Deep asymptotic expansion method for solving singularly perturbed time-dependent reaction-advection-diffusion equations

TL;DR

This paper introduces a deep asymptotic expansion method that enhances the ability of physics-informed neural networks to solve singularly perturbed reaction-advection-diffusion equations with sharp internal layers.

Contribution

The paper proposes a novel deep asymptotic expansion approach that combines asymptotic analysis with neural networks to effectively handle steep gradients in complex PDEs.

Findings

DAE outperforms standard PINN and variants in accuracy

DAE is robust to training configurations

Numerical results validate the effectiveness of DAE

Abstract

Physics-informed neural network (PINN) has shown great potential in solving partial differential equations. However, it faces challenges when dealing with problems involving steep gradients. The solutions to singularly perturbed time-dependent reaction-advection-diffusion equations exhibit internal moving transition layers with sharp gradients, and thus the standard PINN becomes ineffective. In this work, we propose a deep asymptotic expansion (DAE) method, which is inspired by asymptotic analysis and leverages deep learning to approximate the smooth part of the expansion. We first derive the governing equations for transition layers, which are then solved using PINN. Numerical experiments show that the DAE outperforms the standard PINN, gPINN, and PINN with adaptive sampling. We also show its robustness with respect to training point distributions, network architectures, and random seeds.