Physical Covariance Functions for Dynamic Systems with Time-Dependent Parameters

As monitoring data becomes increasingly available, it is natural for structural health monitoring practitioners to turn towards data-driven models. Despite the expressive capability and flexibility of such models, their predictive performance relies on access to suitably represenative training data, which in structural health monitoring, equates to data that span the full environmental and operational envelope of the structure. Additionally, given the black-box nature of such models, there is no guarantee that predictions will adhere to fundamental physical principles. In an attempt to address these limitations, recent attention has been directed towards models that seek to combine physical insight with traditional machine learning techniques, referred to generally as physics-informed machine learning or grey-box modelling. In this paper, we seek to incorporate physical insight into a Gaussian process through the use of covariance functions that explicitly capture the evolution of dynamic systems in time. Specifically, within an autoregressive setting, we begin by deriving the covariance for the approximate response of a single degree of freedom system. Consideration is then given to how such kernels may be used when the governing parameters in the equations of motion vary over time, which is investigated here through a varying temperature, and consequently, structural stiffness. It is demonstrated that the derived grey-box models are able to outperform equivalent physics-based and data-driven models over a number of simulated case studies.