graph_function</span><span class="o">.</span><span class="n">_call_flat</span><span class="p">(</span>
<span class="g g-Whitespace"> </span><span class="mi">2957</span> <span class="n">filtered_flat_args</span><span class="p">,</span> <span class="n">captured_inputs</span><span class="o">=</span><span class="n">graph_function</span><span class="o">.</span><span class="n">captured_inputs</span><span class="p">)</span> <span class="c1"># pylint: disable=protected-access</span>
<span class="g g-Whitespace"> </span><span class="mi">2958</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/function.py</span> in <span class="ni">_call_flat</span><span class="nt">(self, args, captured_inputs, cancellation_manager)</span>
<span class="g g-Whitespace"> </span><span class="mi">1851</span> <span class="ow">and</span> <span class="n">executing_eagerly</span><span class="p">):</span>
<span class="g g-Whitespace"> </span><span class="mi">1852</span> <span class="c1"># No tape is watching; skip to running the function.</span>
<span class="ne">-&gt; </span><span class="mi">1853</span> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_build_call_outputs</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_inference_function</span><span class="o">.</span><span class="n">call</span><span class="p">(</span>
<span class="g g-Whitespace"> </span><span class="mi">1854</span> <span class="n">ctx</span><span class="p">,</span> <span class="n">args</span><span class="p">,</span> <span class="n">cancellation_manager</span><span class="o">=</span><span class="n">cancellation_manager</span><span class="p">))</span>
<span class="g g-Whitespace"> </span><span class="mi">1855</span> <span class="n">forward_backward</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_select_forward_and_backward_functions</span><span class="p">(</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/function.py</span> in <span class="ni">call</span><span class="nt">(self, ctx, args, cancellation_manager)</span>
<span class="g g-Whitespace"> </span><span class="mi">497</span> <span class="k">with</span> <span class="n">_InterpolateFunctionError</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="g g-Whitespace"> </span><span class="mi">498</span> <span class="k">if</span> <span class="n">cancellation_manager</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="ne">--&gt; </span><span class="mi">499</span> <span class="n">outputs</span> <span class="o">=</span> <span class="n">execute</span><span class="o">.</span><span class="n">execute</span><span class="p">(</span>
<span class="g g-Whitespace"> </span><span class="mi">500</span> <span class="nb">str</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">signature</span><span class="o">.</span><span class="n">name</span><span class="p">),</span>
<span class="g g-Whitespace"> </span><span class="mi">501</span> <span class="n">num_outputs</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">_num_outputs</span><span class="p">,</span>
<span class="nn">~/miniforge3/envs/myenv/lib/python3.9/site-packages/tensorflow/python/eager/execute.py</span> in <span class="ni">quick_execute</span><span class="nt">(op_name, num_outputs, inputs, attrs, ctx, name)</span>
<span class="g g-Whitespace"> </span><span class="mi">52</span> <span class="k">try</span><span class="p">:</span>
<span class="g g-Whitespace"> </span><span class="mi">53</span> <span class="n">ctx</span><span class="o">.</span><span class="n">ensure_initialized</span><span class="p">()</span>
<span class="ne">---&gt; </span><span class="mi">54</span> <span class="n">tensors</span> <span class="o">=</span> <span class="n">pywrap_tfe</span><span class="o">.</span><span class="n">TFE_Py_Execute</span><span class="p">(</span><span class="n">ctx</span><span class="o">.</span><span class="n">_handle</span><span class="p">,</span> <span class="n">device_name</span><span class="p">,</span> <span class="n">op_name</span><span class="p">,</span>
<span class="g g-Whitespace"> </span><span class="mi">55</span> <span class="n">inputs</span><span class="p">,</span> <span class="n">attrs</span><span class="p">,</span> <span class="n">num_outputs</span><span class="p">)</span>
<span class="g g-Whitespace"> </span><span class="mi">56</span> <span class="k">except</span> <span class="n">core</span><span class="o">.</span><span class="n">_NotOkStatusException</span> <span class="k">as</span> <span class="n">e</span><span class="p">:</span>
<span class="ne">KeyboardInterrupt</span>:
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="an-extrapolation-example">
<h2><span class="section-number">17.2. </span>An extrapolation example<a class="headerlink" href="#an-extrapolation-example" title="Permalink to this headline"></a></h2>
<p>The following code provides an example of how recurrent neural
networks can be used to extrapolate to unknown values of physics data
sets. Specifically, the data sets used in this program come from
a quantum mechanical many-body calculation of energies as functions of the number of particles.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># For matrices and calculations</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="c1"># For machine learning (backend for keras)</span>
<span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
<span class="c1"># User-friendly machine learning library</span>
<span class="c1"># Front end for TensorFlow</span>
<span class="kn">import</span> <span class="nn">tensorflow.keras</span>
<span class="c1"># Different methods from Keras needed to create an RNN</span>
<span class="c1"># This is not necessary but it shortened function calls </span>
<span class="c1"># that need to be used in the code.</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">datasets</span><span class="p">,</span> <span class="n">layers</span><span class="p">,</span> <span class="n">models</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Input</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">regularizers</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.models</span> <span class="kn">import</span> <span class="n">Model</span><span class="p">,</span> <span class="n">Sequential</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.layers</span> <span class="kn">import</span> <span class="n">Dense</span><span class="p">,</span> <span class="n">SimpleRNN</span><span class="p">,</span> <span class="n">LSTM</span><span class="p">,</span> <span class="n">GRU</span>
<span class="c1"># For timing the code</span>
<span class="kn">from</span> <span class="nn">timeit</span> <span class="kn">import</span> <span class="n">default_timer</span> <span class="k">as</span> <span class="n">timer</span>
<span class="c1"># For plotting</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="c1"># The data set</span>
<span class="n">datatype</span><span class="o">=</span><span class="s1">&#39;VaryDimension&#39;</span>
<span class="n">X_tot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">42</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">y_tot</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="o">-</span><span class="mf">0.03077640549</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.08336233266</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.1446729567</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2116753732</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2830637392</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.3581341341</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.436462435</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5177783846</span><span class="p">,</span>
<span class="o">-</span><span class="mf">0.6019067271</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.6887363571</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.7782028952</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.8702784034</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.9649652536</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.062292565</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.16231451</span><span class="p">,</span>
<span class="o">-</span><span class="mf">1.265109911</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.370782966</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.479465113</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.591317992</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.70653767</span><span class="p">])</span>
</pre></div>
</div>
</div>
</div>
<p>The way the recurrent neural networks are trained in this program
differs from how machine learning algorithms are usually trained.
Typically a machine learning algorithm is trained by learning the
relationship between the x data and the y data. In this program, the
recurrent neural network will be trained to recognize the relationship
in a sequence of y values. This is type of data formatting is
typically used time series forcasting, but it can also be used in any
extrapolation (time series forecasting is just a specific type of
extrapolation along the time axis). This method of data formatting
does not use the x data and assumes that the y data are evenly spaced.</p>
<p>For a standard machine learning algorithm, the training data has the
form of (x,y) so the machine learning algorithm learns to assiciate a
y value with a given x value. This is useful when the test data has x
values within the same range as the training data. However, for this
application, the x values of the test data are outside of the x values
of the training data and the traditional method of training a machine
learning algorithm does not work as well. For this reason, the
recurrent neural network is trained on sequences of y values of the
form ((y1, y2), y3), so that the network is concerned with learning
the pattern of the y data and not the relation between the x and y
data. As long as the pattern of y data outside of the training region
stays relatively stable compared to what was inside the training
region, this method of training can produce accurate extrapolations to
y values far removed from the training data set.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># FORMAT_DATA</span>
<span class="k">def</span> <span class="nf">format_data</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">length_of_sequence</span> <span class="o">=</span> <span class="mi">2</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Inputs:</span>
<span class="sd"> data(a numpy array): the data that will be the inputs to the recurrent neural</span>
<span class="sd"> network</span>
<span class="sd"> length_of_sequence (an int): the number of elements in one iteration of the</span>
<span class="sd"> sequence patter. For a function approximator use length_of_sequence = 2.</span>
<span class="sd"> Returns:</span>
<span class="sd"> rnn_input (a 3D numpy array): the input data for the recurrent neural network. Its</span>
<span class="sd"> dimensions are length of data - length of sequence, length of sequence, </span>
<span class="sd"> dimnsion of data</span>
<span class="sd"> rnn_output (a numpy array): the training data for the neural network</span>
<span class="sd"> Formats data to be used in a recurrent neural network.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="p">[],</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">data</span><span class="p">)</span><span class="o">-</span><span class="n">length_of_sequence</span><span class="p">):</span>
<span class="c1"># Get the next length_of_sequence elements</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="n">length_of_sequence</span><span class="p">]</span>
<span class="c1"># Get the element that immediately follows that</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="n">length_of_sequence</span><span class="p">]</span>
<span class="c1"># Reshape so that each data point is contained in its own array</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">X</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">Y</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="n">rnn_input</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">rnn_output</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
<span class="k">return</span> <span class="n">rnn_input</span><span class="p">,</span> <span class="n">rnn_output</span>
<span class="c1"># ## Defining the Recurrent Neural Network Using Keras</span>
<span class="c1"># </span>
<span class="c1"># The following method defines a simple recurrent neural network in keras consisting of one input layer, one hidden layer, and one output layer.</span>
<span class="k">def</span> <span class="nf">rnn</span><span class="p">(</span><span class="n">length_of_sequences</span><span class="p">,</span> <span class="n">batch_size</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span> <span class="n">stateful</span> <span class="o">=</span> <span class="kc">False</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Inputs:</span>
