Hamiltonian Operator Inference has been introduced in [Sharma, H., Wang, Z., Kramer, B., Physica D: Nonlinear Phenomena, 431, p.133122, 2022] to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. This approach constructs a low-dimensional model using only data and knowledge of the Hamiltonian function. Such ROMs can keep the intrinsic structure of the system, allowing them to capture the physics described by the governing equations. In this work, we extend this approach to more general systems that are either conservative or dissipative in energy, and which possess a gradient structure. We derive the optimization problems for inferring structure-preserving ROMs that preserve the gradient structure. We further derive an {\em a priori} error estimate for the reduced-order approximation. To test the algorithms, we consider semi-discretized partial differential equations with gradient structure, such as the parameterized wave and Korteweg-de-Vries equations in the conservative case and the one- and two-dimensional Allen-Cahn equations in the dissipative case. The numerical results illustrate the accuracy, structure-preservation properties, and predictive capabilities of the gradient-preserving Operator Inference ROMs.

翻译：哈密顿算子推断方法已在[Sharma, H., Wang, Z., Kramer, B., Physica D: Nonlinear Phenomena, 431, p.133122, 2022]中被提出，用于学习哈密顿系统的保结构降阶模型。该方法仅利用数据和哈密顿函数知识构建低维模型，此类降阶模型能保持系统的内在结构，从而捕捉控制方程所描述的物理特性。本文将该方法推广至更一般的系统——包括保守系统与能量耗散系统，并特别针对具有梯度结构的系统。我们推导了推断保梯度结构降阶模型的优化问题，进一步给出了降阶近似的先验误差估计。为验证算法，我们考虑半离散化的梯度结构偏微分方程，包括保守情形下的参数化波动方程和Korteweg-de-Vries方程，以及耗散情形下的一维和二维Allen-Cahn方程。数值结果展示了梯度保持算子推断降阶模型的准确性、保结构特性及预测能力。