Morten Hjorth-Jensen
952 lines
44 KiB
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952 lines
44 KiB
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Applied Data Analysis and Machine Learning
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About the course
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Teachers and Grading
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Review of Statistics with Resampling Techniques and Linear Algebra
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1. Elements of Probability Theory and Statistical Data Analysis
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</a>
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<a class="reference internal" href="linalg.html">
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2. Linear Algebra, Handling of Arrays and more Python Features
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</ul>
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<p aria-level="2" class="caption" role="heading">
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<span class="caption-text">
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From Regression to Support Vector Machines
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</span>
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<li class="toctree-l1">
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3. Linear Regression
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter2.html">
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4. Ridge and Lasso Regression
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter3.html">
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5. Resampling Methods
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter4.html">
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6. Logistic Regression
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapteroptimization.html">
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7. Optimization, the central part of any Machine Learning algortithm
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter5.html">
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8. Support Vector Machines, overarching aims
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</a>
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</li>
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</ul>
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<p aria-level="2" class="caption" role="heading">
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<span class="caption-text">
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Decision Trees, Ensemble Methods and Boosting
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</span>
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9. Decision trees, overarching aims
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</a>
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</li>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter7.html">
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10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
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</a>
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</li>
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</ul>
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<p aria-level="2" class="caption" role="heading">
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<span class="caption-text">
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Dimensionality Reduction
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11. Basic ideas of the Principal Component Analysis (PCA)
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</a>
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12. Clustering and Unsupervised Learning
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<p aria-level="2" class="caption" role="heading">
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<span class="caption-text">
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Deep Learning Methods
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13. Neural networks
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter10.html">
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14. Building a Feed Forward Neural Network
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</a>
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</li>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter11.html">
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15. Solving Differential Equations with Deep Learning
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter12.html">
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16. Convolutional Neural Networks
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</a>
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<a class="reference internal" href="chapter13.html">
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17. Recurrent neural networks: Overarching view
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</a>
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</ul>
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<p aria-level="2" class="caption" role="heading">
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Weekly material, notes and exercises
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Exercises week 34
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Week 34: Introduction to the course, Logistics and Practicalities
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</a>
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Exercises week 35
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</a>
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Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
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<a class="reference internal" href="exercisesweek36.html">
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Exercises week 36
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</a>
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Week 36: Statistical interpretation of Linear Regression and Resampling techniques
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="exercisesweek37.html">
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Exercises week 37
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</a>
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<a class="reference internal" href="week37.html">
