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Despite the growing availability of sensing and data in general, we remain unable to fully characterise many in-service engineering systems and structures from a purely data-driven approach. The vast data and resources available to capture human activity are unmatched in our engineered world, and, even in cases where data could be referred to as ``big,'' they will rarely hold information across operational windows or life spans. This paper pursues the combination of machine learning technology and physics-based reasoning to enhance our ability to make predictive models with limited data. By explicitly linking the physics-based view of stochastic processes with a data-based regression approach, a spectrum of possible Gaussian process models are introduced that enable the incorporation of different levels of expert knowledge of a system. Examples illustrate how these approaches can significantly reduce reliance on data collection whilst also increasing the interpretability of the model, another important consideration in this context.

The use of machine learning in Structural Health Monitoring is becoming more common, as many of the inherent tasks (such as regression and classification) in developing condition-based assessment fall naturally into its remit. This chapter introduces the concept of physics-informed machine learning, where one adapts ML algorithms to account for the physical insight an engineer will often have of the structure they are attempting to model or assess. The chapter will demonstrate how grey-box models, that combine simple physics-based models with data-driven ones, can improve predictive capability in an SHM setting. A particular strength of the approach demonstrated here is the capacity of the models to generalise, with enhanced predictive capability in different regimes. This is a key issue when life-time assessment is a requirement, or when monitoring data do not span the operational conditions a structure will undergo. The chapter will provide an overview of physics-informed ML, introducing a number of new approaches for grey-box modelling in a Bayesian setting. The main ML tool discussed will be Gaussian process regression, we will demonstrate how physical assumptions/models can be incorporated through constraints, through the mean function and kernel design, and finally in a state-space setting. A range of SHM applications will be demonstrated, from loads monitoring tasks for off-shore and aerospace structures, through to performance monitoring for long-span bridges.

The quantification of wave loading on offshore structures and components is a crucial element in the assessment of their useful remaining life. In many applications the well-known Morison's equation is employed to estimate the forcing from waves with assumed particle velocities and accelerations. This paper develops a grey-box modelling approach to improve the predictions of the force on structural members. A grey-box model intends to exploit the enhanced predictive capabilities of data-based modelling whilst retaining physical insight into the behaviour of the system; in the context of the work carried out here, this can be considered as physics-informed machine learning. There are a number of possible approaches to establish a grey-box model. This paper demonstrates two means of combining physics (white box) and data-based (black box) components; one where the model is a simple summation of the two components, the second where the white-box prediction is fed into the black box as an additional input. Here Morison's equation is used as the physics-based component in combination with a data-based Gaussian process NARX - a dynamic variant of the more well-known Gaussian process regression. Two key challenges with employing the GP-NARX formulation that are addressed here are the selection of appropriate lag terms and the proper treatment of uncertainty propagation within the dynamic GP. The best performing grey-box model, the residual modelling GP-NARX, was able to achieve a 29.13\% and 5.48\% relative reduction in NMSE over Morison's Equation and a black-box GP-NARX respectively, alongside significant benefits in extrapolative capabilities of the model, in circumstances of low dataset coverage.