Neural equilibria for long-term prediction of nonlinear conservation laws

2025-01-12

AI Summary

Accurate long-term prediction of compressible/supersonic flows governed by nonlinear conservation laws—particularly shock wave evolution over extended time horizons—remains challenging due to numerical instability, resolution limitations, and computational bottlenecks in traditional solvers. Method: This paper introduces Neural Discrete Equilibrium (NeurDE), a novel paradigm that lifts physical conservation laws into a kinetic framework, decoupling local nonlinear relaxation from linear nonlocal transport. Machine learning is thus focused exclusively on modeling the maximum-entropy equilibrium state. NeurDE integrates BGK-type collision dynamics, lattice Boltzmann discretization, and an operator network grounded in the maximum-entropy principle to construct a data-driven surrogate equilibrium. Results: Experiments demonstrate that NeurDE stably tracks supersonic flows and shocks over hundreds of time steps using sparse velocity lattices, achieving high shock resolution without iterative root-finding. It circumvents numerical stiffness and significantly outperforms state-of-the-art methods in computational efficiency.

Technology Category

• Machine Learning: Learning on the Edge & Model Compression
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work

Abstract

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.

Problem

• Complex Fluid Dynamics
• Long-term Prediction
• Computational Efficiency

Innovation

• NeurDE
• Complex Fluid Dynamics
• Resource-efficient Prediction