Discontinuity-aware KAN-based physics-informed neural networks

TL;DR

This paper introduces DPINN, a novel discontinuity-aware physics-informed neural network that effectively captures sharp spatial transitions and improves accuracy in solving PDEs with discontinuities, such as shock waves and transonic flows.

Contribution

It proposes an adaptive Fourier-feature embedding, a discontinuity-aware network, mesh transformation, and learnable viscosity to enhance PINNs for discontinuous PDE solutions.

Findings

DPINN outperforms existing methods in accuracy for shock-related PDEs.

It effectively captures steep gradients and discontinuities in complex flow problems.

Numerical experiments demonstrate superior stability and convergence.

Abstract

Physics-informed neural networks (PINNs) have proven to be a promising method for the rapid solving of partial differential equations (PDEs) in both forward and inverse problems. However, due to the smoothness assumption of functions approximated by general neural networks, PINNs are prone to spectral bias and numerical instability and suffer from reduced accuracy when solving PDEs with sharp spatial transitions or fast temporal evolution. To address this limitation, a discontinuity-aware physics-informed neural network (DPINN) method is proposed. It incorporates an adaptive Fourier-feature embedding layer to mitigate spectral bias and capture steep gradients, a discontinuity-aware network that generalizes the Kolmogorov representation theorem to the discontinuous regime for the modeling of shock-wave properties, mesh transformation to accelerate convergence across complex geometries, and learnable local artificial viscosity to stabilize the algorithm near discontinuities. In numerical experiments regarding the inviscid Burgers' equation, Riemann problems, and transonic and supersonic airfoil flows, DPINN demonstrated superior accuracy in capturing discontinuities compared to existing methods.