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This work formulates a new approach to reduced modeling of parameterized,time-dependent partial differential equations (PDEs). The method employsOperator Inference, a scientific machine learning framework combiningdata-driven learning and physics-based modeling. The parametric structure ofthe governing equations is embedded directly into the reduced-order model, andparameterized reduced-order operators are learned via a data-driven linearregression problem. The result is a reduced-order model that can be solvedrapidly to map parameter values to approximate PDE solutions. Suchparameterized reduced-order models may be used as physics-based surrogates foruncertainty quantification and inverse problems that require many forwardsolves of parametric PDEs. Numerical issues such as well-posedness and the needfor appropriate regularization in the learning problem are considered, and analgorithm for hyperparameter selection is presented. The method is illustratedfor a parametric heat equation and demonstrated for the FitzHugh-Nagumo neuronmodel.

Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference