A Parameter-Driven Physics-Informed Neural Network Framework for Solving Two-Parameter Singular Perturbation Problems Involving Boundary Layers

TL;DR

This paper develops a parameter-driven physics-informed neural network framework to effectively solve complex two-parameter singular perturbation problems with boundary layers, improving accuracy and robustness over existing methods.

Contribution

It introduces a novel adaptation of PA-PINNs for two-parameter SPPs, demonstrating enhanced robustness and accuracy in handling boundary layers.

Findings

PA-PINNs outperform standard PINNs in accuracy.

The method effectively handles boundary and interior layers.

Validated robustness for two-parameter problems.

Abstract

In this article, our goal is to solve two-parameter singular perturbation problems (SPPs) in one- and two-dimensions using an adapted Physics-Informed Neural Networks (PINNs) approach. Such problems are of major importance in engineering and sciences as it appears in control theory, fluid and gas dynamics, financial modelling and so on. Solutions of such problems exhibit boundary and/or interior layers, which make them difficult to handle. It has been validated in the literature that standard PINNs have low accuracy and can't handle such problems efficiently. Recently Cao et. al \cite{cao2023physics} proposed a new parameter asymptotic PINNs (PA-PINNs) to solve one-parameter singularly perturbed convection-dominated problems. It was observed that PA-PINNs works better than standard PINNs and gPINNs in terms of accuracy, convergence and stability. In this article, for the first time robustness of PA-PINNs will be validated for solving two-parameter SPPs.