Solving Partial Differential Equations with TensorFlow

A.V.S.D.S.Mahesh (Final year B.Tech.)

IIT Kanpur

February 27, 2019

A.V.S.D.S.Mahesh (IIT Kanpur) PDE with TensorFlow February 27, 2019 1 / 29

Introduction

Today we shall see how to solve basic partial differential equations usingPython’s TensorFlow library. At the end of this day you will be able towrite basic PDE solvers in TensorFlow. For this purpose, 2Dwave-equation solver is demonstrated in this module.

Table of Contents

1 Getting started

2 TensorFlow and Numpy

3 TensorFlow objectsSession objectComputation graphVariablesMiscellaneous objects

4 2D wave equation solver

5 Conclusion

Table of Contents

1 Getting started

5 Conclusion

What is TensorFlow?

TensorFlow is an open-source deep learning library by Google

Although originally intended for machine learning and deep neuralnetworks, can be applied in many other domains as solving PDEs.

TensorFlow allows to define functions on tensors and simplifies manyoperations on them.1

1url: https://cs224d.stanford.edu/lectures/CS224d-Lecture7.pdf.

What is TensorFlow?

Setting up TensorFlow

TensorFlow can be installed easily using Anaconda package manager.

What is a Tensor?

A scalar is a tensor (of rank 0)

A vector is a tensor (of rank 1)

A matrix can represent tensor of rank 2.

Given fixed basis, a tensor can be represented as a multidimensionalarray of numbers.

What is a Tensor?

Table of Contents

1 Getting started

5 Conclusion

TensorFlow vs. Numpy

TensorFlow and Numpy are very much similar (Both are N-d arraylibraries)

Numpy additionally doesn’t have method to create functions ontensors and more importantly no GPU support.

TensorFlow and Numpy are very much similar (Both are N-d arraylibraries)

Numpy additionally doesn’t have method to create functions ontensors and more importantly no GPU support.

Figure: Comparison of performances of TensorFlow with other libraries 2

2Work done under Dr. M. K. Verma of dept. of Physics at IITK

Numpy quick review

Same in TensorFlow

Where is the difference?

TensorFlow requires explicit evaluation

TensorFlow computes using a computation graph which needs to beevaluated to get a value.

TensorFlow requires explicit evaluation

TensorFlow computes using a computation graph which needs to beevaluated to get a value.

Table of Contents

1 Getting started

5 Conclusion

TensorFlow Session object

“A Session object encapsulates the environment in which Tensor objectsare evaluated.” (from docs)

tf.InteractiveSession() is to keep a default session open in ipython.

Computation graph

“TensorFlow programs are usually structured into a constructionphase, that assembles a graph, and an execution phase that uses asession to execute ops in the graph.” (docs)

All computations add nodes to global default graph. (docs)

Computation graph

“TensorFlow programs are usually structured into a constructionphase, that assembles a graph, and an execution phase that uses asession to execute ops in the graph.” (docs)

All computations add nodes to global default graph. (docs)

Variables

“When you train a model you use variables to hold and updateparameters. Variables are in-memory buffers containing tensors.”(docs)

We have so far used constants. Now we shall see usage of variables.

Note that variables must be initialized before they have values, unlikewith constant tensors.

Variables

Updating and Fetching Variables

Calling sess.run(var) on a tf.Session() object retrieves its value.Can retrieve multiple variables simultaneously with sess.run([var1,

var2])4

Updating and Fetching Variables

Calling sess.run(var) on a tf.Session() object retrieves its value.Can retrieve multiple variables simultaneously with sess.run([var1,

var2])4

Import from Numpy

External data can be imported from Numpy using convert to tensor:

Placeholders and Feed Dictionaries

Inputting data with tf.convert to tensor() is convenient, but it isnot scalable.

Use tf.placeholder variables (dummy nodes that provide entrypoints for data to computational graph).

A feed dict is a python dictionary mapping from tf.placeholder

vars (or their names) to data (numpy arrays, lists, etc.).5

Placeholders and Feed Dictionaries: Example

Table of Contents

1 Getting started

5 Conclusion

2D wave equation solver

This is a small demonstration of a pde solver6. Initializations:

6url: https://www.tensorflow.org/tutorials/non-ml/pdes.

This is a small demonstration of a pde solver6. Initializations:

6url: https://www.tensorflow.org/tutorials/non-ml/pdes.

Main code:

What is inside laplace(U):

2D wave equation solver: Visualization

Table of Contents

1 Getting started

5 Conclusion

What from here?

TensorFlow uses underlying gpu and cpu for parallelizing operations likeconvolution. If it is to be done not multiple gpu’s then the data has to bedivided and planned so for this purpose. A small demonstration:7

7url: https://www.tensorflow.org/guide/using_gpu.

Thank You

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