<span class="sd"> length_of_sequences (an int): the number of y values in &quot;x data&quot;. This is determined</span>
<span class="sd"> when the data is formatted</span>
<span class="sd"> batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.</span>
<span class="sd"> stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.</span>
<span class="sd"> Returns:</span>
<span class="sd"> model (a Keras model): The recurrent neural network that is built and compiled by this</span>
<span class="sd"> method</span>
<span class="sd"> Builds and compiles a recurrent neural network with one hidden layer and returns the model.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1"># Number of neurons in the input and output layers</span>
<span class="n">in_out_neurons</span> <span class="o">=</span> <span class="mi">1</span>
<span class="c1"># Number of neurons in the hidden layer</span>
<span class="n">hidden_neurons</span> <span class="o">=</span> <span class="mi">200</span>
<span class="c1"># Define the input layer</span>
<span class="n">inp</span> <span class="o">=</span> <span class="n">Input</span><span class="p">(</span><span class="n">batch_shape</span><span class="o">=</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span>
<span class="n">length_of_sequences</span><span class="p">,</span>
<span class="n">in_out_neurons</span><span class="p">))</span>
<span class="c1"># Define the hidden layer as a simple RNN layer with a set number of neurons and add it to </span>
<span class="c1"># the network immediately after the input layer</span>
<span class="n">rnn</span> <span class="o">=</span> <span class="n">SimpleRNN</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="p">,</span>
<span class="n">return_sequences</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">stateful</span> <span class="o">=</span> <span class="n">stateful</span><span class="p">,</span>
<span class="n">name</span><span class="o">=</span><span class="s2">&quot;RNN&quot;</span><span class="p">)(</span><span class="n">inp</span><span class="p">)</span>
<span class="c1"># Define the output layer as a dense neural network layer (standard neural network layer)</span>
<span class="c1">#and add it to the network immediately after the hidden layer.</span>
<span class="n">dens</span> <span class="o">=</span> <span class="n">Dense</span><span class="p">(</span><span class="n">in_out_neurons</span><span class="p">,</span><span class="n">name</span><span class="o">=</span><span class="s2">&quot;dense&quot;</span><span class="p">)(</span><span class="n">rnn</span><span class="p">)</span>
<span class="c1"># Create the machine learning model starting with the input layer and ending with the </span>
<span class="c1"># output layer</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Model</span><span class="p">(</span><span class="n">inputs</span><span class="o">=</span><span class="p">[</span><span class="n">inp</span><span class="p">],</span><span class="n">outputs</span><span class="o">=</span><span class="p">[</span><span class="n">dens</span><span class="p">])</span>
<span class="c1"># Compile the machine learning model using the mean squared error function as the loss </span>
<span class="c1"># function and an Adams optimizer.</span>
<span class="n">model</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s2">&quot;mean_squared_error&quot;</span><span class="p">,</span> <span class="n">optimizer</span><span class="o">=</span><span class="s2">&quot;adam&quot;</span><span class="p">)</span>
<span class="k">return</span> <span class="n">model</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="predicting-new-points-with-a-trained-recurrent-neural-network">
<h2><span class="section-number">17.3. </span>Predicting New Points With A Trained Recurrent Neural Network<a class="headerlink" href="#predicting-new-points-with-a-trained-recurrent-neural-network" title="Permalink to this headline"></a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">test_rnn</span> <span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">y_test</span><span class="p">,</span> <span class="n">plot_min</span><span class="p">,</span> <span class="n">plot_max</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Inputs:</span>
<span class="sd"> x1 (a list or numpy array): The complete x component of the data set</span>
<span class="sd"> y_test (a list or numpy array): The complete y component of the data set</span>
<span class="sd"> plot_min (an int or float): the smallest x value used in the training data</span>
<span class="sd"> plot_max (an int or float): the largest x valye used in the training data</span>
<span class="sd"> Returns:</span>
<span class="sd"> None.</span>
<span class="sd"> Uses a trained recurrent neural network model to predict future points in the </span>
<span class="sd"> series. Computes the MSE of the predicted data set from the true data set, saves</span>
<span class="sd"> the predicted data set to a csv file, and plots the predicted and true data sets w</span>
<span class="sd"> while also displaying the data range used for training.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1"># Add the training data as the first dim points in the predicted data array as these</span>
<span class="c1"># are known values.</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">y_test</span><span class="p">[:</span><span class="n">dim</span><span class="p">]</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
<span class="c1"># Generate the first input to the trained recurrent neural network using the last two </span>
<span class="c1"># points of the training data. Based on how the network was trained this means that it</span>
<span class="c1"># will predict the first point in the data set after the training data. All of the </span>
<span class="c1"># brackets are necessary for Tensorflow.</span>
<span class="n">next_input</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[[</span><span class="n">y_test</span><span class="p">[</span><span class="n">dim</span><span class="o">-</span><span class="mi">2</span><span class="p">]],</span> <span class="p">[</span><span class="n">y_test</span><span class="p">[</span><span class="n">dim</span><span class="o">-</span><span class="mi">1</span><span class="p">]]]])</span>
<span class="c1"># Save the very last point in the training data set. This will be used later.</span>
<span class="n">last</span> <span class="o">=</span> <span class="p">[</span><span class="n">y_test</span><span class="p">[</span><span class="n">dim</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
<span class="c1"># Iterate until the complete data set is created.</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span> <span class="p">(</span><span class="n">dim</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">y_test</span><span class="p">)):</span>
<span class="c1"># Predict the next point in the data set using the previous two points.</span>
<span class="nb">next</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">next_input</span><span class="p">)</span>
<span class="c1"># Append just the number of the predicted data set</span>
<span class="n">y_pred</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">next</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">])</span>
<span class="c1"># Create the input that will be used to predict the next data point in the data set.</span>
<span class="n">next_input</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">last</span><span class="p">,</span> <span class="nb">next</span><span class="p">[</span><span class="mi">0</span><span class="p">]]],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="n">last</span> <span class="o">=</span> <span class="nb">next</span>
<span class="c1"># Print the mean squared error between the known data set and the predicted data set.</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;MSE: &#39;</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">subtract</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">))</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
<span class="c1"># Save the predicted data set as a csv file for later use</span>
<span class="n">name</span> <span class="o">=</span> <span class="n">datatype</span> <span class="o">+</span> <span class="s1">&#39;Predicted&#39;</span><span class="o">+</span><span class="nb">str</span><span class="p">(</span><span class="n">dim</span><span class="p">)</span><span class="o">+</span><span class="s1">&#39;.csv&#39;</span>
<span class="n">np</span><span class="o">.</span><span class="n">savetxt</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">delimiter</span><span class="o">=</span><span class="s1">&#39;,&#39;</span><span class="p">)</span>
<span class="c1"># Plot the known data set and the predicted data set. The red box represents the region that was used</span>
<span class="c1"># for the training data.</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">y_test</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;true&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="s1">&#39;g-.&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s2">&quot;predicted&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="c1"># Created a red region to represent the points used in the training data.</span>
<span class="n">ax</span><span class="o">.</span><span class="n">axvspan</span><span class="p">(</span><span class="n">plot_min</span><span class="p">,</span> <span class="n">plot_max</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1"># Check to make sure the data set is complete</span>
<span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">X_tot</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">y_tot</span><span class="p">)</span>
<span class="c1"># This is the number of points that will be used in as the training data</span>
<span class="n">dim</span><span class="o">=</span><span class="mi">12</span>
<span class="c1"># Separate the training data from the whole data set</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">X_tot</span><span class="p">[:</span><span class="n">dim</span><span class="p">]</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="n">y_tot</span><span class="p">[:</span><span class="n">dim</span><span class="p">]</span>
<span class="c1"># Generate the training data for the RNN, using a sequence of 2</span>
<span class="n">rnn_input</span><span class="p">,</span> <span class="n">rnn_training</span> <span class="o">=</span> <span class="n">format_data</span><span class="p">(</span><span class="n">y_train</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="c1"># Create a recurrent neural network in Keras and produce a summary of the </span>
<span class="c1"># machine learning model</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">rnn</span><span class="p">(</span><span class="n">length_of_sequences</span> <span class="o">=</span> <span class="n">rnn_input</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">model</span><span class="o">.</span><span class="n">summary</span><span class="p">()</span>
<span class="c1"># Start the timer. Want to time training+testing</span>
<span class="n">start</span> <span class="o">=</span> <span class="n">timer</span><span class="p">()</span>
<span class="c1"># Fit the model using the training data genenerated above using 150 training iterations and a 5%</span>
<span class="c1"># validation split. Setting verbose to True prints information about each training iteration.</span>
<span class="n">hist</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">rnn_input</span><span class="p">,</span> <span class="n">rnn_training</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span>
<span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span><span class="n">validation_split</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)</span>
<span class="k">for</span> <span class="n">label</span> <span class="ow">in</span> <span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">,</span><span class="s2">&quot;val_loss&quot;</span><span class="p">]:</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">hist</span><span class="o">.</span><span class="n">history</span><span class="p">[</span><span class="n">label</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="n">label</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;loss&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;epoch&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;The final validation loss: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">hist</span><span class="o">.</span><span class="n">history</span><span class="p">[</span><span class="s2">&quot;val_loss&quot;</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1"># Use the trained neural network to predict more points of the data set</span>