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Week 37: Statistical interpretations and Resampling Methods
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</a>
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</li>
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<li class="toctree-l1">
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<a class="reference internal" href="exercisesweek38.html">
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Exercises week 38
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</a>
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</li>
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<a class="reference internal" href="week38.html">
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Week 38: Logistic Regression and Optimization
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<a class="reference internal" href="exercisesweek39.html">
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Exercises week 39
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Week 39: Optimization and Gradient Methods
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<a class="reference internal" href="week40.html">
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Week 40: Gradient descent methods (continued) and start Neural networks
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<a class="reference internal" href="exercisesweek41.html">
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Exercises week 41
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<a class="reference internal" href="week41.html">
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Week 41 Neural networks and constructing a neural network code
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<a class="reference internal" href="exercisesweek42.html">
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Exercises week 42
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Week 42 Constructing a Neural Network code with introduction to Tensor flow
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Exercises weeks 43 and 44
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<a class="reference internal" href="week43.html">
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Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations
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</a>
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</li>
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<li class="toctree-l1">
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<a class="reference internal" href="week44.html">
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Week 44, Convolutional Neural Networks (CNN)
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</a>
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</li>
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<li class="toctree-l1">
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<a class="reference internal" href="week45.html">
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Week 45, Recurrent Neural Networks
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="week46.html">
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Week 46: Decision Trees, Ensemble methods and Random Forests
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="week47.html">
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Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods and Summary of Course
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<a class="reference internal" href="exercisesweek47.html">
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Exercise week 47
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Projects
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<li class="toctree-l1">
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Project 1 on Machine Learning, deadline October 9 (midnight), 2023
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</a>
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</li>
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<li class="toctree-l1">
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<a class="reference internal" href="project2.html">
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Project 2 on Machine Learning, deadline November 17 (Midnight)
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</a>
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Project 3 on Machine Learning, deadline December 18 (midnight), 2023
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|
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<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#defining-the-data-sets-to-analyze-yourself">
|
||
Defining the data sets to analyze yourself
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-basic-structure-of-your-project">
|
||
The basic structure of your project
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-a">
|
||
Part a)
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-b">
|
||
Part b)
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-c">
|
||
Part c)
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-d">
|
||
Part d)
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-e">
|
||
Part e)
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#solving-partial-differential-equations-with-neural-networks">
|
||
Solving partial differential equations with neural networks
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#alternative-differential-equations">
|
||
Alternative differential equations
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-a-setting-up-the-problem">
|
||
Part a), setting up the problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id1">
|
||
Part b)
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-c-neural-networks">
|
||
Part c) Neural networks
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-d-neural-network-complexity">
|
||
Part d) Neural network complexity
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id2">
|
||
Part e)
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#introduction-to-numerical-projects">
|
||
Introduction to numerical projects
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#format-for-electronic-delivery-of-report-and-programs">
|
||
Format for electronic delivery of report and programs
