A Hybrid Discontinuous Galerkin Neural Network Method for Solving Hyperbolic Conservation Laws with Temporal Progressive Learning

Abstract: For hyperbolic conservation laws, traditional methods and physics-informed neural networks (PINNs) often encounter difficulties in capturing sharp discontinuities and maintaining temporal consistency. To address these challenges, we introduce a hybrid computational framework by coupling discontinuous Galerkin (DG) discretizations with a temporally progressive neural network architecture. Our method incorporates a structure-preserving weak-form loss -- combining DG residuals and Rankine-Hugoniot jump conditions -- with a causality-respecting progressive training strategy. The proposed framework trains neural networks sequentially across temporally decomposed subintervals, leveraging pseudo-label supervision to ensure temporal coherence and solution continuity. This approach mitigates error accumulation and enhances the model's capacity to resolve shock waves and steep gradients without explicit limiters. Besides, a theoretical analysis establishes error bounds for the proposed framework, demonstrating convergence toward the physical solution under mesh refinement and regularized training. Numerical experiments on Burgers and Euler equations show that our method consistently outperforms standard PINNs, PINNs-WE, and first-order DG schemes in both accuracy and robustness, particularly in capturing shocks and steep gradients. These results highlight the promise of combining classical discretization techniques with machine learning to develop robust and accurate solvers for nonlinear hyperbolic systems.