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.

翻译：非线性守恒律支配着科学和工业中一大类重要的物理系统，也是科学机器学习（SciML）的核心。大规模通用模型虽能提供计算速度，但用其替代求解器的数值与物理结构往往会损害稳定性、精度与物理保真度。本文旨在通过一种守恒感知的科学机器学习框架，平衡守恒性的一般归纳偏置与神经网络的灵活性和速度，我们将该框架称为神经离散平衡（NeurDE）。NeurDE通过学习玻尔兹曼公式的局部平衡闭包，将机器学习置于动力学求解器内部。该动力学求解器仍负责输运、松弛、矩恢复与守恒计算；神经网络仅提供非线性平衡目标。我们在6个守恒系统上测试了NeurDE，其中包括三个极具挑战性的亚音速、跨音速和超音速激波系统。结果表明，NeurDE超越了最先进的科学机器学习方法，包括分别比其大$10^4$倍和$10^6$倍的神经算子及预训练科学机器学习基础模型。尤为值得注意的是，NeurDE在其所衍生的数值方法基础上实现了改进。因此，NeurDE为保守模拟中的科学机器学习提供了一个紧凑目标：学习系统所松弛至的平衡定律，而非演化定律本身。