Convergence of weak-SINDy Surrogate Models

Abstract

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 nonlinear 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 linear system 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).

Coarse-Graining Hamiltonian Systems Using WSINDy

“…Ultimately, the main purpose of this work is to demonstrate that the weak form itself has inherent temporal coarse-graining capabilities, which are especially useful in reduced-order Hamiltonian modeling. We note that combinations of weak-form equation learning with other reduced-order modeling paradigms is possible, as exhibited by previous works in other contexts (.e.g POD-based methods [33,34] and Neural Networks [35]), and we leave these synergies to future work.…”

Data-driven inference of complex system dynamics: A mini-review

“…Due to the unavoidable noise in empirical data, the numerical calculation of derivatives Ẋ could be inaccurate. Hence, integral formulations of SINDy and occupation kernel techniques are developed, which are robust against observational noises [54][55][56][57]. For more complex functions which are difficult to infer by SINDy without prior knowledge, such as rational nonlinearity with cross terms, Mangan et al proposed a variant of SINDy by assembling function library with time derivative components, enabling the discovery of an implicit equation [50] (see fig.…”