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#software-and-needed-installations">
|
||
Software and needed installations
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div id="main-content" class="row">
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||
<div class="col-12 col-md-9 pl-md-3 pr-md-0">
|
||
<!-- Table of contents that is only displayed when printing the page -->
|
||
<div id="jb-print-docs-body" class="onlyprint">
|
||
<h1>Project 3 on Machine Learning, deadline December 18 (midnight), 2023</h1>
|
||
<!-- Table of contents -->
|
||
<div id="print-main-content">
|
||
<div id="jb-print-toc">
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<div>
|
||
<h2> Contents </h2>
|
||
</div>
|
||
<nav aria-label="Page">
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||
<ul class="visible nav section-nav flex-column">
|
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<li class="toc-h1 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#">
|
||
Project 3 on Machine Learning, deadline December 18 (midnight), 2023
|
||
</a>
|
||
</li>
|
||
<li class="toc-h1 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#paths-for-project-3">
|
||
Paths for project 3
|
||
</a>
|
||
<ul class="visible nav section-nav flex-column">
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#defining-the-data-sets-to-analyze-yourself">
|
||
Defining the data sets to analyze yourself
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#the-basic-structure-of-your-project">
|
||
The basic structure of your project
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-a">
|
||
Part a)
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-b">
|
||
Part b)
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-c">
|
||
Part c)
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-d">
|
||
Part d)
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-e">
|
||
Part e)
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#solving-partial-differential-equations-with-neural-networks">
|
||
Solving partial differential equations with neural networks
|
||
</a>
|
||
<ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#alternative-differential-equations">
|
||
Alternative differential equations
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-a-setting-up-the-problem">
|
||
Part a), setting up the problem
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id1">
|
||
Part b)
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-c-neural-networks">
|
||
Part c) Neural networks
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#part-d-neural-network-complexity">
|
||
Part d) Neural network complexity
|
||
</a>
|
||
</li>
|
||
<li class="toc-h3 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#id2">
|
||
Part e)
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#introduction-to-numerical-projects">
|
||
Introduction to numerical projects
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#format-for-electronic-delivery-of-report-and-programs">
|
||
Format for electronic delivery of report and programs
|
||
</a>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry">
|
||
<a class="reference internal nav-link" href="#software-and-needed-installations">
|
||
Software and needed installations
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<div>
|
||
|
||
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
|
||
doconce format html Project3.do.txt -->
|
||
<!-- dom:TITLE: Project 3 on Machine Learning, deadline December 18 (midnight), 2023 --><div class="tex2jax_ignore mathjax_ignore section" id="project-3-on-machine-learning-deadline-december-18-midnight-2023">
|
||
<h1>Project 3 on Machine Learning, deadline December 18 (midnight), 2023<a class="headerlink" href="#project-3-on-machine-learning-deadline-december-18-midnight-2023" title="Permalink to this headline">¶</a></h1>
|
||
<p><strong><a class="reference external" href="http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html">Data Analysis and Machine Learning FYS-STK3155/FYS4155</a></strong>, Department of Physics, University of Oslo, Norway</p>
|
||
<p>Date: <strong>Nov 13, 2023</strong></p>
|
||
<p>Copyright 1999-2023, <a class="reference external" href="http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html">Data Analysis and Machine Learning FYS-STK3155/FYS4155</a>. Released under CC Attribution-NonCommercial 4.0 license</p>
|
||
</div>
|
||
<div class="tex2jax_ignore mathjax_ignore section" id="paths-for-project-3">
|
||
<h1>Paths for project 3<a class="headerlink" href="#paths-for-project-3" title="Permalink to this headline">¶</a></h1>
|
||
<div class="section" id="defining-the-data-sets-to-analyze-yourself">
|
||
<h2>Defining the data sets to analyze yourself<a class="headerlink" href="#defining-the-data-sets-to-analyze-yourself" title="Permalink to this headline">¶</a></h2>
|
||
<p>For project 3, you can propose own data sets that relate to your research interests or just use existing data sets from say</p>
|
||
<ol class="simple">
|
||
<li><p><a class="reference external" href="https://www.kaggle.com/datasets">Kaggle</a></p></li>
|
||
<li><p>The <a class="reference external" href="https://archive.ics.uci.edu/ml/index.php">University of California at Irvine (UCI) with its machine learning repository</a>.</p></li>
|
||
<li><p>Or other sources.</p></li>
|
||
</ol>
|
||
<p>The approach to the analysis of these new data sets should follow to a large extent what you did in projects 1 and 2. That is:</p>
|
||
<ol class="simple">
|
||
<li><p>Whether you end up with a regression or a classification problem, you should employ at least two of the methods we have discussed among <strong>linear regression (including Ridge and Lasso)</strong>, <strong>Logistic Regression</strong>, <strong>Neural Networks</strong>, <strong>Convolution Neural Networks</strong>, <strong>Recurrent Neural Networks</strong>, and <strong>Decision Trees, Random Forests, Bagging and Boosting</strong>.</p></li>
|
||
</ol>
|
||
<p>Feel also free to use support vector machines, <span class="math notranslate nohighlight">\(k\)</span>-means and principal components analysis, although the latter have not been covered during the lectures. This material can be found in the lecture notes.</p>
|
||
