This work presents a tensorial approach to constructing data-driven reduced-order models corresponding to semi-discrete partial differential equations with canonical Hamiltonian structure. By expressing parameter-varying operators with affine dependence as contractions of a generalized parameter vector against a constant tensor, this method leverages the operator inference framework to capture parametric dependence in the learned reduced-order model via the solution to a convex, least-squares optimization problem. This leads to a concise and straightforward implementation which compactifies previous parametric operator inference approaches and directly extends to learning parametric operators with symmetry constraints, a key feature required for constructing structure-preserving surrogates of Hamiltonian systems. The proposed approach is demonstrated on both a (non-Hamiltonian) heat equation with variable diffusion coefficient as well as a Hamiltonian wave equation with variable wave speed.

翻译：本研究提出了一种张量方法，用于构建与具有典型哈密顿结构的半离散偏微分方程相对应的数据驱动降阶模型。该方法通过将具有仿射依赖性的参数变化算子表示为广义参数向量与常数张量的缩并，利用算子推断框架，通过求解凸最小二乘优化问题来捕获学习降阶模型中的参数依赖性。这实现了一种简洁直观的实现方式，不仅简化了先前的参数化算子推断方法，而且可直接推广至学习具有对称性约束的参数化算子——这是构建哈密顿系统结构保持代理模型所需的关键特性。所提出的方法在具有可变扩散系数的（非哈密顿）热方程以及具有可变波速的哈密顿波动方程上均得到了验证。