Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation

TL;DR

This paper explores the use of transformed physics-informed neural networks (PINNs) to improve the numerical solutions of the convection-diffusion equation, a singularly perturbed problem with steep boundary layers, by correcting oscillations and modifying reduced solutions.

Contribution

It introduces input transformations for PINNs to enhance accuracy in solving convection-diffusion equations and analyzes their behavior using neural tangent kernels.

Findings

PINNs can correct oscillatory solutions from FDMs.

Input transformations improve PINN accuracy.

Analytical explanation via neural tangent kernels.

Abstract

Singularly perturbed problems are known to have solutions with steep boundary layers that are hard to resolve numerically. Traditional numerical methods, such as Finite Difference Methods (FDMs), require a refined mesh to obtain stable and accurate solutions. As Physics-Informed Neural Networks (PINNs) have been shown to successfully approximate solutions to differential equations from various fields, it is natural to examine their performance on singularly perturbed problems. The convection-diffusion equation is a representative example of such a class of problems, and we consider the use of PINNs to produce numerical solutions of this equation. We study two ways to use PINNS: as a method for correcting oscillatory discrete solutions obtained using FDMs, and as a method for modifying reduced solutions of unperturbed problems. For both methods, we also examine the use of input transformation to enhance accuracy, and we explain the behavior of input transformations analytically, with the help of neural tangent kernels.