Semi-analytic PINN methods for singularly perturbed boundary value problems

TL;DR

This paper introduces semi-analytic physics-informed neural networks (PINNs) that incorporate boundary layer analysis to effectively solve singularly perturbed boundary value problems with sharp solution transitions.

Contribution

The paper develops a semi-analytic PINN framework that integrates corrector functions from boundary layer analysis to improve accuracy in stiff differential equations.

Findings

Accurately solves singular perturbation problems with sharp solution transitions.

Outperforms standard PINNs in stiff differential equations.

Effective for both linear and nonlinear singularly perturbed equations.

Abstract

We propose a new semi-analytic physics informed neural network (PINN) to solve singularly perturbed boundary value problems. The PINN is a scientific machine learning framework that offers a promising perspective for finding numerical solutions to partial differential equations. The PINNs have shown impressive performance in solving various differential equations including time-dependent and multi-dimensional equations involved in a complex geometry of the domain. However, when considering stiff differential equations, neural networks in general fail to capture the sharp transition of solutions, due to the spectral bias. To resolve this issue, here we develop the semi-analytic PINN methods, enriched by using the so-called corrector functions obtained from the boundary layer analysis. Our new enriched PINNs accurately predict numerical solutions to the singular perturbation problems. Numerical experiments include various types of singularly perturbed linear and nonlinear differential equations.