We are interested in machine-learning a PDE and/or a set of governing equations, with all relevant constitutive or closure relations, by combining partial knowledge and data. The initial condition for learning includes all relevant forms and items of existing domain knowledge, be it explicit or implicit, and all equipped with statements of (un)certainty. In areas where knowledge is incomplete, we can discover previously unknown relations of interest while providing statements of (un)certainty. The chosen learning paradigm should encode causal relations and useful problem structures in its own internal structure, ensuring that the learned relations are causal and interpretable for scientific modelling. To approach this vision, we develop B-CUDE, a Bayesian, causal extension for the recent concept of universal differential equations (UDE). UDEs are known to retain the causal structure of a system as far as known, even in aspects delegated to learning, and this aspect helps to ensure interpretability. Building upon our previously developed Finite Volume Neural Network (FINN), our emphasis lies in systematically representing relevant domain knowledge using a Bayesian framework, ensuring algorithmic efficiency for the resulting Bayesian learning task, and translating the findings into symbolic expressions with quantified uncertainty. Additionally, we address specific modelling challenges in demonstrator cases. As our motivating problem, we focus on multi-phase flow (MPF), a complex system where scientific modelling faces challenges such as non-equilibrium laws for hysteretic and dynamic effects. Data for MPF microfluidic experiments are provided by partners from Project Network 1 and CRC 1313.
This motivating problem is representative of many others: PDE-based models exist but still fail, as they rest on simplified/uncertain assumptions. However, experimental data are too scarce to rely solely on data-driven models while ignoring uncertainties. Therefore, it is advisable to incorporate physics-informed, causal learning. Conducting (Bayesian) uncertainty analysis becomes inevitable. The benefits of innovative approaches like B-CUDE are essential in advancing the scientific field.
