Schwarz Waveform Relaxation Physics-Informed Neural Networks for Solving Advection-Diffusion-Reaction Equations

TL;DR

This paper introduces a novel physics-informed neural network approach combining Schwarz waveform relaxation to efficiently solve complex advection-diffusion-reaction equations through domain decomposition and parallel training.

Contribution

The paper develops a new PINN framework based on SWR, enabling parallelized local solutions and adaptive network depth for solving PDEs with proven convergence properties.

Findings

Demonstrates effective parallel training of PINNs for PDEs

Shows adaptability of network depth based on local solution complexity

Provides numerical evidence of convergence and performance

Abstract

This paper develops a physics-informed neural network (PINN) based on the Schwarz waveform relaxation (SWR) method for solving local and nonlocal advection-diffusion-reaction equations. Specifically, we derive the formulation by constructing subdomain-dependent local solutions by minimizing local loss functions, allowing the decomposition of the training process in different domains in an embarrassingly parallel procedure. Provided the convergence of PINN, the overall proposed algorithm is convergent. By constructing local solutions, one can, in particular, adapt the depth of the deep neural networks, depending on the solution's spectral space and time complexity in each subdomain. We present some numerical experiments based on classical and Robin-SWR to illustrate the performance and comment on the convergence of the proposed method.