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、时间依赖偏微分方程（PDE）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物理建模的科学机器学习框架。控制方程的参数化结构被直接嵌入降阶模型中，并通过数据驱动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可快速求解，将参数值映射为近似PDE解。此类参数化降阶模型可作为基于物理的替代模型，用于需要多次正向求解参数化PDE的不确定性量化与反问题研究。本文考虑了学习问题中的适定性、适当正则化需求等数值问题，并提出了超参数选择算法。该方法通过参数化热方程进行说明，并在FitzHugh-Nagumo神经元模型上得到验证。