  We propose a fast probabilistic framework for identifying differential
equations governing the dynamics of observed data. We recast the SINDy method
within a Bayesian framework and use Gaussian approximations for the prior and
likelihood to speed up computation. The resulting method, Bayesian-SINDy, not
only quantifies uncertainty in the parameters estimated but also is more robust
when learning the correct model from limited and noisy data. Using both
synthetic and real-life examples such as Lynx-Hare population dynamics, we
demonstrate the effectiveness of the new framework in learning correct model
equations and compare its computational and data efficiency with existing
methods. Because Bayesian-SINDy can quickly assimilate data and is robust
against noise, it is particularly suitable for biological data and real-time
system identification in control. Its probabilistic framework also enables the
calculation of information entropy, laying the foundation for an active
learning strategy.
