Bayesian Learning of Coupled Biogeochemical-Physical Models

Predictive models for marine ecosystems are used for a variety of needs. Due to sparse measurements and limited understanding of the myriad of ocean processes, there is however uncertainty. There is model uncertainty in the parameter values, functional forms with diverse parameterizations, level of complexity needed, and thus in the state fields. We develop a principled Bayesian model learning methodology that allows interpolation in the space of candidate models and discovery of new models, all while estimating state fields and parameter values, as well as the joint probability distributions of all learned quantities. We address the challenges of high-dimensional and multidisciplinary dynamics governed by partial differential equations (PDEs) by using state augmentation and the computationally efficient Gaussian Mixture Model - Dynamically Orthogonal filter. Our innovations include special stochastic parameters to unify candidate models into a single general model and stochastic piecewise function approximations to generate dense candidate model spaces. They allow handling many candidate models, possibly none of which are accurate, and learning elusive unknown functional forms in compatible and embedded models. Our new methodology is generalizable and interpretable and extrapolates out of the space of models to discover new ones. We perform a series of twin experiments based on flows past a seamount coupled with three-to-five component ecosystem models, including flows with chaotic advection. We quantify learning skills, and evaluate convergence and sensitivity to hyper-parameters. Our PDE framework successfully discriminates among model candidates, learns in the absence of prior knowledge by searching in dense function spaces, and updates joint probabilities while capturing non-Gaussian statistics. The parameter values and model formulations that best explain the data are identified.