# A Bayesian Framework for learning governing Partial Differential Equation from Data

arXiv: https://arxiv.org/abs/2306.04894 · HTML: https://mlsanity.com/paper/2306.04894/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 39 references that are in the map.
- First posted on arXiv 8 Jun 2023.

## Builds on (most cited first, 6 of 6)

- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [On spike-and-slab priors for Bayesian equation discovery of nonlinear dynamical systems via sparse linear regression](https://mlsanity.com/paper/2012.01937/) (2020)
- [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)

## Built on by (most cited first, 8 of 8)

- [MDBench: Benchmarking Data-Driven Methods for Model Discovery](https://mlsanity.com/paper/2509.20529/) (2025)
- [Discovering stochastic partial differential equations from limited data using variational Bayes inference](https://mlsanity.com/paper/2306.15873/) (2023)
- [BINDy -- Bayesian identification of nonlinear dynamics with reversible-jump Markov-chain Monte-Carlo](https://mlsanity.com/paper/2408.08062/) (2024)
- [Statistical Mechanics of Dynamical System Identification](https://mlsanity.com/paper/2403.01723/) (2024)
- [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)
- [Bayesian Reasoning for Physics Informed Neural Networks](https://mlsanity.com/paper/2308.13222/) (2023)
- [See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs](https://mlsanity.com/paper/2608.27869/) (2026)

## Abstract (arXiv)

The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to note that existing methods often struggle to identify the underlying equation accurately in the presence of noise. In this study, we present a new approach to discovering PDEs by combining variational Bayes and sparse linear regression. The problem of PDE discovery has been posed as a problem to learn relevant basis from a predefined dictionary of basis functions. To accelerate the overall process, a variational Bayes-based approach for discovering partial differential equations is proposed. To ensure sparsity, we employ a spike and slab prior. We illustrate the efficacy of our strategy in several examples, including Burgers, Korteweg-de Vries, Kuramoto Sivashinsky, wave equation, and heat equation (1D as well as 2D). Our method offers a promising avenue for discovering PDEs from data and has potential applications in fields such as physics, engineering, and biology.