Many-query computations, in which a computational model for an engineering system must be evaluated many times, are crucial in design and control. For systems governed by partial differential equations (PDEs), typical high-fidelity numerical models are high-dimensional and too computationally expensive for the many-query setting. Thus, efficient surrogate models are required to enable low-cost computations in design and control. This work presents a physics-preserving reduced model learning approach that targets PDEs whose quadratic operators preserve energy, such as those arising in governing equations in many fluids problems. The approach is based on the Operator Inference method, which fits reduced model operators to state snapshot and time derivative data in a least-squares sense. However, Operator Inference does not generally learn a reduced quadratic operator with the energy-preserving property of the original PDE. Thus, we propose a new energy-preserving Operator Inference (EP-OpInf) approach, which imposes this structure on the learned reduced model via constrained optimization. Numerical results using the viscous Burgers' and Kuramoto-Sivashinksy equation (KSE) demonstrate that EP-OpInf learns efficient and accurate reduced models that retain this energy-preserving structure.

翻译：多查询计算（即工程系统计算模型需多次评估）在设计与控制中至关重要。对于由偏微分方程（PDEs）描述的系统，典型的高保真数值模型维度高且计算成本过高，难以适应多查询场景。因此，需要高效的代理模型以实现低成本的设计与控制计算。本文提出一种保物理特性的降阶模型学习方法，针对二次算子具有保能量特性的偏微分方程——如众多流体问题控制方程中的情形。该方法基于算子推断（Operator Inference）技术，通过最小二乘拟合状态快照与时间导数数据来构建降阶模型算子。然而，传统算子推断方法通常无法学习到具有原偏微分方程保能量特性的降阶二次算子。为此，我们提出一种新型保能量算子推断（EP-OpInf）方法，通过约束优化将这一结构施加于所学习的降阶模型。基于粘性Burgers方程与Kuramoto-Sivashinsky方程（KSE）的数值结果表明，EP-OpInf能够学习到兼具高效性与准确性的降阶模型，并保留其保能量结构。