Discontinuity-aware KAN-based physics-informed neural networks

Abstract: Physics-informed neural networks (PINNs) have proved 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 sharp changes, a discontinuity-aware Kolmogorov-Arnold network for the modeling of shockwave properties, mesh transformation to accelerate loss function convergence for complex geometries, and learnable local artificial viscosity to stabilize the algorithm near discontinuities. In numerical experiments regarding the inviscid Burgers' equation and transonic and supersonic airfoil flows, DPINN demonstrated superior accuracy when calculating discontinuities compared to existing methods.