<span class="n">test_rnn</span><span class="p">(</span><span class="n">X_tot</span><span class="p">,</span> <span class="n">y_tot</span><span class="p">,</span> <span class="n">X_tot</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">X_tot</span><span class="p">[</span><span class="n">dim</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="c1"># Stop the timer and calculate the total time needed.</span>
<span class="n">end</span> <span class="o">=</span> <span class="n">timer</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Time: &#39;</span><span class="p">,</span> <span class="n">end</span><span class="o">-</span><span class="n">start</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Changing the size of the recurrent neural network and its parameters
can drastically change the results you get from the model. The below
code takes the simple recurrent neural network from above and adds a
second hidden layer, changes the number of neurons in the hidden
layer, and explicitly declares the activation function of the hidden
layers to be a sigmoid function. The loss function and optimizer can
also be changed but are kept the same as the above network. These
parameters can be tuned to provide the optimal result from the
network. For some ideas on how to improve the performance of a
<a class="reference external" href="https://danijar.com/tips-for-training-recurrent-neural-networks">recurrent neural network</a>.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">rnn_2layers</span><span class="p">(</span><span class="n">length_of_sequences</span><span class="p">,</span> <span class="n">batch_size</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span> <span class="n">stateful</span> <span class="o">=</span> <span class="kc">False</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Inputs:</span>
<span class="sd"> length_of_sequences (an int): the number of y values in &quot;x data&quot;. This is determined</span>
<span class="sd"> when the data is formatted</span>
<span class="sd"> batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.</span>
<span class="sd"> stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.</span>
<span class="sd"> Returns:</span>
<span class="sd"> model (a Keras model): The recurrent neural network that is built and compiled by this</span>
<span class="sd"> method</span>
<span class="sd"> Builds and compiles a recurrent neural network with two hidden layers and returns the model.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1"># Number of neurons in the input and output layers</span>
<span class="n">in_out_neurons</span> <span class="o">=</span> <span class="mi">1</span>
<span class="c1"># Number of neurons in the hidden layer, increased from the first network</span>
<span class="n">hidden_neurons</span> <span class="o">=</span> <span class="mi">500</span>
<span class="c1"># Define the input layer</span>
<span class="n">inp</span> <span class="o">=</span> <span class="n">Input</span><span class="p">(</span><span class="n">batch_shape</span><span class="o">=</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span>
<span class="n">length_of_sequences</span><span class="p">,</span>
<span class="n">in_out_neurons</span><span class="p">))</span>
<span class="c1"># Create two hidden layers instead of one hidden layer. Explicitly set the activation</span>
<span class="c1"># function to be the sigmoid function (the default value is hyperbolic tangent)</span>
<span class="n">rnn1</span> <span class="o">=</span> <span class="n">SimpleRNN</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="p">,</span>
<span class="n">return_sequences</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="c1"># This needs to be True if another hidden layer is to follow</span>
<span class="n">stateful</span> <span class="o">=</span> <span class="n">stateful</span><span class="p">,</span> <span class="n">activation</span> <span class="o">=</span> <span class="s1">&#39;sigmoid&#39;</span><span class="p">,</span>
<span class="n">name</span><span class="o">=</span><span class="s2">&quot;RNN1&quot;</span><span class="p">)(</span><span class="n">inp</span><span class="p">)</span>
<span class="n">rnn2</span> <span class="o">=</span> <span class="n">SimpleRNN</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="p">,</span>
<span class="n">return_sequences</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">activation</span> <span class="o">=</span> <span class="s1">&#39;sigmoid&#39;</span><span class="p">,</span>
<span class="n">stateful</span> <span class="o">=</span> <span class="n">stateful</span><span class="p">,</span>
<span class="n">name</span><span class="o">=</span><span class="s2">&quot;RNN2&quot;</span><span class="p">)(</span><span class="n">rnn1</span><span class="p">)</span>
<span class="c1"># Define the output layer as a dense neural network layer (standard neural network layer)</span>
<span class="c1">#and add it to the network immediately after the hidden layer.</span>
<span class="n">dens</span> <span class="o">=</span> <span class="n">Dense</span><span class="p">(</span><span class="n">in_out_neurons</span><span class="p">,</span><span class="n">name</span><span class="o">=</span><span class="s2">&quot;dense&quot;</span><span class="p">)(</span><span class="n">rnn2</span><span class="p">)</span>
<span class="c1"># Create the machine learning model starting with the input layer and ending with the </span>
<span class="c1"># output layer</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Model</span><span class="p">(</span><span class="n">inputs</span><span class="o">=</span><span class="p">[</span><span class="n">inp</span><span class="p">],</span><span class="n">outputs</span><span class="o">=</span><span class="p">[</span><span class="n">dens</span><span class="p">])</span>
<span class="c1"># Compile the machine learning model using the mean squared error function as the loss </span>
<span class="c1"># function and an Adams optimizer.</span>
<span class="n">model</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s2">&quot;mean_squared_error&quot;</span><span class="p">,</span> <span class="n">optimizer</span><span class="o">=</span><span class="s2">&quot;adam&quot;</span><span class="p">)</span>
<span class="k">return</span> <span class="n">model</span>
<span class="c1"># Check to make sure the data set is complete</span>
<span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">X_tot</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">y_tot</span><span class="p">)</span>
<span class="c1"># This is the number of points that will be used in as the training data</span>
<span class="n">dim</span><span class="o">=</span><span class="mi">12</span>
<span class="c1"># Separate the training data from the whole data set</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">X_tot</span><span class="p">[:</span><span class="n">dim</span><span class="p">]</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="n">y_tot</span><span class="p">[:</span><span class="n">dim</span><span class="p">]</span>
<span class="c1"># Generate the training data for the RNN, using a sequence of 2</span>
<span class="n">rnn_input</span><span class="p">,</span> <span class="n">rnn_training</span> <span class="o">=</span> <span class="n">format_data</span><span class="p">(</span><span class="n">y_train</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="c1"># Create a recurrent neural network in Keras and produce a summary of the </span>
<span class="c1"># machine learning model</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">rnn_2layers</span><span class="p">(</span><span class="n">length_of_sequences</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">summary</span><span class="p">()</span>
<span class="c1"># Start the timer. Want to time training+testing</span>
<span class="n">start</span> <span class="o">=</span> <span class="n">timer</span><span class="p">()</span>
<span class="c1"># Fit the model using the training data genenerated above using 150 training iterations and a 5%</span>
<span class="c1"># validation split. Setting verbose to True prints information about each training iteration.</span>
<span class="n">hist</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">rnn_input</span><span class="p">,</span> <span class="n">rnn_training</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span>
<span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span><span class="n">validation_split</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)</span>
<span class="c1"># This section plots the training loss and the validation loss as a function of training iteration.</span>
<span class="c1"># This is not required for analyzing the couple cluster data but can help determine if the network is</span>
<span class="c1"># being overtrained.</span>
<span class="k">for</span> <span class="n">label</span> <span class="ow">in</span> <span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">,</span><span class="s2">&quot;val_loss&quot;</span><span class="p">]:</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">hist</span><span class="o">.</span><span class="n">history</span><span class="p">[</span><span class="n">label</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="n">label</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;loss&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;epoch&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;The final validation loss: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">hist</span><span class="o">.</span><span class="n">history</span><span class="p">[</span><span class="s2">&quot;val_loss&quot;</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1"># Use the trained neural network to predict more points of the data set</span>
<span class="n">test_rnn</span><span class="p">(</span><span class="n">X_tot</span><span class="p">,</span> <span class="n">y_tot</span><span class="p">,</span> <span class="n">X_tot</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">X_tot</span><span class="p">[</span><span class="n">dim</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="c1"># Stop the timer and calculate the total time needed.</span>
<span class="n">end</span> <span class="o">=</span> <span class="n">timer</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Time: &#39;</span><span class="p">,</span> <span class="n">end</span><span class="o">-</span><span class="n">start</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="other-types-of-recurrent-neural-networks">
<h2><span class="section-number">17.4. </span>Other Types of Recurrent Neural Networks<a class="headerlink" href="#other-types-of-recurrent-neural-networks" title="Permalink to this headline"></a></h2>
<p>Besides a simple recurrent neural network layer, there are two other
commonly used types of recurrent neural network layers: Long Short
Term Memory (LSTM) and Gated Recurrent Unit (GRU). For a short
introduction to these layers see <a class="reference external" href="https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b">https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b</a>
and <a class="reference external" href="https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b">https://medium.com/mindboard/lstm-vs-gru-experimental-comparison-955820c21e8b</a>.</p>
<p>The first network created below is similar to the previous network,
but it replaces the SimpleRNN layers with LSTM layers. The second
network below has two hidden layers made up of GRUs, which are
preceeded by two dense (feeddorward) neural network layers. These
dense layers “preprocess” the data before it reaches the recurrent
layers. This architecture has been shown to improve the performance
of recurrent neural networks (see the link above and also
<a class="reference external" href="https://arxiv.org/pdf/1807.02857.pdf">https://arxiv.org/pdf/1807.02857.pdf</a>.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">lstm_2layers</span><span class="p">(</span><span class="n">length_of_sequences</span><span class="p">,</span> <span class="n">batch_size</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span> <span class="n">stateful</span> <span class="o">=</span> <span class="kc">False</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Inputs:</span>
<span class="sd"> length_of_sequences (an int): the number of y values in &quot;x data&quot;. This is determined</span>
<span class="sd"> when the data is formatted</span>
<span class="sd"> batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.</span>
<span class="sd"> stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.</span>
<span class="sd"> Returns:</span>
<span class="sd"> model (a Keras model): The recurrent neural network that is built and compiled by this</span>
<span class="sd"> method</span>
