Semiparametric Inference and Equation Discovery with the Bayesian Machine Scientist

Hybrid modeling, combining machine learning with physical equations, is promising in many fields of science, in particular for climate and Earth Sciences, but faces challenges like interpretability, inconsistent extrapolation, lack of speed, and robust inference. Here we show that the Bayesian machine scientist, a Bayesian approach to symbolic regression is an ideal choice for the challenges in the hybrid modeling task. We formulate the hybrid Bayesian machine scientist and showcase its potential in the example of modeling ecosystem respiration with the $Q_{10}$ model. Specifically, we show that our proposed hybrid equation discovery method (i) extracts the correct equations, (ii) extrapolates better in different scenarios than the non-hybrid and deep-learning-based baselines, and (iii) is able to infer more accurately parameters of interest, even in the presence of equifinality. We anticipate a spur of development of hybrid equation discovery algorithms in the sciences to approach fully interpretable data-driven models.