This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of canonical Hamiltonian systems. Traditional intrusive projection-based model reduction approaches utilize symplectic Galerkin projection to construct Hamiltonian ROMs by projecting Hamilton's equations of the full model onto a symplectic subspace. This symplectic projection requires complete knowledge about the full model operators and full access to manipulate the computer code. In contrast, the proposed Hamiltonian operator inference approach embeds the physics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying symplectic structure. Our method exploits knowledge of the Hamiltonian functional to define and parametrize a Hamiltonian ROM form which can then be learned from data projected via symplectic projectors. The proposed method is gray-box in that it utilizes knowledge of the Hamiltonian structure at the partial differential equation level, as well as knowledge of spatially local components in the system. However, it does not require access to computer code, only data to learn the models. Our numerical results demonstrate Hamiltonian operator inference on a linear wave equation, the cubic nonlinear Schr\"{o}dinger equation, and a nonpolynomial sine-Gordon equation. Accurate long-time predictions far outside the training time interval for nonlinear examples illustrate the generalizability of our learned models.

翻译：这项工作提出了一种非侵扰性的物理保存方法,以学习汉密尔顿系统的减序模型(ROMs) 。传统的侵扰性投影模型减少方法利用干扰性投影法,通过将汉密尔顿的全模型方程式投射到一个干扰性子空间来建造汉密尔顿式的激光器。这种模拟投影需要完全了解全模型操作员和完全使用计算机代码。相比之下,拟议的汉密尔顿操作员推断方法将物理嵌入操作员推导框架,以开发一种数据驱动的模型减少方法,以保存基本的干扰性结构。我们的方法利用汉密尔密尔顿功能的知识来定义和模拟汉密尔顿式的全模型,然后从通过模拟投影仪预测的数据中学习。拟议的方法是灰色框,它利用部分差异方程式一级对汉密尔密尔顿结构的了解,以及系统中空间局部局部组成部分的知识。然而,它并不需要获得计算机代码,而仅需要数据来学习外部的模拟结构模型。 我们的卡密尔密尔密尔密尔顿操作员模型中一个不长的模型。