In this paper, we give an in-depth error analysis for surrogate models generated by a variant of the Sparse Identification of Nonlinear Dynamics (SINDy) method. We start with an overview of a variety of non-linear system identification techniques, namely, SINDy, weak-SINDy, and the occupation kernel method. Under the assumption that the dynamics are a finite linear combination of a set of basis functions, these methods establish a matrix equation to recover coefficients. We illuminate the structural similarities between these techniques and establish a projection property for the weak-SINDy technique. Following the overview, we analyze the error of surrogate models generated by a simplified version of weak-SINDy. In particular, under the assumption of boundedness of a composition operator given by the solution, we show that (i) the surrogate dynamics converges towards the true dynamics and (ii) the solution of the surrogate model is reasonably close to the true solution. Finally, as an application, we discuss the use of a combination of weak-SINDy surrogate modeling and proper orthogonal decomposition (POD) to build a surrogate model for partial differential equations (PDEs).

翻译：本文深入分析了由稀疏非线性动力学识别（SINDy）方法变体生成的替代模型的误差。我们首先概述了多种非线性系统辨识技术，即SINDy、弱SINDy以及占据核方法。在假设动力学过程是基函数集的有限线性组合的前提下，这些方法通过建立矩阵方程来恢复系数。我们阐明了这些技术之间的结构相似性，并为弱SINDy技术建立了投影性质。在概述之后，我们分析了由简化版弱SINDy生成的替代模型的误差。具体而言，在假设由解给出的复合算子有界性的条件下，我们证明了：(i) 替代动力学趋近于真实动力学；(ii) 替代模型的解与真实解足够接近。最后，作为应用，我们讨论了将弱SINDy替代建模与本征正交分解（POD）相结合，为偏微分方程（PDE）构建替代模型的方法。