# Inferring the Langevin Equation with Uncertainty via Bayesian Neural Networks

arXiv: https://arxiv.org/abs/2402.01338 · HTML: https://mlsanity.com/paper/2402.01338/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 110 references that are in the map.
- First posted on arXiv 2 Feb 2024.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Pitfalls of In-Domain Uncertainty Estimation and Ensembling in Deep Learning](https://mlsanity.com/paper/2002.06470/) (2020)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Adv-BNN: Improved Adversarial Defense through Robust Bayesian Neural Network](https://mlsanity.com/paper/1810.01279/) (2018)
- [Uncertainty Quantification with Statistical Guarantees in End-to-End Autonomous Driving Control](https://mlsanity.com/paper/1909.09884/) (2019)
- [GradDiv: Adversarial Robustness of Randomized Neural Networks via Gradient Diversity Regularization](https://mlsanity.com/paper/2107.02425/) (2021)
- [A Primer on Bayesian Neural Networks: Review and Debates](https://mlsanity.com/paper/2309.16314/) (2023)
- [How Tempering Fixes Data Augmentation in Bayesian Neural Networks](https://mlsanity.com/paper/2205.13900/) (2022)
- [Learning entropy production via neural networks](https://mlsanity.com/paper/2003.04166/) (2020)
- [Detection of Gravitational Waves Using Bayesian Neural Networks](https://mlsanity.com/paper/2007.04176/) (2020)

## Built on by (most cited first, 4 of 4)

- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [Learning Stochastic Thermodynamics Directly from Correlation and Trajectory-Fluctuation Currents](https://mlsanity.com/paper/2504.19007/) (2025)
- [BLADE: Bayesian Langevin Active Discovery with Replica Exchange for Identification of Complex Systems](https://mlsanity.com/paper/2503.02983/) (2025)
- [Identifying parameter couplings and uncertainties of mixed-noise stochastic systems via full-covariance Gaussian mixture network](https://mlsanity.com/paper/2608.15198/) (2026)

## Abstract (arXiv)

Pervasive across diverse domains, stochastic systems exhibit fluctuations in processes ranging from molecular dynamics to climate phenomena. The Langevin equation has served as a common mathematical model for studying such systems, enabling predictions of their temporal evolution and analyses of thermodynamic quantities, including absorbed heat, work done on the system, and entropy production. However, inferring the Langevin equation from observed trajectories is a challenging problem, and assessing the uncertainty associated with the inferred equation has yet to be accomplished. In this study, we present a comprehensive framework that employs Bayesian neural networks for inferring Langevin equations in both overdamped and underdamped regimes. Our framework first provides the drift force and diffusion matrix separately and then combines them to construct the Langevin equation. By providing a distribution of predictions instead of a single value, our approach allows us to assess prediction uncertainties, which can help prevent potential misunderstandings and erroneous decisions about the system. We demonstrate the effectiveness of our framework in inferring Langevin equations for various scenarios including a neuron model and microscopic engine, highlighting its versatility and potential impact.