  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.
