Adaptive regularized hybrid physics-informed neural networks for large-gradient problems

To address the issues of insufficient accuracy and susceptibility to numerical oscillations in traditional physics-informed neural networks (PINNs) when solving large-gradient problems of hyperbolic conservation laws, this paper proposes an adaptive regularized hybrid PINN (ARH-PINN) to accurately capture large-gradient fields such as shock waves and suppress numerical oscillations. The ARH-PINNs method introduces a novel neural network architecture that integrates a Fourier embedding layer, a multi-layer perceptron, and a radial basis function layer to effectively capture both the global features and local structures of solutions to hyperbolic conservation laws. To improve the solution accuracy of large-gradient fields such as shock waves and suppress numerical oscillations, we propose an adaptive regularization strategy based on a compression indicator and a sharpness indicator and further conduct a sensitivity analysis of the associated parameters. To balance convergence speed and numerical stability, we incorporate input symmetric normalization and learning rate annealing into the training pipeline. Verified through extensive numerical examples on classical one-dimensional (1D) and two-dimensional (2D) conservation laws, the proposed ARH-PINNs can accurately resolve large-gradient fields such as shock waves, while effectively suppressing non-physical oscillations and retaining low numerical dissipation.