Transform Learn: A data-driven approach to nonlinear model reduction

摘要

This paper presents Transform & Learn, a physics-informed surrogate modeling approach that unites the perspectives of model reduction and machine learning. The proposed method uses insight from the physics of the problem-in the form of partial differential equation (PDE) models-to derive a state transformation in which the system admits a quadratic representation. Snapshot data from a high-fidelity model simulation are transformed to the new state representation and subsequently are projected onto a low-dimensional basis. The quadratic reduced model is then learned via a least-squares-based operator inference procedure. The state transformation thus plays two key roles in the proposed method: it allows the task of nonlinear model reduction to be reformulated as a structured model learning problem, and it parametrizes the machine learning problem in a way that recovers efficient, generalizable models. The proposed method is demonstrated on two PDE examples. First, we transform the Euler equations in conservative variables to the specific volume state representation, yielding low-dimensional Transform & Learn models that achieve a 0.05% relative state error when compared to a high-fidelity simulation in the conservative variables. Second, we consider a model of the Continuously Variable Resonance Combustor, a single element liquid-fueled rocket engine experiment. We show that the specific volume representation of this model also has quadratic structure and that the learned quadratic reduced models can accurately predict the growing oscillations of an unstable combustor.