Data-driven identification of stable sparse differential operators using constrained regression

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 differential operators that are theoretically linearly stable and have sparsity patterns of common discretization schemes. These differential operators are obtained by solving a constrained regression problem, involving local constraints to ensure the linear stability of the global dynamical system. 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.