Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference

TL;DR

This paper introduces a physics-informed regularizer and structure-preserving constraints for operator inference, enabling the learning of stable, accurate reduced models of physical systems with quadratic nonlinearities from data.

Contribution

It proposes a novel regularizer that incorporates physical insights to promote stability and enforces structural constraints to preserve physical properties in learned models.

Findings

Regularized models are more stable and accurate.

Structure preservation improves model fidelity.

Regularization outperforms Tikhonov in stability.

Abstract

Operator inference learns low-dimensional dynamical-system models with polynomial nonlinear terms from trajectories of high-dimensional physical systems (non-intrusive model reduction). This work focuses on the large class of physical systems that can be well described by models with quadratic nonlinear terms and proposes a regularizer for operator inference that induces a stability bias onto quadratic models. The proposed regularizer is physics informed in the sense that it penalizes quadratic terms with large norms and so explicitly leverages the quadratic model form that is given by the underlying physics. This means that the proposed approach judiciously learns from data and physical insights combined, rather than from either data or physics alone. Additionally, a formulation of operator inference is proposed that enforces model constraints for preserving structure such as symmetry and definiteness in the linear terms. Numerical results demonstrate that models learned with operator inference and the proposed regularizer and structure preservation are accurate and stable even in cases where using no regularization or Tikhonov regularization leads to models that are unstable.