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.

翻译：本文提出了一种能量保持的机器学习方法，用于通过利用偏微分方程的多辛形式推断降阶模型。绝大多数能量保持降阶方法采用辛伽辽金投影，通过将完整模型投影到辛子空间来构建降阶哈密顿模型。然而，辛投影要求完全离散算子的存在，而在许多情况下（如黑箱偏微分方程求解器），这些算子是不可获取的。本工作提出了一种能量保持的机器学习方法，能够仅使用数据推断给定偏微分方程的动力学特性，使得所提框架不依赖于完全离散算子。在此背景下，所提方法具有非侵入性。该方法属于灰箱方法，因其仅需偏微分方程层面多辛模型的基本知识。我们证明了所提方法满足空间离散的局部能量守恒，并保持多辛守恒定律。我们在线性波动方程、Korteweg-de Vries方程和Zakharov-Kuznetsov方程上测试了该方法，并通过在远超出训练时间区间的范围内测试学习模型的泛化能力。