Weak SINDy: Galerkin-Based Data-Driven Model Selection

Abstract

We present a weak formulation and discretization of the system discovery problem from noisy measurement data. This method of learning differential equations from data fits into a new class of algorithms that replace pointwise derivative approximations with linear transformations and a variance reduction technique. Our approach improves on the standard SINDy algorithm by orders of magnitude. We first show that in the noise-free regime, this so-called Weak SINDy (WSINDy) framework is capable of recovering the dynamic coefficients to very high accuracy, with the number of significant digits equal to the tolerance of the data simulation scheme. Next we show that the weak form naturally accounts for white noise by identifying the correct nonlinearities with coefficient error scaling favorably with the signal-to-noise ratio while significantly reducing the size of linear systems in the algorithm. In doing so, we combine the ease of implementation of the SINDy algorithm with the natural noise-reduction of integration to arrive at a more robust and user-friendly method of sparse recovery that correctly identifies systems in both small-noise and large-noise regimes.

Added link to code on Github

References in corpus (3)

• Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems
• Sparse Identification of Nonlinear Dynamical Systems via Reweighted ${\ell }_1$-regularized Least Squares
• Discovery of Dynamics Using Linear Multistep Methods

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• Derivative-based SINDy (DSINDy): Addressing the challenge of discovering governing equations from noisy data
• Learning Mean-Field Equations from Particle Data Using WSINDy
• Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data
• WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming
• Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Population