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.

翻译：从数据中识别微分算子是对复杂物理和生物系统进行数学建模的关键，这些系统通常拥有海量数据集。为精确预测动力学演化问题，这些算子必须具有稳定性。本文提出一种新方法，通过求解约束回归问题来学习理论上线性稳定的稀疏微分算子。这些底层约束源于动力系统的线性稳定性分析。我们进一步将该方法扩展至非线性微分算子的学习：通过确定平衡点附近线性化方程的线性稳定性约束来实现。该方法的适用性通过线性和非线性偏微分方程得到验证，包括一维标量对流扩散方程、一维伯格斯方程和二维对流方程。结果表明，具有线性稳定性约束的约束回归问题的解能够提供精确且线性稳定的稀疏微分算子。