Neural equilibria for long-term prediction of nonlinear conservation laws

Abstract

We introduce Neural Discrete Equilibrium (NeurDE), a machine learning framework for stable and accurate long-term forecasting of nonlinear conservation laws. NeurDE leverages a kinetic lifting that decomposes the dynamics into a fixed linear transport component and a local nonlinear relaxation to equilibrium. This structure provides a natural and principled interface between physics, numerical methods, and machine learning methodologies, enabling NeurDE to be viewed as a ``neural twin'' to Boltzmann-BGK. The transport step can be implemented efficiently in solvers such as lattice Boltzmann (LB), while the equilibrium is modeled by a neural network that maps macroscopic observables to a discrete equilibrium distribution. When integrated into a LB solver, the transport step becomes an efficient lattice streaming operation, and NeurDE yields a hybrid algorithm that robustly captures shock propagation and complex compressible dynamics over long time horizons. The NeurDE method is highly data-efficient: a small network trained on limited data generalizes far beyond the training regime, resolving shocks that evolve well outside the initial training distribution. Unlike traditional kinetic solvers, NeurDE achieves this accuracy without costly root-finding procedures or large velocity lattices. These results establish NeurDE as a scalable, efficient, and physics-informed paradigm for learning-based simulation of nonlinear conservation laws