Neural Equilibria for Long-Term Prediction of Nonlinear Conservation Laws

NeurDE (Neural Discrete Equilibrium) — a machine-learning framework for stable, accurate long-term forecasting of nonlinear conservation laws.

• Uses a kinetic lifting that splits the dynamics into a fixed linear transport step and a local nonlinear relaxation to equilibrium, making NeurDE a “neural twin” of Boltzmann-BGK.
• Integrates directly into a lattice-Boltzmann solver: the transport step becomes an efficient lattice streaming operation, while a neural network maps macroscopic observables to a discrete equilibrium distribution.
• Data-efficient: small networks trained on limited data generalize far beyond the training regime to shocks well outside the initial distribution.
• Avoids costly root-finding procedures and large velocity lattices used in traditional kinetic solvers.