  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.
