Neural equilibria for long-term prediction of nonlinear conservation laws

We introduce Neural Discrete Equilibrium (NeurDE), a machine learning (ML) approach for long-term forecasting of flow phenomena that relies on a "lifting" of physical conservation laws into the framework of kinetic theory. The kinetic formulation provides an excellent structure for ML algorithms by separating nonlinear, non-local physics into a nonlinear but local relaxation to equilibrium and a linear non-local transport. This separation allows the ML to focus on the local nonlinear components while addressing the simpler linear transport with efficient classical numerical algorithms. To accomplish this, we design an operator network that maps macroscopic observables to equilibrium states in a manner that maximizes entropy, yielding expressive BGK-type collisions. By incorporating our surrogate equilibrium into the lattice Boltzmann (LB) algorithm, we achieve accurate flow forecasts for a wide range of challenging flows. We show that NeurDE enables accurate prediction of compressible flows, including supersonic flows, while tracking shocks over hundreds of time steps, using a small velocity lattice-a heretofore unattainable feat without expensive numerical root finding.