# Discovering stochastic partial differential equations from limited data using variational Bayes inference

arXiv: https://arxiv.org/abs/2306.15873 · HTML: https://mlsanity.com/paper/2306.15873/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 51 references that are in the map.
- First posted on arXiv 28 Jun 2023.

## Builds on (most cited first, 22 of 22)

- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [Automated discovery of generalized standard material models with EUCLID](https://mlsanity.com/paper/2211.04453/) (2022)
- [On spike-and-slab priors for Bayesian equation discovery of nonlinear dynamical systems via sparse linear regression](https://mlsanity.com/paper/2012.01937/) (2020)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [Bayesian-EUCLID: discovering hyperelastic material laws with uncertainties](https://mlsanity.com/paper/2203.07422/) (2022)
- [Adaptable Hamiltonian neural networks](https://mlsanity.com/paper/2102.13235/) (2021)
- [A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise](https://mlsanity.com/paper/2005.03769/) (2020)
- [Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data](https://mlsanity.com/paper/2108.04085/) (2021)
- [Automated Learning of Interpretable Models with Quantified Uncertainty](https://mlsanity.com/paper/2205.01626/) (2022)
- [A Bayesian Framework for learning governing Partial Differential Equation from Data](https://mlsanity.com/paper/2306.04894/) (2023)
- [Model-agnostic stochastic model predictive control](https://mlsanity.com/paper/2211.13012/) (2022)
- [MAntRA: A framework for model agnostic reliability analysis](https://mlsanity.com/paper/2212.06303/) (2022)
- [Probabilistic machine learning based predictive and interpretable digital twin for dynamical systems](https://mlsanity.com/paper/2212.09240/) (2022)
- [Learning governing physics from output only measurements](https://mlsanity.com/paper/2208.05609/) (2022)

## Built on by (most cited first, 6 of 6)

- [Statistical Mechanics of Dynamical System Identification](https://mlsanity.com/paper/2403.01723/) (2024)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [A Bayesian framework for discovering interpretable Lagrangian of dynamical systems from data](https://mlsanity.com/paper/2310.06241/) (2023)
- [Discovering governing equation in structural dynamics from acceleration-only measurements](https://mlsanity.com/paper/2407.13704/) (2024)
- [Principled model selection for stochastic dynamics](https://mlsanity.com/paper/2501.10339/) (2025)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)

## Abstract (arXiv)

We propose a novel framework for discovering Stochastic Partial Differential Equations (SPDEs) from data. The proposed approach combines the concepts of stochastic calculus, variational Bayes theory, and sparse learning. We propose the extended Kramers-Moyal expansion to express the drift and diffusion terms of an SPDE in terms of state responses and use Spike-and-Slab priors with sparse learning techniques to efficiently and accurately discover the underlying SPDEs. The proposed approach has been applied to three canonical SPDEs, (a) stochastic heat equation, (b) stochastic Allen-Cahn equation, and (c) stochastic Nagumo equation. Our results demonstrate that the proposed approach can accurately identify the underlying SPDEs with limited data. This is the first attempt at discovering SPDEs from data, and it has significant implications for various scientific applications, such as climate modeling, financial forecasting, and chemical kinetics.