Physics-Informed Machine Learning for Hyperbolic PDEs

Overview

Traffic jams, floods, and compressible gas flows are all governed by hyperbolic partial differential equations (PDEs), also known as hyperbolic conservation laws. These equations can develop sharp discontinuities, or shocks, even from smooth initial conditions, which makes them hard to solve quickly and reliably. Classical numerical solvers capture shocks accurately but must be re-run for every new scenario, while standard machine learning approaches are fast but tend to blur sharp fronts or produce physically meaningless solutions. Our lab develops learning-based solvers that combine the speed of machine learning with the accuracy and physical guarantees of classical numerical methods.

Graph neural operators for discontinuity-dominated flows (Cassie An Jeng)

Neural operators such as the Fourier Neural Operator (FNO) generalize across initial conditions, but they mix information globally rather than locally, which blurs shocks. Our recent work, HypNO, closes this gap with a graph neural operator that passes messages only along physically causal directions, gated by the same principles classical solvers use: upwind direction, entropy admissibility, and finite propagation speed. This architectural bias reduces prediction error on one-dimensional traffic-flow models by several-fold compared with both numerical solvers and standard neural operators. We are extending this approach to multidimensional systems, including shallow-water flow and compressible gas dynamics, where several wave types interact.

Neural finite-volume solvers with entropy guarantees (Abdessamad El Kabid)

When shocks form, a conservation law can have multiple mathematically valid solutions, not all of which are physically meaningful. We use neural networks to learn the numerical fluxes inside finite-volume schemes, so that models can evolve complex dynamical systems without needing reference solutions during training. To ensure the learned solutions are physically admissible, we introduce unsupervised training objectives based on Kružkov entropy inequalities. Using tools from numerical analysis, approximation theory, and statistical learning, we establish guarantees on the approximation capability, convergence, and generalization of these models, and we show why enforcing conservation laws alone, as in standard physics-informed learning, can be insufficient. On nonlinear traffic-flow models, these methods substantially reduce entropy violations and spurious oscillations while remaining competitive in accuracy with classical solvers.

Why it matters

Together, this work aims to make discontinuity-dominated systems fast, accurate, and trustworthy to simulate, with applications ranging from traffic and flood forecasting to large-scale transportation networks, aerospace, and climate modeling.