Weak SINDy: A Data-Driven Galerkin Method for System Identification
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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 subsequent variance reduction techniques and improves on the standard SINDy algorithm by orders of magnitude. We first show that in the noise-free regime, this so-called Weak SINDy 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 measurement noise and recovers approximately twice the significant digits of the standard SINDy algorithm 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.
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