<p>You could for example explore all of the approaches from decision trees, via bagging and voting classifiers, to random forests, boosting and finally XGboost. If you wish to venture into <strong>convolutional neural networks</strong> or <strong>recurrent neural networks</strong>, or extensions of neural networkds, feel free to do so. You can also study unsupervised methods, although we in this course have mainly paid attention to supervised learning.</p>
|
||
<p>For Boosting, feel also free to write your own codes.</p>
|
||
<ol class="simple">
|
||
<li><p>For project 3, you should feel free to use your own codes from projects 1 and 2, eventually write your own for Decision trees/random forests/bagging/boosting’ or use the available functionality of <strong>Scikit-Learn</strong>, <strong>Tensorflow</strong>, PyTorch etc.</p></li>
|
||
<li><p>The estimates you used and tested in projects 1 and 2 should also be included, that is the <span class="math notranslate nohighlight">\(R2\)</span>-score, <strong>MSE</strong>, confusion matrix, accuracy score, information gain, ROC and Cumulative gains curves and other, cross-validation and/or bootstrap if these are relevant.</p></li>
|
||
<li><p>Similarly, feel free to explore various activations functions in deep learning and various approachs to stochastic gradient descent approaches.</p></li>
|
||
<li><p>If possible, you should link the data sets with exisiting research and analyses thereof. Scientific articles which have used Machine Learning algorithms to analyze the data are highly welcome. Perhaps you can improve previous analyses and even publish a new article?</p></li>
|
||
<li><p>A critical assessment of the methods with ditto perspectives and recommendations is also something you need to include.</p></li>
|
||
</ol>
|
||
<p>All in all, the report should follow the same pattern as the two previous ones, with abstract, introduction, methods, code, results, conclusions etc..</p>
|
||
<p>We propose also an alternative to the above. This is a project on using machine learning methods (neural networks mainly) to the solution of ordinary differential equations and partial differential equations, with a final twist on how to diagonalize a symmetric matrix with neural networks.</p>
|
||
<p>This is a field with large scientific interest, spanning from studies of turbulence in fluid mechanics and meteorology to the solution of quantum mechanical systems. As reading background you can use the slides <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week42.html">from week 43</a> and/or the textbook by <a class="reference external" href="https://www.springer.com/gp/book/9789401798150">Yadav et al</a>.</p>
|
||
</div>
|
||
<div class="section" id="the-basic-structure-of-your-project">
|
||
<h2>The basic structure of your project<a class="headerlink" href="#the-basic-structure-of-your-project" title="Permalink to this headline">¶</a></h2>
|
||
<p>Here follows a set up on how to structure your report and analyze the data you have opted for.</p>
|
||
<div class="section" id="part-a">
|
||
<h3>Part a)<a class="headerlink" href="#part-a" title="Permalink to this headline">¶</a></h3>
|
||
<p>The first part deals with structuring and reading the data, much along the same lines as done in projects 1 and 2. Explain how the data are produced and place them in a proper context.</p>
|
||
</div>
|
||
<div class="section" id="part-b">
|
||
<h3>Part b)<a class="headerlink" href="#part-b" title="Permalink to this headline">¶</a></h3>
|
||
<p>You need to include at least two central algorithms, or as an alternative explore methods from decisions tree to bagging, random forests and boosting. Explain the basics of the methods you have chosen to work with. This would be your theory part.</p>
|
||
</div>
|
||
<div class="section" id="part-c">
|
||
<h3>Part c)<a class="headerlink" href="#part-c" title="Permalink to this headline">¶</a></h3>
|
||
<p>Then describe your algorithm and its implementation and tests you have performed.</p>
|
||
</div>
|
||
<div class="section" id="part-d">
|
||
<h3>Part d)<a class="headerlink" href="#part-d" title="Permalink to this headline">¶</a></h3>
|
||
<p>Then presents your results and findings, link with existing literature and more.</p>
|
||
</div>
|
||
<div class="section" id="part-e">
|
||
<h3>Part e)<a class="headerlink" href="#part-e" title="Permalink to this headline">¶</a></h3>
|
||
<p>Finally, here you should present a critical assessment of the methods you have studied and link your results with the existing literature.</p>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="solving-partial-differential-equations-with-neural-networks">
|
||
<h2>Solving partial differential equations with neural networks<a class="headerlink" href="#solving-partial-differential-equations-with-neural-networks" title="Permalink to this headline">¶</a></h2>
|
||
<p>For this variant of project 3, we will assume that you have some
|
||
background in the solution of partial differential equations using
|
||
finite difference schemes. We will study the solution of the diffusion
|
||
equation in one dimension using a standard explicit scheme and neural
|
||
networks to solve the same equations.</p>
|
||
<p>For the explicit scheme, you can study for example chapter 10 of the lecture notes in <a class="reference external" href="https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Lectures/lectures2015.pdf">Computational Physics, FYS3150/4150</a> or alternative sources from courses like <a class="reference external" href="https://www.uio.no/studier/emner/matnat/math/MAT-MEK4270/index.html">MAT-MEK4270</a>. For the solution of ordinary and partial differential equations using neural networks, the lectures by <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/pub/week42/html/week43.html">included in the lectures of week 43</a> at this course are highly recommended.</p>
|
||
<p>For the machine learning part you can use your own code from project 2 or the functionality of for example <strong>Tensorflow/Keras</strong>, <strong>PyTorch</strong> or other libraries such <a class="reference external" href="https://maziarraissi.github.io/PINNs/">Physics informed machine learning</a>.</p>
|
||
<div class="section" id="alternative-differential-equations">
|
||
<h3>Alternative differential equations<a class="headerlink" href="#alternative-differential-equations" title="Permalink to this headline">¶</a></h3>
|
||
<p>Note that you can replace the one-dimensional diffusion equation discussed below with other sets of either ordinary differential equations or partial differential equations.
