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We consider the data-driven discovery of governing equations from time-seriesdata in the limit of high noise. The algorithms developed describe an extensivetoolkit of methods for circumventing the deleterious effects of noise in thecontext of the sparse identification of nonlinear dynamics (SINDy) framework.We offer two primary contributions, both focused on noisy data acquired from asystem x' = f(x). First, we propose, for use in high-noise settings, anextensive toolkit of critically enabling extensions for the SINDy regressionmethod, to progressively cull functionals from an over-complete library andyield a set of sparse equations that regress to the derivate x'. Theseinnovations can extract sparse governing equations and coefficients fromhigh-noise time-series data (e.g. 300% added noise). For example, it discoversthe correct sparse libraries in the Lorenz system, with median coefficientestimate errors equal to 1% - 3% (for 50% noise), 6% - 8% (for 100% noise); and23% - 25% (for 300% noise). The enabling modules in the toolkit are combinedinto a single method, but the individual modules can be tactically applied inother equation discovery methods (SINDy or not) to improve results onhigh-noise data. Second, we propose a technique, applicable to any modeldiscovery method based on x' = f(x), to assess the accuracy of a discoveredmodel in the context of non-unique solutions due to noisy data. Currently, thisnon-uniqueness can obscure a discovered model's accuracy and thus a discoverymethod's effectiveness. We describe a technique that uses linear dependenciesamong functionals to transform a discovered model into an equivalent form thatis closest to the true model, enabling more accurate assessment of a discoveredmodel's accuracy.

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