This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning and physics-based modeling. The parametric structure of the governing equations is embedded directly into the reduced-order model, and parameterized reduced-order operators are learned via a data-driven linear regression problem. The result is a reduced-order model that can be solved rapidly to map parameter values to approximate PDE solutions. Such parameterized reduced-order models may be used as physics-based surrogates for uncertainty quantification and inverse problems that require many forward solves of parametric PDEs. Numerical issues such as well-posedness and the need for appropriate regularization in the learning problem are considered, and an algorithm for hyperparameter selection is presented. The method is illustrated for a parametric heat equation and demonstrated for the FitzHugh-Nagumo neuron model.

翻译：本文提出了一种针对参数化、时间相关偏微分方程（PDEs）降阶建模的新方法。该方法采用算子推断——一种融合数据驱动学习与基于物理建模的科学机器学习框架——将控制方程的参数化结构直接嵌入降阶模型中，并通过数据驱动的线性回归问题学习参数化的降阶算子。由此得到的降阶模型能够快速求解，将参数值映射为近似的PDE解。此类参数化降阶模型可作为基于物理的替代模型，用于不确定性量化和反问题（需多次求解参数化PDE正问题）中。本文考虑了学习问题中的适定性及适当正则化需求等数值问题，并提出了一种超参数选择算法。该方法以参数化热方程为例进行说明，并针对FitzHugh-Nagumo神经元模型进行了验证。