Deep Learning of High-Dimensional Partial Differential Equations

This library allows you to experiment with the Deep Galerkin algorithm. For finding a PDE or ODE solution you simply define a loss function. Then, by calling train(), the neural network learns the solution. It outputs several useful information: <br> <br> 1- Loss function value (for the differential operator, boundary condition, etc.) <br> 2- Neural Network solution for the given equation <br> 3- Layer by Layer mean activation value (during training) for the neural network <br> <br> You can also find implementation code for Free Boundry PDE (American Option) up to 7 assets (9 dimensions) with the method discussed at https://arxiv.org/abs/1708.07469. There is also a finite-difference Matlab code that is useful for measuring the accuracy of your result. <br> In this repository, there are two low-dimensional examples: the heat equation and the advection equation. The following animation illustrates the two equations as they are trained: <br>

<p align="center"> <img src="https://github.com/pooyasf/DGM/blob/main/Docs/advection_anim.gif?raw=true" width="320"> <img src="https://github.com/pooyasf/DGM/blob/main/Docs/heat_anim.gif?raw=true" width="320"> </p> <br>

Mean activation value for different layers of the neural net (during training): <br><br>

<p align="center"> <img src="https://github.com/pooyasf/DGM/blob/main/Docs/heat_layers_activ_value.png?raw=true" width="400" > </p>

<br><br>

Here are the building blocks of this code: <br><br>

<p align="center"> <img src="https://github.com/pooyasf/DGM/blob/main/Docs/LibraryDiagram.png?raw=true" width="400" > </p>

<br><br>

Requirements

Python 3.7.7 <br> Pytorch 1.6