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.

中文翻译：

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使用约束回归数据驱动识别稳定稀疏微分算子


从数据中识别微分算子对于拥有大量数据集的复杂物理和生物系统的数学建模至关重要。这些算子必须稳定才能准确预测动态预测问题。在本文中，我们提出了一种学习微分算子的新方法，该算子在理论上是线性稳定的，并且具有常见离散化方案的稀疏模式。这些微分算子是通过求解约束回归问题获得的，涉及局部约束以确保全局动力系统的线性稳定性。我们通过确定平衡点周围线性方程的线性稳定性约束，进一步扩展了这种学习非线性微分算子的方法。证明了该方法对于线性和非线性偏微分方程（如一维标量平流扩散方程、一维 Burgers 方程和二维平流方程）的适用性。结果表明，具有线性稳定性约束的约束回归问题的解提供了精确且线性稳定的稀疏微分算子。