<span class="sd"> Builds and compiles a recurrent neural network with two LSTM hidden layers and returns the model.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1"># Number of neurons on the input/output layer and the number of neurons in the hidden layer</span>
<span class="n">in_out_neurons</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">hidden_neurons</span> <span class="o">=</span> <span class="mi">250</span>
<span class="c1"># Input Layer</span>
<span class="n">inp</span> <span class="o">=</span> <span class="n">Input</span><span class="p">(</span><span class="n">batch_shape</span><span class="o">=</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span>
<span class="n">length_of_sequences</span><span class="p">,</span>
<span class="n">in_out_neurons</span><span class="p">))</span>
<span class="c1"># Hidden layers (in this case they are LSTM layers instead if SimpleRNN layers)</span>
<span class="n">rnn</span><span class="o">=</span> <span class="n">LSTM</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="p">,</span>
<span class="n">return_sequences</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="n">stateful</span> <span class="o">=</span> <span class="n">stateful</span><span class="p">,</span>
<span class="n">name</span><span class="o">=</span><span class="s2">&quot;RNN&quot;</span><span class="p">,</span> <span class="n">use_bias</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;tanh&#39;</span><span class="p">)(</span><span class="n">inp</span><span class="p">)</span>
<span class="n">rnn1</span> <span class="o">=</span> <span class="n">LSTM</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="p">,</span>
<span class="n">return_sequences</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">stateful</span> <span class="o">=</span> <span class="n">stateful</span><span class="p">,</span>
<span class="n">name</span><span class="o">=</span><span class="s2">&quot;RNN1&quot;</span><span class="p">,</span> <span class="n">use_bias</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;tanh&#39;</span><span class="p">)(</span><span class="n">rnn</span><span class="p">)</span>
<span class="c1"># Output layer</span>
<span class="n">dens</span> <span class="o">=</span> <span class="n">Dense</span><span class="p">(</span><span class="n">in_out_neurons</span><span class="p">,</span><span class="n">name</span><span class="o">=</span><span class="s2">&quot;dense&quot;</span><span class="p">)(</span><span class="n">rnn1</span><span class="p">)</span>
<span class="c1"># Define the midel</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Model</span><span class="p">(</span><span class="n">inputs</span><span class="o">=</span><span class="p">[</span><span class="n">inp</span><span class="p">],</span><span class="n">outputs</span><span class="o">=</span><span class="p">[</span><span class="n">dens</span><span class="p">])</span>
<span class="c1"># Compile the model</span>
<span class="n">model</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s1">&#39;mean_squared_error&#39;</span><span class="p">,</span> <span class="n">optimizer</span><span class="o">=</span><span class="s1">&#39;adam&#39;</span><span class="p">)</span>
<span class="c1"># Return the model</span>
<span class="k">return</span> <span class="n">model</span>
<span class="k">def</span> <span class="nf">dnn2_gru2</span><span class="p">(</span><span class="n">length_of_sequences</span><span class="p">,</span> <span class="n">batch_size</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span> <span class="n">stateful</span> <span class="o">=</span> <span class="kc">False</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Inputs:</span>
<span class="sd"> length_of_sequences (an int): the number of y values in &quot;x data&quot;. This is determined</span>
<span class="sd"> when the data is formatted</span>
<span class="sd"> batch_size (an int): Default value is None. See Keras documentation of SimpleRNN.</span>
<span class="sd"> stateful (a boolean): Default value is False. See Keras documentation of SimpleRNN.</span>
<span class="sd"> Returns:</span>
<span class="sd"> model (a Keras model): The recurrent neural network that is built and compiled by this</span>
<span class="sd"> method</span>
<span class="sd"> Builds and compiles a recurrent neural network with four hidden layers (two dense followed by</span>
<span class="sd"> two GRU layers) and returns the model.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1"># Number of neurons on the input/output layers and hidden layers</span>
<span class="n">in_out_neurons</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">hidden_neurons</span> <span class="o">=</span> <span class="mi">250</span>
<span class="c1"># Input layer</span>
<span class="n">inp</span> <span class="o">=</span> <span class="n">Input</span><span class="p">(</span><span class="n">batch_shape</span><span class="o">=</span><span class="p">(</span><span class="n">batch_size</span><span class="p">,</span>
<span class="n">length_of_sequences</span><span class="p">,</span>
<span class="n">in_out_neurons</span><span class="p">))</span>
<span class="c1"># Hidden Dense (feedforward) layers</span>
<span class="n">dnn</span> <span class="o">=</span> <span class="n">Dense</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;relu&#39;</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;dnn&#39;</span><span class="p">)(</span><span class="n">inp</span><span class="p">)</span>
<span class="n">dnn1</span> <span class="o">=</span> <span class="n">Dense</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span> <span class="n">activation</span><span class="o">=</span><span class="s1">&#39;relu&#39;</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;dnn1&#39;</span><span class="p">)(</span><span class="n">dnn</span><span class="p">)</span>
<span class="c1"># Hidden GRU layers</span>
<span class="n">rnn1</span> <span class="o">=</span> <span class="n">GRU</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="p">,</span>
<span class="n">return_sequences</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="n">stateful</span> <span class="o">=</span> <span class="n">stateful</span><span class="p">,</span>
<span class="n">name</span><span class="o">=</span><span class="s2">&quot;RNN1&quot;</span><span class="p">,</span> <span class="n">use_bias</span><span class="o">=</span><span class="kc">True</span><span class="p">)(</span><span class="n">dnn1</span><span class="p">)</span>
<span class="n">rnn</span> <span class="o">=</span> <span class="n">GRU</span><span class="p">(</span><span class="n">hidden_neurons</span><span class="p">,</span>
<span class="n">return_sequences</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">stateful</span> <span class="o">=</span> <span class="n">stateful</span><span class="p">,</span>
<span class="n">name</span><span class="o">=</span><span class="s2">&quot;RNN&quot;</span><span class="p">,</span> <span class="n">use_bias</span><span class="o">=</span><span class="kc">True</span><span class="p">)(</span><span class="n">rnn1</span><span class="p">)</span>
<span class="c1"># Output layer</span>
<span class="n">dens</span> <span class="o">=</span> <span class="n">Dense</span><span class="p">(</span><span class="n">in_out_neurons</span><span class="p">,</span><span class="n">name</span><span class="o">=</span><span class="s2">&quot;dense&quot;</span><span class="p">)(</span><span class="n">rnn</span><span class="p">)</span>
<span class="c1"># Define the model</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Model</span><span class="p">(</span><span class="n">inputs</span><span class="o">=</span><span class="p">[</span><span class="n">inp</span><span class="p">],</span><span class="n">outputs</span><span class="o">=</span><span class="p">[</span><span class="n">dens</span><span class="p">])</span>
<span class="c1"># Compile the mdoel</span>
<span class="n">model</span><span class="o">.</span><span class="n">compile</span><span class="p">(</span><span class="n">loss</span><span class="o">=</span><span class="s1">&#39;mean_squared_error&#39;</span><span class="p">,</span> <span class="n">optimizer</span><span class="o">=</span><span class="s1">&#39;adam&#39;</span><span class="p">)</span>
<span class="c1"># Return the model</span>
<span class="k">return</span> <span class="n">model</span>
<span class="c1"># Check to make sure the data set is complete</span>
<span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">X_tot</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">y_tot</span><span class="p">)</span>
<span class="c1"># This is the number of points that will be used in as the training data</span>
<span class="n">dim</span><span class="o">=</span><span class="mi">12</span>
<span class="c1"># Separate the training data from the whole data set</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">X_tot</span><span class="p">[:</span><span class="n">dim</span><span class="p">]</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="n">y_tot</span><span class="p">[:</span><span class="n">dim</span><span class="p">]</span>
<span class="c1"># Generate the training data for the RNN, using a sequence of 2</span>
<span class="n">rnn_input</span><span class="p">,</span> <span class="n">rnn_training</span> <span class="o">=</span> <span class="n">format_data</span><span class="p">(</span><span class="n">y_train</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="c1"># Create a recurrent neural network in Keras and produce a summary of the </span>
<span class="c1"># machine learning model</span>
<span class="c1"># Change the method name to reflect which network you want to use</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">dnn2_gru2</span><span class="p">(</span><span class="n">length_of_sequences</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">summary</span><span class="p">()</span>
<span class="c1"># Start the timer. Want to time training+testing</span>
<span class="n">start</span> <span class="o">=</span> <span class="n">timer</span><span class="p">()</span>
<span class="c1"># Fit the model using the training data genenerated above using 150 training iterations and a 5%</span>
<span class="c1"># validation split. Setting verbose to True prints information about each training iteration.</span>
<span class="n">hist</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">rnn_input</span><span class="p">,</span> <span class="n">rnn_training</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span>
<span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span><span class="n">validation_split</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)</span>
<span class="c1"># This section plots the training loss and the validation loss as a function of training iteration.</span>
<span class="c1"># This is not required for analyzing the couple cluster data but can help determine if the network is</span>
<span class="c1"># being overtrained.</span>
<span class="k">for</span> <span class="n">label</span> <span class="ow">in</span> <span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">,</span><span class="s2">&quot;val_loss&quot;</span><span class="p">]:</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">hist</span><span class="o">.</span><span class="n">history</span><span class="p">[</span><span class="n">label</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="n">label</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;loss&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;epoch&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;The final validation loss: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">hist</span><span class="o">.</span><span class="n">history</span><span class="p">[</span><span class="s2">&quot;val_loss&quot;</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1"># Use the trained neural network to predict more points of the data set</span>
<span class="n">test_rnn</span><span class="p">(</span><span class="n">X_tot</span><span class="p">,</span> <span class="n">y_tot</span><span class="p">,</span> <span class="n">X_tot</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">X_tot</span><span class="p">[</span><span class="n">dim</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="c1"># Stop the timer and calculate the total time needed.</span>
<span class="n">end</span> <span class="o">=</span> <span class="n">timer</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Time: &#39;</span><span class="p">,</span> <span class="n">end</span><span class="o">-</span><span class="n">start</span><span class="p">)</span>
<span class="c1"># ### Training Recurrent Neural Networks in the Standard Way (i.e. learning the relationship between the X and Y data)</span>
<span class="c1"># </span>
<span class="c1"># Finally, comparing the performace of a recurrent neural network using the standard data formatting to the performance of the network with time sequence data formatting shows the benefit of this type of data formatting with extrapolation.</span>
