Spatio-Temporal Uncertainty-Modulated Physics-Informed Neural Networks for Solving Hyperbolic Conservation Laws with Strong Shocks

Abstract: Physics-Informed Neural Networks (PINNs) frequently encounter difficulties in accurately resolving shock waves within high-speed compressible flows, a failure largely attributed to the "gradient pathology" arising from extreme stiffness at discontinuities. To overcome this limitation, we propose the Spatio-Temporal Uncertainty-Modulated PINN (UM-PINN), a probabilistic framework that reinterprets the training process as a multi-task learning problem governed by homoscedastic aleatoric uncertainty. By integrating a gradient-based spatial mask with learnable variance parameters, our method dynamically balances the conflicting contributions of Partial Differential Equation (PDE) residuals and initial conditions across the spatiotemporal domain, further stabilized by Quasi-Monte Carlo Sobol sampling. We validate the framework against challenging benchmarks, including the one-dimensional (1D) Sod shock tube, the high-frequency Shu-Osher problem, and the complex two-dimensional (2D) Riemann interaction, where standard gradient-based weighting schemes typically fail. Experimental results demonstrate that UM-PINN achieves orders of magnitude improvement in accuracy and shock resolution compared to baseline methods, establishing a robust new paradigm for mesh-free Computational Fluid Dynamics in hyperbolic systems.