论文标题
虚弱的替代模型的收敛
Convergence of weak-SINDy Surrogate Models
论文作者
论文摘要
在本文中，我们为由非线性动力学稀疏识别（Sindy）方法的变体生成的替代模型提供了深入的误差分析。我们首先概述各种非线性系统识别技术，即Sindy，Sindy和Sivectation kernel方法。假设动力学是一组基础函数的有限线性组合，这些方法建立了矩阵方程以恢复系数。我们阐明了这些技术之间的结构相似性，并为弱势技术建立了投影特性。概述之后，我们分析了由弱源的简化版本生成的替代模型的误差。特别是，在解决方案给出的组成算子的界限的假设下，我们表明（i）替代动力学收敛于真实动力学，以及（ii）替代模型的解相当接近真实的解决方案。最后，作为应用程序，我们讨论了使用弱替代替代建模和正确的正交分解（POD）的组合来构建偏微分方程（PDE）的替代模型。
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).
