Data-driven identification of stable differential operators using constrained regression

TL;DR

This paper introduces a data-driven method to identify stable differential operators by solving constrained regression problems, ensuring stability and sparsity for modeling complex systems.

Contribution

It presents a novel approach that incorporates linear stability constraints into regression for learning both linear and nonlinear differential operators from data.

Findings

Accurate and linearly stable sparse differential operators were obtained.

Method successfully applied to linear and nonlinear PDEs like advection-diffusion and Burgers equations.

Constrained regression improves stability and accuracy of learned operators.

Abstract

Identifying differential operators from data is essential for the mathematical modeling of complex physical and biological systems where massive datasets are available. These operators must be stable for accurate predictions for dynamics forecasting problems. In this article, we propose a novel methodology for learning sparse differential operators that are theoretically linearly stable by solving a constrained regression problem. These underlying constraints are obtained following linear stability for dynamical systems. We further extend this approach for learning nonlinear differential operators by determining linear stability constraints for linearized equations around an equilibrium point. The applicability of the proposed method is demonstrated for both linear and nonlinear partial differential equations such as 1-D scalar advection-diffusion equation, 1-D Burgers equation and 2-D advection equation. The results indicated that solutions to constrained regression problems with linear stability constraints provide accurate and linearly stable sparse differential operators.