<span class="c1"># Check to make sure the data set is complete</span>
<span class="k">assert</span> <span class="nb">len</span><span class="p">(</span><span class="n">X_tot</span><span class="p">)</span> <span class="o">==</span> <span class="nb">len</span><span class="p">(</span><span class="n">y_tot</span><span class="p">)</span>
<span class="c1"># This is the number of points that will be used in as the training data</span>
<span class="n">dim</span><span class="o">=</span><span class="mi">12</span>
<span class="c1"># Separate the training data from the whole data set</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">X_tot</span><span class="p">[:</span><span class="n">dim</span><span class="p">]</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="n">y_tot</span><span class="p">[:</span><span class="n">dim</span><span class="p">]</span>
<span class="c1"># Reshape the data for Keras specifications</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="n">X_train</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">dim</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="n">y_train</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">dim</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="c1"># Create a recurrent neural network in Keras and produce a summary of the </span>
<span class="c1"># machine learning model</span>
<span class="c1"># Set the sequence length to 1 for regular data formatting </span>
<span class="n">model</span> <span class="o">=</span> <span class="n">rnn</span><span class="p">(</span><span class="n">length_of_sequences</span> <span class="o">=</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">summary</span><span class="p">()</span>
<span class="c1"># Start the timer. Want to time training+testing</span>
<span class="n">start</span> <span class="o">=</span> <span class="n">timer</span><span class="p">()</span>
<span class="c1"># Fit the model using the training data genenerated above using 150 training iterations and a 5%</span>
<span class="c1"># validation split. Setting verbose to True prints information about each training iteration.</span>
<span class="n">hist</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">batch_size</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">epochs</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span>
<span class="n">verbose</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span><span class="n">validation_split</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)</span>
<span class="c1"># This section plots the training loss and the validation loss as a function of training iteration.</span>
<span class="c1"># This is not required for analyzing the couple cluster data but can help determine if the network is</span>
<span class="c1"># being overtrained.</span>
<span class="k">for</span> <span class="n">label</span> <span class="ow">in</span> <span class="p">[</span><span class="s2">&quot;loss&quot;</span><span class="p">,</span><span class="s2">&quot;val_loss&quot;</span><span class="p">]:</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">hist</span><span class="o">.</span><span class="n">history</span><span class="p">[</span><span class="n">label</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="n">label</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;loss&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;epoch&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;The final validation loss: </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">hist</span><span class="o">.</span><span class="n">history</span><span class="p">[</span><span class="s2">&quot;val_loss&quot;</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1"># Use the trained neural network to predict the remaining data points</span>
<span class="n">X_pred</span> <span class="o">=</span> <span class="n">X_tot</span><span class="p">[</span><span class="n">dim</span><span class="p">:]</span>
<span class="n">X_pred</span> <span class="o">=</span> <span class="n">X_pred</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">X_pred</span><span class="p">),</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">y_model</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_pred</span><span class="p">)</span>
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">y_tot</span><span class="p">[:</span><span class="n">dim</span><span class="p">],</span> <span class="n">y_model</span><span class="o">.</span><span class="n">flatten</span><span class="p">()))</span>
<span class="c1"># Plot the known data set and the predicted data set. The red box represents the region that was used</span>
<span class="c1"># for the training data.</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_tot</span><span class="p">,</span> <span class="n">y_tot</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;true&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_tot</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="s1">&#39;g-.&#39;</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s2">&quot;predicted&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="c1"># Created a red region to represent the points used in the training data.</span>
<span class="n">ax</span><span class="o">.</span><span class="n">axvspan</span><span class="p">(</span><span class="n">X_tot</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">X_tot</span><span class="p">[</span><span class="n">dim</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1"># Stop the timer and calculate the total time needed.</span>
<span class="n">end</span> <span class="o">=</span> <span class="n">timer</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Time: &#39;</span><span class="p">,</span> <span class="n">end</span><span class="o">-</span><span class="n">start</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
</div>
</div>
<div class="tex2jax_ignore mathjax_ignore section" id="generative-models">
<h1><span class="section-number">18. </span>Generative Models<a class="headerlink" href="#generative-models" title="Permalink to this headline"></a></h1>
<p><strong>Generative models</strong> describe a class of statistical models that are a contrast
to <strong>discriminative models</strong>. Informally we say that generative models can
generate new data instances while discriminative models discriminate between
different kinds of data instances. A generative model could generate new photos
of animals that look like ‘real’ animals while a discriminative model could tell
a dog from a cat. More formally, given a data set <span class="math notranslate nohighlight">\(x\)</span> and a set of labels /
targets <span class="math notranslate nohighlight">\(y\)</span>. Generative models capture the joint probability <span class="math notranslate nohighlight">\(p(x, y)\)</span>, or
just <span class="math notranslate nohighlight">\(p(x)\)</span> if there are no labels, while discriminative models capture the
conditional probability <span class="math notranslate nohighlight">\(p(y | x)\)</span>. Discriminative models generally try to draw
boundaries in the data space (often high dimensional), while generative models
try to model how data is placed throughout the space.</p>
<p><strong>Note</strong>: this material is thanks to Linus Ekstrøm.</p>
<div class="section" id="generative-adversarial-networks">
<h2><span class="section-number">18.1. </span>Generative Adversarial Networks<a class="headerlink" href="#generative-adversarial-networks" title="Permalink to this headline"></a></h2>
<p><strong>Generative Adversarial Networks</strong> are a type of unsupervised machine learning
algorithm proposed by <a class="reference external" href="https://arxiv.org/pdf/1406.2661.pdf">Goodfellow et. al</a>
in 2014 (short and good article).</p>
<p>The simplest formulation of
the model is based on a game theoretic approach, <em>zero sum game</em>, where we pit
two neural networks against one another. We define two rival networks, one
generator <span class="math notranslate nohighlight">\(g\)</span>, and one discriminator <span class="math notranslate nohighlight">\(d\)</span>. The generator directly produces
samples</p>
<!-- Equation labels as ordinary links -->
<div id="_auto1"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
x = g(z; \theta^{(g)})
\label{_auto1} \tag{1}
\end{equation}
\]</div>
<p>The discriminator attempts to distinguish between samples drawn from the
training data and samples drawn from the generator. In other words, it tries to
tell the difference between the fake data produced by <span class="math notranslate nohighlight">\(g\)</span> and the actual data
samples we want to do prediction on. The discriminator outputs a probability
value given by</p>
<!-- Equation labels as ordinary links -->
<div id="_auto2"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
d(x; \theta^{(d)})
\label{_auto2} \tag{2}
\end{equation}
\]</div>
<p>indicating the probability that <span class="math notranslate nohighlight">\(x\)</span> is a real training example rather than a
fake sample the generator has generated. The simplest way to formulate the
learning process in a generative adversarial network is a zero-sum game, in
which a function</p>
<!-- Equation labels as ordinary links -->
<div id="_auto3"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
v(\theta^{(g)}, \theta^{(d)})
\label{_auto3} \tag{3}
\end{equation}
\]</div>
<p>determines the reward for the discriminator, while the generator gets the
conjugate reward</p>
<!-- Equation labels as ordinary links -->
<div id="_auto4"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
-v(\theta^{(g)}, \theta^{(d)})
\label{_auto4} \tag{4}
\end{equation}
\]</div>
<p>During learning both of the networks maximize their own reward function, so that
the generator gets better and better at tricking the discriminator, while the
discriminator gets better and better at telling the difference between the fake
and real data. The generator and discriminator alternate on which one trains at
one time (i.e. for one epoch). In other words, we keep the generator constant
and train the discriminator, then we keep the discriminator constant to train
the generator and repeat. It is this back and forth dynamic which lets GANs
tackle otherwise intractable generative problems. As the generator improves with
training, the discriminator’s performance gets worse because it cannot easily
tell the difference between real and fake. If the generator ends up succeeding
perfectly, the the discriminator will do no better than random guessing i.e.
50%. This progression in the training poses a problem for the convergence
criteria for GANs. The discriminator feedback gets less meaningful over time,
if we continue training after this point then the generator is effectively
training on junk data which can undo the learning up to that point. Therefore,
we stop training when the discriminator starts outputting <span class="math notranslate nohighlight">\(1/2\)</span> everywhere.</p>
<p>At convergence we have</p>
<!-- Equation labels as ordinary links -->
<div id="_auto5"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
g^* = \underset{g}{\mathrm{argmin}}\hspace{2pt}
\underset{d}{\mathrm{max}}v(\theta^{(g)}, \theta^{(d)})
\label{_auto5} \tag{5}
\end{equation}
\]</div>
<p>The default choice for <span class="math notranslate nohighlight">\(v\)</span> is</p>
<!-- Equation labels as ordinary links -->
<div id="_auto6"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
v(\theta^{(g)}, \theta^{(d)}) = \mathbb{E}_{x\sim p_\mathrm{data}}\log d(x)
+ \mathbb{E}_{x\sim p_\mathrm{model}}
\log (1 - d(x))
\label{_auto6} \tag{6}
\end{equation}
\]</div>
<p>The main motivation for the design of GANs is that the learning process requires
neither approximate inference (variational autoencoders for example) nor
approximation of a partition function. In the case where</p>
<!-- Equation labels as ordinary links -->
<div id="_auto7"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}
\underset{d}{\mathrm{max}}v(\theta^{(g)}, \theta^{(d)})
\label{_auto7} \tag{7}
\end{equation}
\]</div>
<p>is convex in $\theta^{(g)} then the procedure is guaranteed to converge and is
asymptotically consistent
( <a class="reference external" href="https://arxiv.org/pdf/1804.09139.pdf">Seth Lloyd on QuGANs</a> ).</p>
<p>This is in
general not the case and it is possible to get situations where the training
process never converges because the generator and discriminator chase one
another around in the parameter space indefinitely. A much deeper discussion on
the currently open research problem of GAN convergence is available
<a class="reference external" href="https://www.deeplearningbook.org/contents/generative_models.html">here</a>. To
anyone interested in learning more about GANs it is a highly recommended read.