|
||
Please discuss such a change with us at the lab.</p>
|
||
</div>
|
||
<div class="section" id="part-a-setting-up-the-problem">
|
||
<h3>Part a), setting up the problem<a class="headerlink" href="#part-a-setting-up-the-problem" title="Permalink to this headline">¶</a></h3>
|
||
<p>The physical problem can be that of the temperature gradient in a rod of length <span class="math notranslate nohighlight">\(L=1\)</span> at <span class="math notranslate nohighlight">\(x=0\)</span> and <span class="math notranslate nohighlight">\(x=1\)</span>.
|
||
We are looking at a one-dimensional
|
||
problem</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial^2 u(x,t)}{\partial x^2} =\frac{\partial u(x,t)}{\partial t}, t> 0, x\in [0,L]
|
||
\]</div>
|
||
<p>or</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
u_{xx} = u_t,
|
||
\]</div>
|
||
<p>with initial conditions, i.e., the conditions at <span class="math notranslate nohighlight">\(t=0\)</span>,</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
u(x,0)= \sin{(\pi x)} \hspace{0.5cm} 0 < x < L,
|
||
\]</div>
|
||
<p>with <span class="math notranslate nohighlight">\(L=1\)</span> the length of the <span class="math notranslate nohighlight">\(x\)</span>-region of interest. The
|
||
boundary conditions are</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
u(0,t)= 0 \hspace{0.5cm} t \ge 0,
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
u(L,t)= 0 \hspace{0.5cm} t \ge 0.
|
||
\]</div>
|
||
<p>The function <span class="math notranslate nohighlight">\(u(x,t)\)</span> can be the temperature gradient of a rod.
|
||
As time increases, the velocity approaches a linear variation with <span class="math notranslate nohighlight">\(x\)</span>.</p>
|
||
<p>We will limit ourselves to the so-called explicit forward Euler algorithm with discretized versions of time given by a forward formula and a centered difference in space resulting in</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
u_t\approx \frac{u(x,t+\Delta t)-u(x,t)}{\Delta t}=\frac{u(x_i,t_j+\Delta t)-u(x_i,t_j)}{\Delta t}
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
u_{xx}\approx \frac{u(x+\Delta x,t)-2u(x,t)+u(x-\Delta x,t)}{\Delta x^2},
|
||
\]</div>
|
||
<p>or</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
u_{xx}\approx \frac{u(x_i+\Delta x,t_j)-2u(x_i,t_j)+u(x_i-\Delta x,t_j)}{\Delta x^2}.
|
||
\]</div>
|
||
<p>Write down the algorithm and the equations you need to implement.
