Deep Ritz PDE Solver: Deep Ritz Method for Partial Differetial Equations

🪐 A project for solving Partial Differential Equations with Deep Neural Network!

About The Project

This project is mainly inspired by Deep Ritz Method proposed by E Weinan, et al. And some codes are adopted from

https://github.com/ZeyuJia/DeepRitzMethod

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Getting Started

How to use this project?

Prerequisites

• Clone the whole repo from GitHub

  git clone https://github.com/JiantingFeng/DRM-PDE.git && cd ./DRM-PDE

• Install Python and related packages

  conda install --file requirements.txt

Usage

You should know a little bit about Finite Element Method or Functional Analysis for using this project. The basic procedure is following:

Reorganize the Equation

$$ \begin{equation} \begin{split} -\sum_{i, k}\partial_i(a_{ik}\partial_ku)+a_0u &= f\quad x\in\Omega,\\ u &= 0\quad x\in\partial\Omega \end{split} \end{equation} $$

the corresponding functional optimization problem is

$$ J[v]=\int_\Omega\Big[\frac{1}{2}\sum_{i,k}a_{ik}\partial_iv\partial_kv +\frac{1}{2}a_0v^2-fv\Big]dx $$

and the solution satisfies

$$ u = \arg\min_{v\in H} J[v] $$

Here, $H$ is DNN functional space

Insert Boundary Condition

For a typical machine learning algorithm, loss function is essential, here, we use

$$ \begin{split} l(u) &= J[u] + \beta \int_{\partial\Omega}|u-u_{\text{bnd}}|^2\textrm dx\\ &= \int_\Omega\Big(\frac{1}{2}\sum_{i,k}a_{i,k}\partial_i u\partial_j u +\frac{1}{2}a_0u^2-fu\Big)\textrm dx + \beta \int_{\partial\Omega}|u-u_{\text{bnd}}|^2\textrm ds \end{split} $$

as our loss function

Define the architecture of DNN

We use basic ResNet-like architecture to avoid gradient vanishing, with GELU activation.

In dimension is as same as $\dim \Omega$, out dimension is set as $1$ (For real-valued equations).

Training

Randomly sample points inside $\Omega$ and $\partial \Omega$, pass through the NN, and get the output and loss value, use Adam or SGD to take gradient descent.

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Contributing

No further maintenance for this project, you can just fort this project and make changes without contacting me.

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Contact

Jianting Feng - gianting01@gmail.com

Homepage - [https://jiantingfeng.vip]

Project Link - [https://github.com/JiantingFeng/ Deep-Ritz-PDE-Solver]

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