DMIS (AAAI2023)

Official code for "DMIS: Dynamic Mesh-based Importance Sampling for Training Physics-Informed Neural Networks" (AAAI 2023)

Modeling dynamics in the form of partial differential equations (PDEs) is an effectual way to understand real-world physics processes. For complex physics systems, analytical solutions are not available and numerical solutions are widely-used. However, traditional numerical algorithms are computationally expensive and challenging in handling multiphysics systems. Recently, using neural networks to solve PDEs has made significant progress, called physics-informed neural networks (PINNs). PINNs encode physical laws into neural networks and learn the continuous solutions of PDEs. For the training of PINNs, existing methods suffer from the problems of inefficiency and unstable convergence, since the PDE residuals require calculating automatic differentiation. In this paper, we propose Dynamic Mesh-based Importance Sampling (DMIS) to tackle these problems. DMIS is a novel sampling scheme based on importance sampling, which constructs a dynamic triangular mesh to estimate sample weights efficiently. DMIS has broad applicability and can be easily integrated into existing methods. The evaluation of DMIS on three widely-used benchmarks shows that DMIS improves the convergence speed and accuracy in the meantime. Especially in solving the highly nonlinear Schrödinger Equation, compared with state-of-the-art methods, DMIS shows up to 46% smaller root mean square error and five times faster convergence speed.

Quick Start

Installation

Setup environment

Dependencies:

• PyTorch == 1.11.0
• hydra == 1.2.0
• tensorboard == 2.9.0
• sympy == 1.10.1
• scipy == 1.8.1
• pandas == 1.4.3
• numpy == 1.22.4
• matplotlib == 3.5.2

conda create --name DMIS python=3.7
conda activate DMIS
conda install --file requirements.txt

All the code has been tested on Ubuntu 16.04, Python 3.7.12, PyTorch 1.11.0, and CUDA 11.3

Clone this repository

git clone git@github.com:MatrixBrain/DMIS.git
cd DMIS

Training

To train PINNs with DMIS for solving Schrödinger Equation:

python train.py --config-name=Schrodinger train_conf.pde_sampler=SamplerWithDMIS train_conf.pde_reweighting=BiasedReWeighting hydra.job.chdir=True

To train PINNs with uniform sampling for solving Schrödinger Equation:

python train.py --config-name=Schrodinger train_conf.pde_sampler=UniformSampler train_conf.pde_reweighting=NoReWeighting hydra.job.chdir=True

For other equations, you can replace Schrodinger with KDV(KdV Equation), Burgers(Burgers' Equation), Diffusion(Diffusion Equation) and ACEquation (Allen-Cahn Equation)

Evaluation

To evaluate PINN-O, PINN-N, DMIS used in our paper, please run:

python evaluate.py hydra.job.chdir=True

Results

• Schrödinger Equation

Method	ME	MAE	RMSE
PINN-O	1.360	0.186	0.4092
PINN-N	0.948	0.149	0.2906
xPINN	0.546	0.045	0.0089
cPINN	0.591	0.069	0.0169
PINN-DMIS(ours)	0.647	0.127	0.2196
xPINN-DMIS(ours)	0.867	0.036	0.0129
cPINN-DMIS(ours)	0.358	0.025	0.0033

• Burgers' Equation

Method	ME	MAE	RMSE
PINN-O	0.451	0.0738	0.1100
PINN-N	0.358	0.0579	0.0859
xPINN	0.261	0.0099	0.0010
cPINN	0.324	0.0084	0.0007
PINN-DMIS(ours)	0.225	0.0294	0.0495
xPINN-DMIS(ours)	0.420	0.0115	0.0017
cPINN-DMIS(ours)	0.397	0.0111	0.0016

• KdV Equation

Method	ME	MAE	RMSE
PINN-O	2.140	0.363	0.520
PINN-N	1.860	0.292	0.441
xPINN	2.462	0.272	0.230
cPINN	2.925	0.258	0.248
PINN-DMIS(ours)	1.170	0.391	0.492
xPINN-DMIS(ours)	2.380	0.233	0.196
cPINN-DMIS(ours)	2.680	0.230	0.200

Note: The results of PINN-O are different from the provided results in the original PINN paper because we use extrapolation precision and the original PINN paper uses interpolation precision.

Citation

If you find the code and pre-trained models useful for your research, please consider citing our paper. 😊

@InProceedings{yang2022dmis,
author = {Yang, Zijiang and Qiu, Zhongwei and Fu, Dongmei},
title = {DMIS: Dynamic Mesh-based Importance Sampling for Training Physics-Informed Neural Networks},
booktitle = {AAAI},
year = {2023},
}