Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks

TL;DR

This paper introduces a non-overlapping Schwarz-type domain decomposition method for physics-informed neural networks, improving interface learning and reducing communication overhead for solving PDEs like Poisson and Helmholtz equations.

Contribution

It proposes a novel domain decomposition approach with a generalized interface condition and an augmented Lagrangian method, enhancing solution accuracy and scalability in physics-informed neural networks.

Findings

Effective for high-wavenumber Helmholtz equations

Maintains accuracy with up to 64 subdomains

Reduces communication overhead in domain decomposition

Abstract

We present a non-overlapping, Schwarz-type domain decomposition method with a generalized interface condition, designed for physics-informed machine learning of partial differential equations (PDEs) in both forward and inverse contexts. Our approach employs physics and equality-constrained artificial neural networks (PECANN) within each subdomain. Unlike the original PECANN method, which relies solely on initial and boundary conditions to constrain PDEs, our method uses both boundary conditions and the governing PDE to constrain a unique interface loss function for each subdomain. This modification improves the learning of subdomain-specific interface parameters while reducing communication overhead by delaying information exchange between neighboring subdomains. To address the constrained optimization in each subdomain, we apply an augmented Lagrangian method with a conditionally adaptive update strategy, transforming the problem into an unconstrained dual optimization. A distinct advantage of our domain decomposition method is its ability to learn solutions to both Poisson's and Helmholtz equations, even in cases with high-wavenumber and complex-valued solutions. Through numerical experiments with up to 64 subdomains, we demonstrate that our method consistently generalizes well as the number of subdomains increases.