🤖 AI Summary
This work addresses the high computational cost of traditional numerical methods for solving parametric hyperbolic conservation laws and the tendency of existing neural surrogates to produce non-physical solutions, suffer from unstable rollouts, and exhibit poor generalization due to neglecting underlying physical structures. The authors propose a structure-preserving graph neural network (GNN) solver that explicitly formulates the GNN as a learnable conservative reconstruction and upwind flux operator, inherently satisfying local conservation and upwinding properties. By integrating this formulation within an ADER framework, the method enables high-order spatiotemporal predictions while maintaining physical consistency and supporting stable long-time integration with large time steps. On challenging supersonic flow benchmarks, the approach significantly outperforms both low-order numerical schemes and current neural surrogates in long-term prediction accuracy and achieves speedups of several orders of magnitude over conventional high-resolution simulations.
📝 Abstract
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.