ASPINN: An asymptotic strategy for solving singularly perturbed differential equations

TL;DR

The paper introduces ASPINN, a novel asymptotic physics-informed neural network method that improves the accuracy and efficiency of solving singularly perturbed differential equations, especially near boundary layers.

Contribution

It proposes ASPINN, a decomposition-based neural network approach that enhances boundary layer solution accuracy and reduces training costs compared to existing PINN and GKPINN methods.

Findings

ASPINN outperforms PINN and GKPINN in boundary layer accuracy.

ASPINN reduces training cost by fewer layers.

Chebyshev-KAN improves performance over MLP.

Abstract

Solving Singularly Perturbed Differential Equations (SPDEs) presents challenges due to the rapid change of their solutions at the boundary layer. In this manuscript, We propose Asymptotic Physics-Informed Neural Networks (ASPINN), a generalization of Physics-Informed Neural Networks (PINN) and General-Kindred Physics-Informed Neural Networks (GKPINN) approaches. This is a decomposition method based on the idea of asymptotic analysis. Compared to PINN, the ASPINN method has a strong fitting ability for solving SPDEs due to the placement of exponential layers at the boundary layer. Unlike GKPINN, ASPINN lessens the number of fully connected layers, thereby reducing the training cost more effectively. Moreover, ASPINN theoretically approximates the solution at the boundary layer more accurately, which accuracy is also improved compared to GKPINN. We demonstrate the effect of ASPINN by solving diverse classes of SPDEs, which clearly shows that the ASPINN method is promising in boundary layer problems. Furthermore, we introduce Chebyshev Kolmogorov-Arnold Networks (Chebyshev-KAN) instead of MLP, achieving better performance in various experiments.