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

arXiv: https://arxiv.org/abs/2211.10575 · HTML: https://mlsanity.com/paper/2211.10575/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 118 references that are in the map.
- First posted on arXiv 19 Nov 2022.

## Builds on (most cited first, 30 of 40)

- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Learning Sparse Neural Networks through L0 Regularization](https://mlsanity.com/paper/1712.01312/) (2017)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [Embed to Control: A Locally Linear Latent Dynamics Model for Control from Raw Images](https://mlsanity.com/paper/1506.07365/) (2015)
- [Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning](https://mlsanity.com/paper/1902.03932/) (2019)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Evaluating Scalable Bayesian Deep Learning Methods for Robust Computer Vision](https://mlsanity.com/paper/1906.01620/) (2019)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [Training Sparse Neural Networks](https://mlsanity.com/paper/1611.06694/) (2016)
- [A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1702.05575/) (2017)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Posterior Concentration for Sparse Deep Learning](https://mlsanity.com/paper/1803.09138/) (2018)
- [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018)
- [Quantifying Uncertainty in Deep Spatiotemporal Forecasting](https://mlsanity.com/paper/2105.11982/) (2021)
- [Efficient Variational Inference for Sparse Deep Learning with Theoretical Guarantee](https://mlsanity.com/paper/2011.07439/) (2020)
- [A Unified Framework for Sparse Relaxed Regularized Regression: SR3](https://mlsanity.com/paper/1807.05411/) (2018)
- [Non-convex Learning via Replica Exchange Stochastic Gradient MCMC](https://mlsanity.com/paper/2008.05367/) (2020)

## Built on by (most cited first, 24 of 24)

- [GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder](https://mlsanity.com/paper/2308.05882/) (2023)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2501.13329/) (2025)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Multi-fidelity reduced-order surrogate modeling](https://mlsanity.com/paper/2309.00325/) (2023)
- [Convergence of uncertainty estimates in Ensemble and Bayesian sparse model discovery](https://mlsanity.com/paper/2301.12649/) (2023)
- [VENI, VINDy, VICI: a generative reduced-order modeling framework with uncertainty quantification](https://mlsanity.com/paper/2405.20905/) (2024)
- [Shallow Recurrent Decoder for Reduced Order Modeling of Plasma Dynamics](https://mlsanity.com/paper/2405.11955/) (2024)
- [Statistical Mechanics of Dynamical System Identification](https://mlsanity.com/paper/2403.01723/) (2024)
- [Machine Learning Applications to Computational Plasma Physics and Reduced-Order Plasma Modeling: A Perspective](https://mlsanity.com/paper/2409.02349/) (2024)
- [Mesh-free sparse identification of nonlinear dynamics](https://mlsanity.com/paper/2505.16058/) (2025)
- [T-SHRED: Symbolic Regression for Regularization and Model Discovery with Transformer Shallow Recurrent Decoders](https://mlsanity.com/paper/2506.15881/) (2025)
- [Interpretable and Efficient Data-driven Discovery and Control of Distributed Systems](https://mlsanity.com/paper/2411.04098/) (2024)
- [Recurrent Deep Kernel Learning of Dynamical Systems](https://mlsanity.com/paper/2405.19785/) (2024)
- [Discovering Car-following Dynamics from Trajectory Data through Deep Learning](https://mlsanity.com/paper/2408.00251/) (2024)
- [SODAs: Sparse Optimization for the Discovery of Differential and Algebraic Equations](https://mlsanity.com/paper/2503.05993/) (2025)
- [UQ-SHRED: uncertainty quantification of shallow recurrent decoder networks for sparse sensing via engression](https://mlsanity.com/paper/2604.01305/) (2026)
- [Discovering Latent Response Laws in Forced Physical Systems](https://mlsanity.com/paper/2607.09801/) (2026)
- [Physics-informed Discovery of State Variables in Second-Order and Hamiltonian Systems](https://mlsanity.com/paper/2408.11691/) (2024)
- [Optimizing Hard Thresholding for Sparse Model Discovery](https://mlsanity.com/paper/2504.20256/) (2025)
- [A joint optimization approach to identifying sparse dynamics using least squares kernel collocation](https://mlsanity.com/paper/2511.18555/) (2025)
- [Learning Control-Affine Reduced-Order Models via Autoencoders](https://mlsanity.com/paper/2606.05045/) (2026)
- [An Introduction to Sparse Identification of Nonlinear Dynamics for Engineering Applications](https://mlsanity.com/paper/2607.15077/) (2026)
- [ViBR-WM: Visual Bayesian Regression for World Modeling](https://mlsanity.com/paper/2609.33844/) (2026)

## Abstract (arXiv)

Recent progress in autoencoder-based sparse identification of nonlinear dynamics (SINDy) under ℓ1 constraints allows joint discoveries of governing equations and latent coordinate systems from spatio-temporal data, including simulated video frames. However, it is challenging for ℓ1-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.