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：在许多工程系统的设计与控制中，需要基于计算模型进行大量重复计算（多查询计算），这至关重要。对于由偏微分方程（PDEs）控制的系统，典型的高保真数值模型维数过高，在多查询场景下计算成本过大。因此，需要高效的替代模型来实现设计与控制中的低成本计算。本文提出了一种保持物理特性的降阶模型学习方法，针对其二次算子具有能量保持特性的偏微分方程，例如许多流体问题控制方程中的算子。该方法基于算子推断技术，该技术通过最小二乘拟合，利用状态快照和时间导数数据学习降阶模型算子。然而，标准算子推断通常无法学习到保持原始偏微分方程能量保持特性的降阶二次算子。为此，我们提出了一种新的能量保持算子推断（EP-OpInf）方法，通过约束优化将这一结构强加于所学习的降阶模型。基于粘性Burgers方程和Kuramoto-Sivashinksy方程（KSE）的数值结果表明，EP-OpInf能够学习高效且准确的降阶模型，并保持其能量保持结构。