|
||
Find also the analytical solution to the problem.</p>
|
||
</div>
|
||
<div class="section" id="id1">
|
||
<h3>Part b)<a class="headerlink" href="#id1" title="Permalink to this headline">¶</a></h3>
|
||
<p>Implement the explicit scheme algorithm and perform tests of the solution
|
||
for <span class="math notranslate nohighlight">\(\Delta x=1/10\)</span>, <span class="math notranslate nohighlight">\(\Delta x=1/100\)</span> using <span class="math notranslate nohighlight">\(\Delta t\)</span> as dictated by the stability limit of the explicit scheme. The stability criterion for the explicit scheme requires that <span class="math notranslate nohighlight">\(\Delta t/\Delta x^2 \leq 1/2\)</span>.</p>
|
||
<p>Study the solutions at two time points <span class="math notranslate nohighlight">\(t_1\)</span> and <span class="math notranslate nohighlight">\(t_2\)</span> where <span class="math notranslate nohighlight">\(u(x,t_1)\)</span> is smooth but still significantly curved
|
||
and <span class="math notranslate nohighlight">\(u(x,t_2)\)</span> is almost linear, close to the stationary state.</p>
|
||
</div>
|
||
<div class="section" id="part-c-neural-networks">
|
||
<h3>Part c) Neural networks<a class="headerlink" href="#part-c-neural-networks" title="Permalink to this headline">¶</a></h3>
|
||
<p>Study now the lecture notes on solving ODEs and PDEs with neural
|
||
network and use either your own code from project 2 or the
|
||
functionality of tensorflow/keras to solve the same equation as in
|
||
part b). Discuss your results and compare them with the standard
|
||
explicit scheme. Include also the analytical solution and compare with
|
||
that.</p>
|
||
</div>
|
||
<div class="section" id="part-d-neural-network-complexity">
|
||
<h3>Part d) Neural network complexity<a class="headerlink" href="#part-d-neural-network-complexity" title="Permalink to this headline">¶</a></h3>
|
||
<p>Here we study the stability of the results of the results as functions of the number of hidden nodes, layers and activation functions for the hidden layers.
|
||
Increase the number of hidden nodes and layers in order to see if this improves your results. Try also different activation functions for the hidden layers, such as the <strong>tanh</strong>, <strong>ReLU</strong>, and other activation functions.
|
||
Discuss your results.</p>
|
||
</div>
|
||
<div class="section" id="id2">
|
||
<h3>Part e)<a class="headerlink" href="#id2" title="Permalink to this headline">¶</a></h3>
|
||
<p>Finally, present a critical assessment of the methods you have studied
|
||
and discuss the potential for the solving differential equations with machine learning methods.</p>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="introduction-to-numerical-projects">
|
||
<h2>Introduction to numerical projects<a class="headerlink" href="#introduction-to-numerical-projects" title="Permalink to this headline">¶</a></h2>
|
||
<p>Here follows a brief recipe and recommendation on how to write a report for each
|
||
project.</p>
|
||
<ul class="simple">
|
||
<li><p>Give a short description of the nature of the problem and the eventual numerical methods you have used.</p></li>
|
||
<li><p>Describe the algorithm you have used and/or developed. Here you may find it convenient to use pseudocoding. In many cases you can describe the algorithm in the program itself.</p></li>
|
||
<li><p>Include the source code of your program. Comment your program properly.</p></li>
|
||
<li><p>If possible, try to find analytic solutions, or known limits in order to test your program when developing the code.</p></li>
|
||
<li><p>Include your results either in figure form or in a table. Remember to label your results. All tables and figures should have relevant captions and labels on the axes.</p></li>
|
||
<li><p>Try to evaluate the reliabilty and numerical stability/precision of your results. If possible, include a qualitative and/or quantitative discussion of the numerical stability, eventual loss of precision etc.</p></li>
|
||
<li><p>Try to give an interpretation of you results in your answers to the problems.</p></li>
|
||
<li><p>Critique: if possible include your comments and reflections about the exercise, whether you felt you learnt something, ideas for improvements and other thoughts you’ve made when solving the exercise. We wish to keep this course at the interactive level and your comments can help us improve it.</p></li>
|
||
<li><p>Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you don’t properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning.</p></li>
