Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws

Physics-Informed Neural Networks (PINNs) provide a meshless approach for solving partial differential equations (PDEs), but suffer severe degradation in stiff and shock-dominated problems, where small PDE residuals can correspond to globally inaccurate solutions. We show these failures are multi-causal, arising from the concurrent interplay of (i) spectral bias against sharp features, (ii) imbalanced multi-term optimization and loss-weight collapse, (iii) violation of temporal causality, and (iv) under-resolved collocation. We present SPARC-Net, a unified architecture and training framework that jointly addresses all four pathologies. SPARC-Net leverages an adaptive multi-scale spectral encoder with a learnable spectral gate, a gated residual backbone, adaptive activations, and a hard-constraint output ansatz that exactly enforces initial and boundary conditions, structurally eliminating loss-weight collapse. Training employs stabilized gradient-norm loss balancing, floored causality-respecting residual weighting, and residual-based adaptive collocation (RAD). Validated against exact analytic and high-order spectral reference solutions across four canonical benchmarks -- viscous Burgers', Allen-Cahn, convection (beta=30), and reaction -- SPARC-Net yields substantial improvements over vanilla PINNs: relative L2 error drops from 1.47e-1 to 1.14e-1 on Burgers' (22% reduction), 9.93e-1 to 5.78e-2 on Allen-Cahn (94% reduction), and 9.82e-1 to 3.54e-3 on reaction (100% reduction). A characteristic-coordinate encoder for hyperbolic transport further reduces convection error from 5.14e-1 to 9.88e-5 (100% reduction). We report five-seed mean +/- standard deviation errors, Wilcoxon significance tests, full ablation studies, hyperparameter sensitivities, an extension to the 2D heat equation, and comparisons against parameter-matched baselines.

Hyperbolic conservation laws govern a wide range of transport-driven dynamics featuring shocks, contact discontinuities, and complex wave interactions, posing distinct challenges for deep-learning-based surrogate modeling. While classical numerical methods provide robust and physically admissible solutions, their computational cost restricts applicability in many-query tasks such as parametric studies and design optimization. Conversely, existing neural surrogates offer rapid inference but often fail to respect intrinsic PDE structures, leading to non-physical artifacts, rollout instability, and poor generalization. We present an interpretable, structure-preserving graph neural solver that bridges classical numerical principles with graph neural networks (GNNs). The network is designed as a learned reconstruction-and-flux operator rather than a black-box state updater, thereby inherently preserving key properties such as local conservation and upwinding. Inspired by Arbitrary high-order DERivatives schemes, we further recast message-passing GNNs as high-order space-time predictors, enabling conservative and stable neural updates with large time steps. Evaluation is performed on challenging supersonic flow benchmarks spanning broad parametric variations in geometry, initial/boundary conditions, and flow regimes. The neural solver achieves superior long-horizon rollout stability and accuracy compared with strong surrogate baselines, outperforms low-order discretizations, and delivers orders-of-magnitude runtime speedups over high-resolution simulations.

We explore the possibility of solving Partial Differential Equations (PDEs) using discrete weak formulations. We propose a programming environment for defining a discrete computational domain, introducing discrete functions defined over a set of points, constructing discrete inner products, and introducing discrete weak formulations employing Kronecker delta test functions. Building on this setup, we propose a discrete neural network representation, training the solution function defined over a discrete set of points and employing discrete finite difference derivatives in the automatic differentiation procedures. As a challenging computational model example, we focus on Stokes equations in two-dimensions, defined over a discrete set of points. We train the solution using the discrete weak residual and the Adamax algorithm with discrete automatic differentiation of the discrete gradients. Despite introducing the python environment, we also provide a rigorous mathematical formulation based on discrete weak formulations, proving the well-posedness and robustness of the loss function. The solution of the discrete weak formulations is based on neural network training employing a robust loss function that is related to the true error. In this way, we have a robust control of the numerical error during the training of the neural networks. Besides the Stokes formulation, we also explain the functionality of the proposed library using the Laplace problem formulation.