Enforcing structure in data-driven reduced modeling through nested Operator Inference

Nicole Aretz, Karen Willcox

Abstract: We introduce the data-driven nested Operator Inference method for learning projection-based reduced-order models (ROMs) from snapshot data of high-dimensional dynamical systems. These ROMs achieve significant computational speed-up by exploiting the intrinsic low-dimensionality of the full-order solution trajectory through projection onto a low-dimensional subspace. Our nested Operator Inference approach builds upon a nested structure of the projection-based reduced-order matrices and a hierarchy within the subspace’s basis vectors to partition the Operator Inference learning problem into multiple regression problems defined on subspaces. Each regression problem is provably better conditioned than when all reduced-order operators are learned together, reducing the need for additional regularization. Since only $\mathcal{O}(1)$ unknowns are learned at a time, nested Operator Inference is particularly applicable to higher-order polynomial systems. We demonstrate our method for the shallow ice equations with eighth order polynomial operators.