# Bayesian Symbolic Regression for Missing Physics

arXiv: https://arxiv.org/abs/2603.14918 · HTML: https://mlsanity.com/paper/2603.14918/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 20 references that are in the map.
- First posted on arXiv 16 Mar 2026.

## Builds on (most cited first, 10 of 10)

- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Bayesian Symbolic Regression](https://mlsanity.com/paper/1910.08892/) (2019)
- [Bayesian Neural Ordinary Differential Equations](https://mlsanity.com/paper/2012.07244/) (2020)
- [Machine Learning for Prediction with Missing Dynamics](https://mlsanity.com/paper/1910.05861/) (2019)
- [Learning orbital dynamics of binary black hole systems from gravitational wave measurements](https://mlsanity.com/paper/2102.12695/) (2021)
- [Non-Negative Universal Differential Equations With Applications in Systems Biology](https://mlsanity.com/paper/2406.14246/) (2024)
- [Efficient hybrid modeling and sorption model discovery for non-linear advection-diffusion-sorption systems: A systematic scientific machine learning approach](https://mlsanity.com/paper/2303.13555/) (2023)
- [Non-reversible jump algorithms for Bayesian nested model selection](https://mlsanity.com/paper/1911.01340/) (2019)
- [Learning COVID-19 Regional Transmission Using Universal Differential Equations in a SIR model](https://mlsanity.com/paper/2310.16804/) (2023)

## Abstract (arXiv)

Model-based approaches for (bio)process systems often suffer from incomplete knowledge of the underlying physical, chemical, or biological laws. Universal differential equations, which embed neural networks within differential equations, have emerged as powerful tools to learn this missing physics from experimental data. However, neural networks are inherently opaque, motivating their post-processing via symbolic regression to obtain interpretable mathematical expressions. Genetic algorithm-based symbolic regression is a popular approach for this post-processing step, but provides only point estimates and cannot quantify the confidence we should place in a discovered equation. We address this limitation by applying Bayesian symbolic regression, which uses Reversible Jump Markov Chain Monte Carlo to sample from the posterior distribution over symbolic expression trees. This approach naturally quantifies uncertainty in the recovered model structure. We demonstrate the methodology on a Lotka-Volterra predator-prey system and then show how a well-designed experiment leads to lower uncertainty in a fed-batch bioreactor case study.