Bayesian Symbolic Regression for Missing Physics

TL;DR

This paper introduces Bayesian symbolic regression using Reversible Jump MCMC to quantify uncertainty in symbolic models derived from neural network-embedded differential equations, enhancing interpretability and confidence in scientific modeling.

Contribution

It presents a novel Bayesian symbolic regression method that provides uncertainty quantification for models obtained from neural differential equations, improving interpretability and reliability.

Findings

Successfully applied to Lotka-Volterra system

Lower uncertainty achieved with well-designed experiments in bioreactor case study

Quantifies confidence in symbolic expressions derived from experimental data

Abstract

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.