Solution of Advection Equation with Discontinuous Initial and Boundary Conditions via Physics-Informed Neural Networks

TL;DR

This paper advances physics-informed neural networks (PINNs) for solving 1D advection equations with discontinuous initial and boundary conditions by introducing Fourier features, adaptive training, filtering, and a modified loss to improve accuracy and handle discontinuities.

Contribution

It presents novel techniques including Fourier feature mapping, adaptive loss weighting, and a modified loss function inspired by upwind schemes to better model discontinuous solutions with PINNs.

Findings

Fourier features mitigate spectral bias in PINNs.

Adaptive loss weighting improves convergence.

Modified loss function reduces smoothing of discontinuities.

Abstract

In this paper, we investigate several techniques for modeling the one-dimensional advection equation for a specific class of problems with discontinuous initial and boundary conditions using physics-informed neural networks (PINNs). To mitigate the spectral bias phenomenon, we employ a Fourier feature mapping layer as the input representation, adopt a two-stage training strategy in which the Fourier feature parameters and the neural network weights are optimized sequentially, and incorporate adaptive loss weighting. To further enhance the approximation accuracy, a median filter is applied to the spatial data, and the predicted solution is constrained through a bounded linear mapping. Moreover, for certain nonlinear problems, we introduce a modified loss function inspired by the upwind numerical scheme to alleviate the excessive smoothing of discontinuous solutions typically observed in neural network approximations.