A toolkit for data-driven discovery of governing equations in high-noise regimes

TL;DR

This paper introduces a comprehensive toolkit for accurately discovering governing equations from high-noise time-series data using the SINDy framework, including methods to improve robustness and assess model accuracy.

Contribution

The paper presents novel extensions to the SINDy method for high-noise data and a technique to evaluate model accuracy amidst non-uniqueness caused by noise.

Findings

Successfully extracts sparse governing equations from data with up to 300% noise.

Achieves low coefficient estimate errors (1-8%) in high-noise regimes.

Provides a model accuracy assessment method for noisy data scenarios.

Abstract

We consider the data-driven discovery of governing equations from time-series data in the limit of high noise. The algorithms developed describe an extensive toolkit of methods for circumventing the deleterious effects of noise in the context of the sparse identification of nonlinear dynamics (SINDy) framework. We offer two primary contributions, both focused on noisy data acquired from a system x' = f(x). First, we propose, for use in high-noise settings, an extensive toolkit of critically enabling extensions for the SINDy regression method, to progressively cull functionals from an over-complete library and yield a set of sparse equations that regress to the derivate x'. These innovations can extract sparse governing equations and coefficients from high-noise time-series data (e.g. 300% added noise). For example, it discovers the correct sparse libraries in the Lorenz system, with median coefficient estimate errors equal to 1% - 3% (for 50% noise), 6% - 8% (for 100% noise); and 23% - 25% (for 300% noise). The enabling modules in the toolkit are combined into a single method, but the individual modules can be tactically applied in other equation discovery methods (SINDy or not) to improve results on high-noise data. Second, we propose a technique, applicable to any model discovery method based on x' = f(x), to assess the accuracy of a discovered model in the context of non-unique solutions due to noisy data. Currently, this non-uniqueness can obscure a discovered model's accuracy and thus a discovery method's effectiveness. We describe a technique that uses linear dependencies among functionals to transform a discovered model into an equivalent form that is closest to the true model, enabling more accurate assessment of a discovered model's accuracy.