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

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.