Direct quote: “In this best-performing formulation, the generator aims to
increase the log probability that the discriminator makes a mistake, rather than
aiming to decrease the log probability that the discriminator makes the correct
prediction.” <a class="reference external" href="https://arxiv.org/abs/1701.00160">Another interesting read</a></p>
</div>
<div class="section" id="writing-our-first-generative-adversarial-network">
<h2><span class="section-number">18.2. </span>Writing Our First Generative Adversarial Network<a class="headerlink" href="#writing-our-first-generative-adversarial-network" title="Permalink to this headline"></a></h2>
<p>Let us now move on to actually implementing a GAN in tensorflow. We will study
the performance of our GAN on the MNIST dataset. This code is based on and
adapted from the
<a class="reference external" href="https://www.tensorflow.org/tutorials/generative/dcgan">google tutorial</a></p>
<p>First we import our libraries</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
<span class="kn">import</span> <span class="nn">time</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras</span> <span class="kn">import</span> <span class="n">layers</span>
<span class="kn">from</span> <span class="nn">tensorflow.keras.utils</span> <span class="kn">import</span> <span class="n">plot_model</span>
</pre></div>
</div>
</div>
</div>
<p>Next we define our hyperparameters and import our data the usual way</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">BUFFER_SIZE</span> <span class="o">=</span> <span class="mi">60000</span>
<span class="n">BATCH_SIZE</span> <span class="o">=</span> <span class="mi">256</span>
<span class="n">EPOCHS</span> <span class="o">=</span> <span class="mi">30</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">keras</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">mnist</span><span class="o">.</span><span class="n">load_data</span><span class="p">()</span>
<span class="p">(</span><span class="n">train_images</span><span class="p">,</span> <span class="n">train_labels</span><span class="p">),</span> <span class="p">(</span><span class="n">test_images</span><span class="p">,</span> <span class="n">test_labels</span><span class="p">)</span> <span class="o">=</span> <span class="n">data</span>
<span class="n">train_images</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">train_images</span><span class="p">,</span> <span class="p">(</span><span class="n">train_images</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span>
<span class="mi">28</span><span class="p">,</span>
<span class="mi">28</span><span class="p">,</span>
<span class="mi">1</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="s1">&#39;float32&#39;</span><span class="p">)</span>
<span class="c1"># we normalize between -1 and 1</span>
<span class="n">train_images</span> <span class="o">=</span> <span class="p">(</span><span class="n">train_images</span> <span class="o">-</span> <span class="mf">127.5</span><span class="p">)</span> <span class="o">/</span> <span class="mf">127.5</span>
<span class="n">training_dataset</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">Dataset</span><span class="o">.</span><span class="n">from_tensor_slices</span><span class="p">(</span>
<span class="n">train_images</span><span class="p">)</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">BUFFER_SIZE</span><span class="p">)</span><span class="o">.</span><span class="n">batch</span><span class="p">(</span><span class="n">BATCH_SIZE</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="section" id="mnist-and-gans">
<h3><span class="section-number">18.2.1. </span>MNIST and GANs<a class="headerlink" href="#mnist-and-gans" title="Permalink to this headline"></a></h3>
<p>Let’s have a quick look</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">train_images</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Greys&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<p>Now we define our two models. This is where the ‘magic’ happens. There are a
huge amount of possible formulations for both models. A lot of engineering and
trial and error can be done here to try to produce better performing models. For
more advanced GANs this is by far the step where you can ‘make or break’ a
model.</p>
<p>We start with the generator. As stated in the introductory text the generator
<span class="math notranslate nohighlight">\(g\)</span> upsamples from a random sample to the shape of what we want to predict. In
our case we are trying to predict MNIST images (<span class="math notranslate nohighlight">\(28\times 28\)</span> pixels).</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">generator_model</span><span class="p">():</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> The generator uses upsampling layers tf.keras.layers.Conv2DTranspose() to</span>
<span class="sd"> produce an image from a random seed. We start with a Dense layer taking this</span>
<span class="sd"> random sample as an input and subsequently upsample through multiple</span>
<span class="sd"> convolutional layers.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1"># we define our model</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">keras</span><span class="o">.</span><span class="n">Sequential</span><span class="p">()</span>
<span class="c1"># adding our input layer. Dense means that every neuron is connected and</span>
<span class="c1"># the input shape is the shape of our random noise. The units need to match</span>
<span class="c1"># in some sense the upsampling strides to reach our desired output shape.</span>
<span class="c1"># we are using 100 random numbers as our seed</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Dense</span><span class="p">(</span><span class="n">units</span><span class="o">=</span><span class="mi">7</span><span class="o">*</span><span class="mi">7</span><span class="o">*</span><span class="n">BATCH_SIZE</span><span class="p">,</span>
<span class="n">use_bias</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">input_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="p">)))</span>
<span class="c1"># we normalize the output form the Dense layer</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">BatchNormalization</span><span class="p">())</span>
<span class="c1"># and add an activation function to our &#39;layer&#39;. LeakyReLU avoids vanishing</span>
<span class="c1"># gradient problem</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">LeakyReLU</span><span class="p">())</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Reshape</span><span class="p">((</span><span class="mi">7</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="n">BATCH_SIZE</span><span class="p">)))</span>
<span class="k">assert</span> <span class="n">model</span><span class="o">.</span><span class="n">output_shape</span> <span class="o">==</span> <span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="n">BATCH_SIZE</span><span class="p">)</span>
<span class="c1"># even though we just added four keras layers we think of everything above</span>
<span class="c1"># as &#39;one&#39; layer</span>
<span class="c1"># next we add our upscaling convolutional layers</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Conv2DTranspose</span><span class="p">(</span><span class="n">filters</span><span class="o">=</span><span class="mi">128</span><span class="p">,</span>
<span class="n">kernel_size</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span>
<span class="n">strides</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span>
<span class="n">padding</span><span class="o">=</span><span class="s1">&#39;same&#39;</span><span class="p">,</span>
<span class="n">use_bias</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">BatchNormalization</span><span class="p">())</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">LeakyReLU</span><span class="p">())</span>
<span class="k">assert</span> <span class="n">model</span><span class="o">.</span><span class="n">output_shape</span> <span class="o">==</span> <span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">128</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Conv2DTranspose</span><span class="p">(</span><span class="n">filters</span><span class="o">=</span><span class="mi">64</span><span class="p">,</span>
<span class="n">kernel_size</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span>
<span class="n">strides</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
<span class="n">padding</span><span class="o">=</span><span class="s1">&#39;same&#39;</span><span class="p">,</span>
<span class="n">use_bias</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">BatchNormalization</span><span class="p">())</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">LeakyReLU</span><span class="p">())</span>
<span class="k">assert</span> <span class="n">model</span><span class="o">.</span><span class="n">output_shape</span> <span class="o">==</span> <span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">14</span><span class="p">,</span> <span class="mi">64</span><span class="p">)</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Conv2DTranspose</span><span class="p">(</span><span class="n">filters</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="n">kernel_size</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span>
<span class="n">strides</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
<span class="n">padding</span><span class="o">=</span><span class="s1">&#39;same&#39;</span><span class="p">,</span>
<span class="n">use_bias</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">activation</span><span class="o">=</span><span class="s1">&#39;tanh&#39;</span><span class="p">))</span>
<span class="k">assert</span> <span class="n">model</span><span class="o">.</span><span class="n">output_shape</span> <span class="o">==</span> <span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="mi">28</span><span class="p">,</span> <span class="mi">28</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">model</span>
</pre></div>
</div>
</div>
</div>
<p>And there we have our ‘simple’ generator model. Now we move on to defining our
discriminator model <span class="math notranslate nohighlight">\(d\)</span>, which is a convolutional neural network based image
classifier.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">discriminator_model</span><span class="p">():</span>
<span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> The discriminator is a convolutional neural network based image classifier</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1"># we define our model</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">keras</span><span class="o">.</span><span class="n">Sequential</span><span class="p">()</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Conv2D</span><span class="p">(</span><span class="n">filters</span><span class="o">=</span><span class="mi">64</span><span class="p">,</span>
<span class="n">kernel_size</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span>
<span class="n">strides</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
<span class="n">padding</span><span class="o">=</span><span class="s1">&#39;same&#39;</span><span class="p">,</span>
<span class="n">input_shape</span><span class="o">=</span><span class="p">[</span><span class="mi">28</span><span class="p">,</span> <span class="mi">28</span><span class="p">,</span> <span class="mi">1</span><span class="p">]))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">LeakyReLU</span><span class="p">())</span>
<span class="c1"># adding a dropout layer as you do in conv-nets</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Dropout</span><span class="p">(</span><span class="mf">0.3</span><span class="p">))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Conv2D</span><span class="p">(</span><span class="n">filters</span><span class="o">=</span><span class="mi">128</span><span class="p">,</span>
<span class="n">kernel_size</span><span class="o">=</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span>
<span class="n">strides</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
<span class="n">padding</span><span class="o">=</span><span class="s1">&#39;same&#39;</span><span class="p">))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">LeakyReLU</span><span class="p">())</span>
<span class="c1"># adding a dropout layer as you do in conv-nets</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Dropout</span><span class="p">(</span><span class="mf">0.3</span><span class="p">))</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Flatten</span><span class="p">())</span>
<span class="n">model</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">layers</span><span class="o">.</span><span class="n">Dense</span><span class="p">(</span><span class="mi">1</span><span class="p">))</span>
<span class="k">return</span> <span class="n">model</span>
</pre></div>
</div>
</div>
</div>
<p>Let us take a look at our models.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">generator</span> <span class="o">=</span> <span class="n">generator_model</span><span class="p">()</span>
<span class="n">plot_model</span><span class="p">(</span><span class="n">generator</span><span class="p">,</span> <span class="n">show_shapes</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">rankdir</span><span class="o">=</span><span class="s1">&#39;LR&#39;</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">discriminator</span> <span class="o">=</span> <span class="n">discriminator_model</span><span class="p">()</span>
