Bayesian Inference for Missing Physics
This work provides a method for quantifying uncertainty in interpretable models for process systems, which is incremental as it builds on existing symbolic regression techniques.
The paper tackles the problem of learning missing physics in model-based systems using neural networks embedded in differential equations, but addresses the opacity of neural networks by applying Bayesian symbolic regression to quantify uncertainty in discovered equations, demonstrating lower uncertainty in a fed-batch bioreactor case study.
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.