Overview
Research introduces an application of Bayesian symbolic regression for addressing missing physics within model-based approaches, particularly in (bio)process systems. The methodology employs Reversible Jump Markov Chain Monte Carlo (RJMCMC) to sample from the posterior distribution of symbolic expression trees, thereby quantifying uncertainty in the recovered model structure. This contrasts with genetic algorithm-based symbolic regression, which provides point estimates without confidence quantification.
Research Context
Model-based approaches in (bio)process systems often contend with incomplete knowledge concerning underlying physical, chemical, or biological laws. Universal differential equations (UDEs), which embed neural networks within differential equations, serve as tools for learning this missing physics directly from experimental data. However, neural networks are characterized by their inherent opacity. This characteristic motivates post-processing using symbolic regression techniques to derive interpretable mathematical expressions from the UDEs' output. Traditional genetic algorithm-based symbolic regression, while a popular post-processing method, produces only point estimates, lacking a mechanism to quantify the confidence associated with a discovered equation.
Approach
The research addresses the limitation of quantifying confidence by applying Bayesian symbolic regression. This approach leverages Reversible Jump Markov Chain Monte Carlo (RJMCMC) to perform sampling from the posterior distribution over symbolic expression trees. Sampling from this posterior distribution enables the quantification of uncertainty in the identified model structure. The methodology's application was demonstrated across two distinct case studies.
Findings
- Bayesian symbolic regression quantifies uncertainty in the recovered model structure by sampling from the posterior distribution over symbolic expression trees.
- This methodology was demonstrated on a Lotka-Volterra predator-prey system.
- In a fed-batch bioreactor case study, a direct relationship was observed between a well-designed experiment and lower uncertainty in the recovered model.