  Recent progress in autoencoder-based sparse identification of nonlinear
dynamics (SINDy) under $\ell_1$ constraints allows joint discoveries of
governing equations and latent coordinate systems from spatio-temporal data,
including simulated video frames. However, it is challenging for $\ell_1$-based
sparse inference to perform correct identification for real data due to the
noisy measurements and often limited sample sizes. To address the data-driven
discovery of physics in the low-data and high-noise regimes, we propose
Bayesian SINDy autoencoders, which incorporate a hierarchical Bayesian
sparsifying prior: Spike-and-slab Gaussian Lasso. Bayesian SINDy autoencoder
enables the joint discovery of governing equations and coordinate systems with
a theoretically guaranteed uncertainty estimate. To resolve the challenging
computational tractability of the Bayesian hierarchical setting, we adapt an
adaptive empirical Bayesian method with Stochatic gradient Langevin dynamics
(SGLD) which gives a computationally tractable way of Bayesian posterior
sampling within our framework. Bayesian SINDy autoencoder achieves better
physics discovery with lower data and fewer training epochs, along with valid
uncertainty quantification suggested by the experimental studies. The Bayesian
SINDy autoencoder can be applied to real video data, with accurate physics
discovery which correctly identifies the governing equation and provides a
close estimate for standard physics constants like gravity $g$, for example, in
videos of a pendulum.