<span class="n">plot_model</span><span class="p">(</span><span class="n">discriminator</span><span class="p">,</span> <span class="n">show_shapes</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">rankdir</span><span class="o">=</span><span class="s1">&#39;LR&#39;</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Next we need a few helper objects we will use in training</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">cross_entropy</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">keras</span><span class="o">.</span><span class="n">losses</span><span class="o">.</span><span class="n">BinaryCrossentropy</span><span class="p">(</span><span class="n">from_logits</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">generator_optimizer</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">keras</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="mf">1e-4</span><span class="p">)</span>
<span class="n">discriminator_optimizer</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">keras</span><span class="o">.</span><span class="n">optimizers</span><span class="o">.</span><span class="n">Adam</span><span class="p">(</span><span class="mf">1e-4</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>The first object, <em>cross_entropy</em> is our loss function and the two others are
our optimizers. Notice we use the same learning rate for both <span class="math notranslate nohighlight">\(g\)</span> and <span class="math notranslate nohighlight">\(d\)</span>. This
is because they need to improve their accuracy at approximately equal speeds to
get convergence (not necessarily exactly equal). Now we define our loss
functions</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">generator_loss</span><span class="p">(</span><span class="n">fake_output</span><span class="p">):</span>
<span class="n">loss</span> <span class="o">=</span> <span class="n">cross_entropy</span><span class="p">(</span><span class="n">tf</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">fake_output</span><span class="p">),</span> <span class="n">fake_output</span><span class="p">)</span>
<span class="k">return</span> <span class="n">loss</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">discriminator_loss</span><span class="p">(</span><span class="n">real_output</span><span class="p">,</span> <span class="n">fake_output</span><span class="p">):</span>
<span class="n">real_loss</span> <span class="o">=</span> <span class="n">cross_entropy</span><span class="p">(</span><span class="n">tf</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">real_output</span><span class="p">),</span> <span class="n">real_output</span><span class="p">)</span>
<span class="n">fake_loss</span> <span class="o">=</span> <span class="n">cross_entropy</span><span class="p">(</span><span class="n">tf</span><span class="o">.</span><span class="n">zeros_liks</span><span class="p">(</span><span class="n">fake_output</span><span class="p">),</span> <span class="n">fake_output</span><span class="p">)</span>
<span class="n">total_loss</span> <span class="o">=</span> <span class="n">real_loss</span> <span class="o">+</span> <span class="n">fake_loss</span>
<span class="k">return</span> <span class="n">total_loss</span>
</pre></div>
</div>
</div>
</div>
<p>Next we define a kind of seed to help us compare the learning process over
multiple training epochs.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">noise_dimension</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">n_examples_to_generate</span> <span class="o">=</span> <span class="mi">16</span>
<span class="n">seed_images</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">([</span><span class="n">n_examples_to_generate</span><span class="p">,</span> <span class="n">noise_dimension</span><span class="p">])</span>
</pre></div>
</div>
</div>
</div>
<p>Now we have everything we need to define our training step, which we will apply
for every step in our training loop. Notice the &#64;tf.function flag signifying
that the function is tensorflow ‘compiled’. Removing this flag doubles the
computation time.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nd">@tf</span><span class="o">.</span><span class="n">function</span>
<span class="k">def</span> <span class="nf">train_step</span><span class="p">(</span><span class="n">images</span><span class="p">):</span>
<span class="n">noise</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">([</span><span class="n">BATCH_SIZE</span><span class="p">,</span> <span class="n">noise_dimension</span><span class="p">])</span>
<span class="k">with</span> <span class="n">tf</span><span class="o">.</span><span class="n">GradientTape</span><span class="p">()</span> <span class="k">as</span> <span class="n">gen_tape</span><span class="p">,</span> <span class="n">tf</span><span class="o">.</span><span class="n">GradientTape</span><span class="p">()</span> <span class="k">as</span> <span class="n">disc_tape</span><span class="p">:</span>
<span class="n">generated_images</span> <span class="o">=</span> <span class="n">generator</span><span class="p">(</span><span class="n">noise</span><span class="p">,</span> <span class="n">training</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">real_output</span> <span class="o">=</span> <span class="n">discriminator</span><span class="p">(</span><span class="n">images</span><span class="p">,</span> <span class="n">training</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">fake_output</span> <span class="o">=</span> <span class="n">discriminator</span><span class="p">(</span><span class="n">generated_images</span><span class="p">,</span> <span class="n">training</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">gen_loss</span> <span class="o">=</span> <span class="n">generator_loss</span><span class="p">(</span><span class="n">fake_output</span><span class="p">)</span>
<span class="n">disc_loss</span> <span class="o">=</span> <span class="n">discriminator_loss</span><span class="p">(</span><span class="n">real_output</span><span class="p">,</span> <span class="n">fake_output</span><span class="p">)</span>
<span class="n">gradients_of_generator</span> <span class="o">=</span> <span class="n">gen_tape</span><span class="o">.</span><span class="n">gradient</span><span class="p">(</span><span class="n">gen_loss</span><span class="p">,</span>
<span class="n">generator</span><span class="o">.</span><span class="n">trainable_variables</span><span class="p">)</span>
<span class="n">gradients_of_discriminator</span> <span class="o">=</span> <span class="n">disc_tape</span><span class="o">.</span><span class="n">gradient</span><span class="p">(</span><span class="n">disc_loss</span><span class="p">,</span>
<span class="n">discriminator</span><span class="o">.</span><span class="n">trainable_variables</span><span class="p">)</span>
<span class="n">generator_optimizer</span><span class="o">.</span><span class="n">apply_gradients</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">gradients_of_generator</span><span class="p">,</span>
<span class="n">generator</span><span class="o">.</span><span class="n">trainable_variables</span><span class="p">))</span>
<span class="n">discriminator_optimizer</span><span class="o">.</span><span class="n">apply_gradients</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">gradients_of_discriminator</span><span class="p">,</span>
<span class="n">discriminator</span><span class="o">.</span><span class="n">trainable_variables</span><span class="p">))</span>
<span class="k">return</span> <span class="n">gen_loss</span><span class="p">,</span> <span class="n">disc_loss</span>
</pre></div>
</div>
</div>
</div>
<p>Next we define a helper function to produce an output over our training epochs
to see the predictive progression of our generator model. <strong>Note</strong>: I am including
this code here, but comment it out in the training loop.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">generate_and_save_images</span><span class="p">(</span><span class="n">model</span><span class="p">,</span> <span class="n">epoch</span><span class="p">,</span> <span class="n">test_input</span><span class="p">):</span>
<span class="c1"># we&#39;re making inferences here</span>
<span class="n">predictions</span> <span class="o">=</span> <span class="n">model</span><span class="p">(</span><span class="n">test_input</span><span class="p">,</span> <span class="n">training</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">predictions</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">predictions</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="mf">127.5</span> <span class="o">+</span> <span class="mf">127.5</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;gray&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;off&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;./images_from_seed_images/image_at_epoch_</span><span class="si">{</span><span class="nb">str</span><span class="p">(</span><span class="n">epoch</span><span class="p">)</span><span class="o">.</span><span class="n">zfill</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="si">}</span><span class="s1">.png&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
<span class="c1">#plt.show()</span>
</pre></div>
</div>
</div>
</div>
<p>Setting up checkpoints to periodically save our model during training so that
everything is not lost even if the program were to somehow terminate while
training.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Setting up checkpoints to save model during training</span>
<span class="n">checkpoint_dir</span> <span class="o">=</span> <span class="s1">&#39;./training_checkpoints&#39;</span>
<span class="n">checkpoint_prefix</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">checkpoint_dir</span><span class="p">,</span> <span class="s1">&#39;ckpt&#39;</span><span class="p">)</span>
<span class="n">checkpoint</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">train</span><span class="o">.</span><span class="n">Checkpoint</span><span class="p">(</span><span class="n">generator_optimizer</span><span class="o">=</span><span class="n">generator_optimizer</span><span class="p">,</span>
<span class="n">discriminator_optimizer</span><span class="o">=</span><span class="n">discriminator_optimizer</span><span class="p">,</span>
<span class="n">generator</span><span class="o">=</span><span class="n">generator</span><span class="p">,</span>
<span class="n">discriminator</span><span class="o">=</span><span class="n">discriminator</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Now we define our training loop</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">train</span><span class="p">(</span><span class="n">dataset</span><span class="p">,</span> <span class="n">epochs</span><span class="p">):</span>
<span class="n">generator_loss_list</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">discriminator_loss_list</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">epochs</span><span class="p">):</span>
<span class="n">start</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="k">for</span> <span class="n">image_batch</span> <span class="ow">in</span> <span class="n">dataset</span><span class="p">:</span>
<span class="n">gen_loss</span><span class="p">,</span> <span class="n">disc_loss</span> <span class="o">=</span> <span class="n">train_step</span><span class="p">(</span><span class="n">image_batch</span><span class="p">)</span>
<span class="n">generator_loss_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">gen_loss</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
<span class="n">discriminator_loss_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">disc_loss</span><span class="o">.</span><span class="n">numpy</span><span class="p">())</span>
<span class="c1">#generate_and_save_images(generator, epoch + 1, seed_images)</span>
<span class="k">if</span> <span class="p">(</span><span class="n">epoch</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">15</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">checkpoint</span><span class="o">.</span><span class="n">save</span><span class="p">(</span><span class="n">file_prefix</span><span class="o">=</span><span class="n">checkpoint_prefix</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Time for epoch </span><span class="si">{</span><span class="n">epoch</span><span class="si">}</span><span class="s1"> is </span><span class="si">{</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span> <span class="o">-</span> <span class="n">start</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
<span class="c1">#generate_and_save_images(generator, epochs, seed_images)</span>
<span class="n">loss_file</span> <span class="o">=</span> <span class="s1">&#39;./data/lossfile.txt&#39;</span>
<span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="n">loss_file</span><span class="p">,</span> <span class="s1">&#39;w&#39;</span><span class="p">)</span> <span class="k">as</span> <span class="n">outfile</span><span class="p">:</span>
<span class="n">outfile</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">generator_loss_list</span><span class="p">))</span>
<span class="n">outfile</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span><span class="p">)</span>
<span class="n">outfile</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span><span class="p">)</span>
<span class="n">outfile</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="nb">str</span><span class="p">(</span><span class="n">discriminator_loss_list</span><span class="p">))</span>
<span class="n">outfile</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span><span class="p">)</span>
<span class="n">outfile</span><span class="o">.</span><span class="n">write</span><span class="p">(</span><span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>To train simply call this function. <strong>Warning</strong>: this might take a long time so
there is a folder of a pretrained network already included in the repository.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">train</span><span class="p">(</span><span class="n">train_dataset</span><span class="p">,</span> <span class="n">EPOCHS</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Now to avoid having to train and everything, which will take a while depending
on your computer setup we now load in the model which produced the above gif.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">checkpoint</span><span class="o">.</span><span class="n">restore</span><span class="p">(</span><span class="n">tf</span><span class="o">.</span><span class="n">train</span><span class="o">.</span><span class="n">latest_checkpoint</span><span class="p">(</span><span class="n">checkpoint_dir</span><span class="p">))</span>
<span class="n">restored_generator</span> <span class="o">=</span> <span class="n">checkpoint</span><span class="o">.</span><span class="n">generator</span>
<span class="n">restored_discriminator</span> <span class="o">=</span> <span class="n">checkpoint</span><span class="o">.</span><span class="n">discriminator</span>
<span class="nb">print</span><span class="p">(</span><span class="n">restored_generator</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">restored_discriminator</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>We have successfully loaded in our latest model. Let us now play around a bit
and see what kind of things we can learn about this model. Our generator takes
an array of 100 numbers. One idea can be to try to systematically change our
input. Let us try and see what we get</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">generate_latent_points</span><span class="p">(</span><span class="n">number</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">scale_means</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">scale_stds</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
<span class="n">latent_dim</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">means</span> <span class="o">=</span> <span class="n">scale_means</span> <span class="o">*</span> <span class="n">tf</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">latent_dim</span><span class="p">)</span>
<span class="n">stds</span> <span class="o">=</span> <span class="n">scale_stds</span> <span class="o">*</span> <span class="n">tf</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">latent_dim</span><span class="p">)</span>
<span class="n">latent_space_value_range</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">([</span><span class="n">number</span><span class="p">,</span> <span class="n">latent_dim</span><span class="p">],</span>
<span class="n">means</span><span class="p">,</span>
<span class="n">stds</span><span class="p">,</span>
<span class="n">dtype</span><span class="o">=</span><span class="n">tf</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="k">return</span> <span class="n">latent_space_value_range</span>
<span class="k">def</span> <span class="nf">generate_images</span><span class="p">(</span><span class="n">latent_points</span><span class="p">):</span>
<span class="c1"># notice we set training to false because we are making inferences</span>
<span class="n">generated_images</span> <span class="o">=</span> <span class="n">restored_generator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">latent_points</span><span class="p">)</span>
<span class="k">return</span> <span class="n">generated_images</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_result</span><span class="p">(</span><span class="n">generated_images</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="mi">100</span><span class="p">):</span>
<span class="c1"># obviously this assumes sqrt number is an int</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axs</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">number</span><span class="p">)),</span> <span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">number</span><span class="p">)),</span>
<span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">number</span><span class="p">))):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">number</span><span class="p">))):</span>
<span class="n">axs</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">generated_images</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="n">j</span><span class="p">],</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;Greys&#39;</span><span class="p">)</span>
<span class="n">axs</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;off&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">generated_images</span> <span class="o">=</span> <span class="n">generate_images</span><span class="p">(</span><span class="n">generate_latent_points</span><span class="p">())</span>
<span class="n">plot_result</span><span class="p">(</span><span class="n">generated_images</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>We see that the generator generates images that look like MNIST
numbers: <span class="math notranslate nohighlight">\(1, 4, 7, 9\)</span>. Let’s try to tweak it a bit more to see if we are able
to generate a similar plot where we generate every MNIST number. Let us now try
to ‘move’ a bit around in the latent space. <strong>Note</strong>: decrease the plot number if
these following cells take too long to run on your computer.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plot_number</span> <span class="o">=</span> <span class="mi">225</span>
<span class="n">generated_images</span> <span class="o">=</span> <span class="n">generate_images</span><span class="p">(</span><span class="n">generate_latent_points</span><span class="p">(</span><span class="n">number</span><span class="o">=</span><span class="n">plot_number</span><span class="p">,</span>
<span class="n">scale_means</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
<span class="n">scale_stds</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<span class="n">plot_result</span><span class="p">(</span><span class="n">generated_images</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="n">plot_number</span><span class="p">)</span>
<span class="n">generated_images</span> <span class="o">=</span> <span class="n">generate_images</span><span class="p">(</span><span class="n">generate_latent_points</span><span class="p">(</span><span class="n">number</span><span class="o">=</span><span class="n">plot_number</span><span class="p">,</span>
<span class="n">scale_means</span><span class="o">=-</span><span class="mi">5</span><span class="p">,</span>
<span class="n">scale_stds</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<span class="n">plot_result</span><span class="p">(</span><span class="n">generated_images</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="n">plot_number</span><span class="p">)</span>
<span class="n">generated_images</span> <span class="o">=</span> <span class="n">generate_images</span><span class="p">(</span><span class="n">generate_latent_points</span><span class="p">(</span><span class="n">number</span><span class="o">=</span><span class="n">plot_number</span><span class="p">,</span>
<span class="n">scale_means</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="n">scale_stds</span><span class="o">=</span><span class="mi">5</span><span class="p">))</span>
<span class="n">plot_result</span><span class="p">(</span><span class="n">generated_images</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="n">plot_number</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Again, we have found something interesting. <em>Moving</em> around using our means
takes us from digit to digit, while <em>moving</em> around using our standard
deviations seem to increase the number of different digits! In the last image
above, we can barely make out every MNIST digit. Let us make on last plot using
this information by upping the standard deviation of our Gaussian noises.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plot_number</span> <span class="o">=</span> <span class="mi">400</span>
<span class="n">generated_images</span> <span class="o">=</span> <span class="n">generate_images</span><span class="p">(</span><span class="n">generate_latent_points</span><span class="p">(</span><span class="n">number</span><span class="o">=</span><span class="n">plot_number</span><span class="p">,</span>
<span class="n">scale_means</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="n">scale_stds</span><span class="o">=</span><span class="mi">10</span><span class="p">))</span>
<span class="n">plot_result</span><span class="p">(</span><span class="n">generated_images</span><span class="p">,</span> <span class="n">number</span><span class="o">=</span><span class="n">plot_number</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>A pretty cool result! We see that our generator indeed has learned a
distribution which qualitatively looks a whole lot like the MNIST dataset.</p>
<p>Another interesting way to explore the latent space of our generator model is by
interpolating between the MNIST digits. This section is largely based on
<a class="reference external" href="https://machinelearningmastery.com/how-to-interpolate-and-perform-vector-arithmetic-with-faces-using-a-generative-adversarial-network/">this excellent blogpost</a>
by Jason Brownlee.</p>
<p>So let us start by defining a function to interpolate between two points in the
latent space.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">interpolation</span><span class="p">(</span><span class="n">point_1</span><span class="p">,</span> <span class="n">point_2</span><span class="p">,</span> <span class="n">n_steps</span><span class="o">=</span><span class="mi">10</span><span class="p">):</span>
<span class="n">ratios</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">n_steps</span><span class="p">)</span>
<span class="n">vectors</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">ratio</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ratios</span><span class="p">):</span>
<span class="n">vectors</span><span class="o">.</span><span class="n">append</span><span class="p">(((</span><span class="mf">1.0</span> <span class="o">-</span> <span class="n">ratio</span><span class="p">)</span> <span class="o">*</span> <span class="n">point_1</span> <span class="o">+</span> <span class="n">ratio</span> <span class="o">*</span> <span class="n">point_2</span><span class="p">))</span>
<span class="k">return</span> <span class="n">tf</span><span class="o">.</span><span class="n">stack</span><span class="p">(</span><span class="n">vectors</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Now we have all we need to do our interpolation analysis.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">plot_number</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">latent_points</span> <span class="o">=</span> <span class="n">generate_latent_points</span><span class="p">(</span><span class="n">number</span><span class="o">=</span><span class="n">plot_number</span><span class="p">)</span>
<span class="n">results</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">plot_number</span><span class="p">),</span> <span class="mi">2</span><span class="p">):</span>
<span class="n">interpolated</span> <span class="o">=</span> <span class="n">interpolation</span><span class="p">(</span><span class="n">latent_points</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">latent_points</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">])</span>
<span class="n">generated_images</span> <span class="o">=</span> <span class="n">generate_images</span><span class="p">(</span><span class="n">interpolated</span><span class="p">)</span>
<span class="k">if</span> <span class="n">results</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">results</span> <span class="o">=</span> <span class="n">generated_images</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">results</span> <span class="o">=</span> <span class="n">tf</span><span class="o">.</span><span class="n">stack</span><span class="p">((</span><span class="n">results</span><span class="p">,</span> <span class="n">generated_images</span><span class="p">))</span>
<span class="n">plot_results</span><span class="p">(</span><span class="n">results</span><span class="p">,</span> <span class="n">plot_number</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
</div>
</div>
</div>
<script type="text/x-thebe-config">
{
requestKernel: true,
binderOptions: {
repo: "binder-examples/jupyter-stacks-datascience",
ref: "master",
},
codeMirrorConfig: {
theme: "abcdef",
mode: "python"
},
kernelOptions: {
kernelName: "python3",
path: "./."
},
predefinedOutput: true
}
</script>
<script>kernelName = 'python3'</script>
</div>
<div class='prev-next-bottom'>
<div id="prev">
<a class="left-prev" href="chapter12.html" title="previous page">
<i class="prevnext-label fas fa-angle-left"></i>
<div class="prevnext-info">
<p class="prevnext-label">previous</p>
<p class="prevnext-title"><span class="section-number">16. </span>Convolutional Neural Networks</p>
</div>
</a>
</div>
</div>
</div>
</div>
<footer class="footer">
<div class="container">
<p>
By Morten Hjorth-Jensen<br/>
&copy; Copyright 2021.<br/>
</p>
</div>
</footer>
</main>
</div>
</div>
<script src="_static/js/index.1c5a1a01449ed65a7b51.js"></script>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink