HyperSINDy: Deep Generative Modeling of Nonlinear Stochastic Governing
  Equations
Mozes Jacobs Bingni W. Brunton Steven L. Brunton J. Nathan Kutz Ryan V. Raut
The discovery of governing differential equations from data is an open
frontier in machine learning. The sparse identification of nonlinear dynamics
(SINDy) \citep{brunton_discovering_2016} framework enables data-driven
discovery of interpretable models in the form of sparse, deterministic
governing laws. Recent works have sought to adapt this approach to the
stochastic setting, though these adaptations are severely hampered by the curse
of dimensionality. On the other hand, Bayesian-inspired deep learning methods
have achieved widespread success in high-dimensional probabilistic modeling via
computationally efficient approximate inference techniques, suggesting the use
of these techniques for efficient stochastic equation discovery. Here, we
introduce HyperSINDy, a framework for modeling stochastic dynamics via a deep
generative model of sparse governing equations whose parametric form is
discovered from data. HyperSINDy employs a variational encoder to approximate
the distribution of observed states and derivatives. A hypernetwork
\citep{ha_hypernetworks_2016} transforms samples from this distribution into
the coefficients of a differential equation whose sparse form is learned
simultaneously using a trainable binary mask \citep{louizos_learning_2018}.
Once trained, HyperSINDy generates stochastic dynamics via a differential
equation whose coefficients are driven by a Gaussian white noise. In
experiments, HyperSINDy accurately recovers ground truth stochastic governing
equations, with learned stochasticity scaling to match that of the data.
Finally, HyperSINDy provides uncertainty quantification that scales to
high-dimensional systems. Taken together, HyperSINDy offers a promising
framework for model discovery and uncertainty quantification in real-world
systems, integrating sparse equation discovery methods with advances in
statistical machine learning and deep generative modeling.