Neural equilibria for long-term prediction of nonlinear conservation laws

Nonlinear conservation laws govern a broad class of important physical systems in science and industry and are central to scientific machine learning (SciML). Large general-purpose models offer speed, but replacing the numerical and physical structure of solvers often compromises stability, accuracy, and physical faithfulness. Here, we aim to balance the general inductive bias of conservation with the flexibility and speed of neural networks through a conservation-aware SciML backbone, which we call Neural Discrete Equilibrium (NeurDE). NeurDE places machine learning inside a kinetic solver by learning the local equilibrium closure of a Boltzmann formulation. The kinetic solver still performs transport, relaxation, moment recovery, and conservation; the neural network provides only the nonlinear equilibrium target. We test NeurDE on $6$ conserved systems, including three very challenging subsonic, transonic, and supersonic shock systems. NeurDE outperforms state-of-the-art SciML methods, including neural operators and pretrained SciML foundation models that are $10^4$ and $10^6$ times larger, respectively. Most notably, NeurDE improves upon the numerical method from which it is derived. NeurDE therefore provides a compact target for scientific machine learning in conservative simulation: learn the equilibrium law toward which the system relaxes, not the evolution law itself.