Robust equation discovery considering model discrepancy: a sparse Bayesian and Gaussian process approach

Computational models are widely used to describe engineering systems and to predict their behaviour. However, in many applications, these computational models do not capture the complete physics of the real system, leading to model discrepancy. Model discrepancy causes bias in the inferred model parameters when it is not properly accounted for. This paper proposes a novel approach that seeks to capture the functional form of the model discrepancy and reduce bias in the estimated model parameters through and equation discovery procedure. A sparse Bayesian model is proposed, where sparsity is introduced through a hierarchical prior structure, providing a mechanism for removing erroneous candidate model terms from a series of potential equations as a part of an equation discovery procedure. At the same time, a Gaussian process model is used to account for model discrepancy. These two modelling assumptions are combined in a Bayesian formulation, allowing the system parameters and model discrepancy to be inferred in a probabilistic manner with associated uncertainties quantified based on their posterior distributions. The resulting method is capable of simultaneously providing physical insights into the system behaviour, by selecting the appropriate candidate model components and their respective system parameters, whilst compensating for model discrepancy that may occur due to an incomplete set of candidate terms. In order to efficiently solve the statistical model, an expectation maximisation algorithm is proposed for performing inference, and illustrative examples are presented to validate the proposed method. It is shown that compared to using a conventional sparse Bayesian approach for performing equation discovery, such as the Relevance Vector Machine, the proposed method provides better equation selection and parameter estimation, with less bias in the parameter estimates.