Data-driven identification of stable differential operators using constrained regression

TL;DR

The paper tackles the challenge of learning discrete differential operators from data with a guarantee of linear stability, which is essential for reliable dynamics forecasting. It introduces a constrained regression framework (S-LDOs) that enforces stability constraints derived from linear dynamical system theory, using Gershgorin-based local conditions for linear operators and linearization-based constraints for nonlinear operators. Across 1-D advection-diffusion, 1-D Burgers, and 2-D advection tests, S-LDOs yield sparse, interpretable operators that are provably stable and accurately reproduce reference dynamics, in contrast to standard unconstrained LDOs which frequently blow up or oscillate. The approach has practical implications for nonintrusive reduced order modeling and stable coarse-grained discretizations, enabling scalable and stable data-driven PDE discretizations with theoretical guarantees.

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.

Figures (18)

• Figure 1: Eigenvalues of the differential operator $\bm{L}$ obtained using (a) the $1^{st}$-order forward difference and (b) the $1^{st}$-order backward difference. The shaded region in red indicates the stable region.
• Figure 2: Diffusion problem ($c = 0$, $\nu = 0.02$): Eigenvalues of LDOs ($\bm{L}^{m}_2$) with stencil size of (a) 3, (b) 5, (c) 7, (d) 11, (e) 21 and (f) 41 and regularization parameter $\beta_1 = 10^{-3}$. The shaded region in red indicates the stable region.
• Figure 3: Diffusion problem ($c = 0$, $\nu = 0.02$): Eigenvalues for LDO ($-\bm{L}^{m}_1$) for stencil size of (a) 5 and (b) 11 with several regularization parameters. The stable region is not shown as the $x$-axis is in the log scale.
• Figure 4: Diffusion problem ($c = 0$, $\nu = 0.02$): Error in the regression objective function $e_{\text{train}}$ for LDOs of different stencil sizes.
• Figure 5: Diffusion problem ($c = 0$, $\nu = 0.02$): Eigenvalues of S-LDOs ($\bm{L}^{m}_2$) for stencil size of (a) 3, (b) 5, (c) 7, (d) 11, (e) 21 and (f) 41 with a regularization parameter of $\beta_1 = 10^{-3}$. The shaded region in red indicates the stable region.