A CFL-type Condition and Theoretical Insights for Discrete-Time Sparse Full-Order Model Inference

TL;DR

This paper develops a CFL-type condition and provides theoretical insights into the stability and consistency of discrete-time sparse full-order models inferred from data, with applications to linear and nonlinear dynamical systems.

Contribution

It introduces a sampling CFL condition for stable inference of sFOMs and connects regularization choices with stability properties, supported by theoretical analysis and numerical examples.

Findings

Derived a sampling CFL condition for linear advection stability.

Established links between regularization and model stability.

Validated theoretical insights with nonlinear system examples.

Abstract

In this work, we investigate the data-driven inference of a discrete-time dynamical system via a sparse Full-Order Model (sFOM). We first formulate the involved Least Squares (LS) problem and discuss the need for regularization, indicating a connection between the typically employed $l_2$ regularization and the stability of the inferred discrete-time sFOM. We then provide theoretical insights considering the consistency and stability properties of the inferred numerical schemes that form the sFOM and exemplify them via illustrative, 1D test cases of linear diffusion and linear advection. For linear advection, we analytically derive a "sampling CFL" condition, which dictates a bound for the ratio of spatial and temporal discretization steps in the training data that ensures stability of the inferred sFOM. Finally, we investigate the sFOM inference for two nonlinear problems, namely a 2D Burgers' test case and the incompressible flow in an oscillating lid driven cavity, and draw connections between the theoretical findings and the properties of the inferred, nonlinear sFOMs.