This paper presents an energy-preserving machine learning method for inferring reduced-order models (ROMs) by exploiting the multi-symplectic form of partial differential equations (PDEs). The vast majority of energy-preserving reduced-order methods use symplectic Galerkin projection to construct reduced-order Hamiltonian models by projecting the full models onto a symplectic subspace. However, symplectic projection requires the existence of fully discrete operators, and in many cases, such as black-box PDE solvers, these operators are inaccessible. In this work, we propose an energy-preserving machine learning method that can infer the dynamics of the given PDE using data only, so that the proposed framework does not depend on the fully discrete operators. In this context, the proposed method is non-intrusive. The proposed method is grey box in the sense that it requires only some basic knowledge of the multi-symplectic model at the partial differential equation level. We prove that the proposed method satisfies spatially discrete local energy conservation and preserves the multi-symplectic conservation laws. We test our method on the linear wave equation, the Korteweg-de Vries equation, and the Zakharov-Kuznetsov equation. We test the generalization of our learned models by testing them far outside the training time interval.

本研究解决了现有能量保持降阶模型方法依赖完全离散算子的局限性，通过提出一种基于数据的能量保持机器学习方法，能够推断多哈密顿偏微分方程的动态。这种非侵入性方法仅需对多哈密顿模型有基本知识，并在多个经典方程中证实了空间离散局部能量守恒和多哈密顿守恒定律的保持。

多哈密顿偏微分方程的结构保持学习