Discontinuity-aware KAN-based physics-informed neural networks

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.

Figures (19)

• Figure 1: Flowchart of the DPINN method applied to a two-dimensional (2D) unsteady flow.
• Figure 2: Schematic of the adaptive Fourier embedding layer with red and blue channels representing high- and low-frequency embeddings, respectively.
• Figure 3: Left: Schematic of the DKAN architecture. Right: The discontinuous-aware activation function consists of parameterized B-splines and DyT functions.
• Figure 4: Schematic of the body-fitted coordinate system. (a) Physical space. (b) Computational space.
• Figure 5: Evolution of the loss function with training epochs.