On Higher Order Drift and Diffusion Estimates for Stochastic SINDy

TL;DR

This paper enhances the SINDy algorithm for stochastic differential equations by developing higher-order estimates for drift and diffusion functions, improving accuracy and feasibility in finite data scenarios.

Contribution

It introduces higher-order estimation methods for drift and diffusion in stochastic SINDy, reducing error and improving practical applicability with finite data.

Findings

Higher-order estimates improve accuracy of drift and diffusion functions.

Enhanced estimates enable more feasible system identification with limited data.

The proposed methods outperform previous lower-order approaches.

Abstract

The Sparse Identification of Nonlinear Dynamics (SINDy) algorithm can be applied to stochastic differential equations to estimate the drift and the diffusion function using data from a realization of the SDE. The SINDy algorithm requires sample data from each of these functions, which is typically estimated numerically from the data of the state. We analyze the performance of the previously proposed estimates for the drift and diffusion function to give bounds on the error for finite data. However, since this algorithm only converges as both the sampling frequency and the length of trajectory go to infinity, obtaining approximations within a certain tolerance may be infeasible. To combat this, we develop estimates with higher orders of accuracy for use in the SINDy framework. For a given sampling frequency, these estimates give more accurate approximations of the drift and diffusion functions, making SINDy a far more feasible system identification method.