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Recent progress in autoencoder-based sparse identification of nonlineardynamics (SINDy) under $\ell_1$ constraints allows joint discoveries ofgoverning equations and latent coordinate systems from spatio-temporal data,including simulated video frames. However, it is challenging for $\ell_1$-basedsparse inference to perform correct identification for real data due to thenoisy measurements and often limited sample sizes. To address the data-drivendiscovery of physics in the low-data and high-noise regimes, we proposeBayesian SINDy autoencoders, which incorporate a hierarchical Bayesiansparsifying prior: Spike-and-slab Gaussian Lasso. Bayesian SINDy autoencoderenables the joint discovery of governing equations and coordinate systems witha theoretically guaranteed uncertainty estimate. To resolve the challengingcomputational tractability of the Bayesian hierarchical setting, we adapt anadaptive empirical Bayesian method with Stochatic gradient Langevin dynamics(SGLD) which gives a computationally tractable way of Bayesian posteriorsampling within our framework. Bayesian SINDy autoencoder achieves betterphysics discovery with lower data and fewer training epochs, along with validuncertainty quantification suggested by the experimental studies. The BayesianSINDy autoencoder can be applied to real video data, with accurate physicsdiscovery which correctly identifies the governing equation and provides aclose estimate for standard physics constants like gravity $g$, for example, invideos of a pendulum.

Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants