Bayesian autoencoders for physics learning

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. 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 Spike-and-slab Gaussian Lasso prior. Bayesian SINDy autoencoder enables the joint discovery of governing equations and coordinate systems with 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. We further demonstrate the power of Bayesian SINDy deep learning from a broader range of physics discovery, including global temperature data, synthetic Kolmogorov flow, and real video recordings of flow over a cylinder.