|
||
</ul>
|
||
</div>
|
||
<div class="section" id="format-for-electronic-delivery-of-report-and-programs">
|
||
<h2>Format for electronic delivery of report and programs<a class="headerlink" href="#format-for-electronic-delivery-of-report-and-programs" title="Permalink to this headline">¶</a></h2>
|
||
<p>The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report:</p>
|
||
<ul class="simple">
|
||
<li><p>Use Canvas to hand in your projects, log in at <a class="reference external" href="https://www.uio.no/english/services/it/education/canvas/">https://www.uio.no/english/services/it/education/canvas/</a> with your normal UiO username and password.</p></li>
|
||
<li><p>Upload <strong>only</strong> the report file or the link to your GitHub/GitLab or similar typo of repos! For the source code file(s) you have developed please provide us with your link to your GitHub/GitLab or similar domain. The report file should include all of your discussions and a list of the codes you have developed. Do not include library files which are available at the course homepage, unless you have made specific changes to them.</p></li>
|
||
<li><p>In your GitHub/GitLab or similar repository, please include a folder which contains selected results. These can be in the form of output from your code for a selected set of runs and input parameters.</p></li>
|
||
</ul>
|
||
<p>Finally,
|
||
we encourage you to collaborate. Optimal working groups consist of
|
||
2-3 students. You can then hand in a common report.</p>
|
||
</div>
|
||
<div class="section" id="software-and-needed-installations">
|
||
<h2>Software and needed installations<a class="headerlink" href="#software-and-needed-installations" title="Permalink to this headline">¶</a></h2>
|
||
<p>If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages,
|
||
we recommend that you install the following Python packages via <strong>pip</strong> as</p>
|
||
<ol class="simple">
|
||
<li><p>pip install numpy scipy matplotlib ipython scikit-learn tensorflow sympy pandas pillow</p></li>
|
||
</ol>
|
||
<p>For Python3, replace <strong>pip</strong> with <strong>pip3</strong>.</p>
|
||
<p>See below for a discussion of <strong>tensorflow</strong> and <strong>scikit-learn</strong>.</p>
|
||
<p>For OSX users we recommend also, after having installed Xcode, to install <strong>brew</strong>. Brew allows
|
||
for a seamless installation of additional software via for example</p>
|
||
<ol class="simple">
|
||
<li><p>brew install python3</p></li>
|
||
</ol>
|
||
<p>For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution
|
||
you can use <strong>pip</strong> as well and simply install Python as</p>
|
||
<ol class="simple">
|
||
<li><p>sudo apt-get install python3 (or python for python2.7)</p></li>
|
||
</ol>
|
||
<p>etc etc.</p>
|
||
<p>If you don’t want to install various Python packages with their dependencies separately, we recommend two widely used distrubutions which set up all relevant dependencies for Python, namely</p>
|
||
<ol class="simple">
|
||
<li><p><a class="reference external" href="https://docs.anaconda.com/">Anaconda</a> Anaconda is an open source distribution of the Python and R programming languages for large-scale data processing, predictive analytics, and scientific computing, that aims to simplify package management and deployment. Package versions are managed by the package management system <strong>conda</strong></p></li>
|
||
<li><p><a class="reference external" href="https://www.enthought.com/product/canopy/">Enthought canopy</a> is a Python distribution for scientific and analytic computing distribution and analysis environment, available for free and under a commercial license.</p></li>
|
||
</ol>
|
||
<p>Popular software packages written in Python for ML are</p>
|
||
<ul class="simple">
|
||
<li><p><a class="reference external" href="http://scikit-learn.org/stable/">Scikit-learn</a>,</p></li>
|
||
<li><p><a class="reference external" href="https://www.tensorflow.org/">Tensorflow</a>,</p></li>
|
||
<li><p><a class="reference external" href="http://pytorch.org/">PyTorch</a> and</p></li>
|
||
<li><p><a class="reference external" href="https://keras.io/">Keras</a>.</p></li>
|
||
</ul>
|
||
<p>These are all freely available at their respective GitHub sites. They
|
||
encompass communities of developers in the thousands or more. And the number
|
||
of code developers and contributors keeps increasing.</p>
|
||
</div>
|
||
</div>
|
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