Convergence of weak-SINDy Surrogate Models

TL;DR

This work analyzes error behavior of surrogate models produced by weak-SINDy, proving convergence of surrogate dynamics to the true dynamics under a bounded composition operator and providing quantitative bounds on the surrogate trajectory error. It frames weak-SINDy as a projection onto a finite-dimensional basis in a Hilbert space and extends the theory from scalar ODEs to systems and POD-based reduced-order models for PDEs. The paper couples rigorous error estimates with extensive numerical experiments, including scalar ODEs of varying regularity and POD discretizations of diffusion equations with both constant and discontinuous diffusivity, demonstrating spectral convergence when the basis and test spaces are aligned. The results support the use of weak-SINDy as a robust surrogate modeling approach for nonlinear dynamics and POD-reduced PDE representations, with error dictated by the regularity of the dynamics and the composition operator.

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 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).

Figures (15)

• Figure 1: The total approximation error (L) [solid blue lines] and the upper bound (R1)$+$(R2)$+$(R3) [dashed orange lines] plotted versus the degree of basis functions $J=1,\dots,30$, for various degrees of test functions $K=$ 5, 10, and 20. A vertical black dashed line is plotted at $J=K$.
• Figure 2: The error terms (R1), (R2), and (R3) plotted versus the degree of basis functions $J=1,\dots,30$ in green, blue, and red lines, respectively, for various degrees of test functions $K=$ 5, 10, and 20. A vertical black dashed line is plotted at $J=K$.
• Figure 3: Total approximation error (L), the individual terms (R1), (R2), (R3), and the upper bound (R1)$+$(R2)$+$(R3) plotted versus the degree of basis functions $J=1,\dots,30$ with the degree of Fourier basis $K=10$, which results in $2K+1=21$ Fourier test functions.
• Figure 4: Solution error $|x(t)-\hat{x}(t)|$ (blue) and the estimate (red) given in Proposition \ref{['prop:Lipschitz']} are plotted on time interval $[0,0.4]$.
• Figure 5: Figure \ref{['fig:function-plots']} plots $g_\alpha$ over the domain $[-2, 2]$ for $\alpha \in \{1,2,3,4\}$. Figure \ref{['fig:varying_alpha-L2 diff']} plots the projection error (R2) at $K = 20$ and varying degree $J= 1,2,\ldots, 20$ for each $\alpha$, with dashed lines indicating the expected convergence rate of (R2) in $J$ from Proposition \ref{['prop:minproblem']}.

Theorems & Definitions (68)

• Remark 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• Proof 1
• Proposition 2.7
• Proof 2
• Definition 2.8: Degree of a multi-variable polynomial