From Grey-Box to Green-Box: When can Physics-Informed Machine Learning Reduce Carbon Footprints in Structural Health Monitoring?

Machine learning continues to emerge as an important tool to be utilised within structural engineering and structural health monitoring, due to its ability to accurately and quickly perform both regression and classification tasks. However, a purely data driven approach has its limitations, particularly where we lack data from relevant environmental and operational conditions, a situation that has led to the development of physics-informed machine learners for structural health monitoring. These "grey-box" models take into account the physical insight that an engineer would have about the structure they are modelling and have shown promising results in the structural engineering field among many others. This work compares black and grey-box models through a "green" lens, comparing them in terms of their environmental impact, and investigating how the high extrapolative performance of grey-box models can reduce their runtimes and therefore carbon emissions. The authors aim to develop physics-informed models with reduced computational costs, while maintaining high performance, illustrated through a structural health monitoring case study.

In modern industry, keeping complex systems reliable, safe, and efficient hinges on Prognostics and Health Management (PHM). Machine Learning (ML) has largely driven advancements in diagnostics and prognostics, yet purely data-driven models face inherent limitations, such as poor generalization, an inability to infer causal relationships, and a lack of interpretability. Physics-Informed Machine Learning (PIML) helps mitigate these limitations by incorporating prior physical knowledge directly into the ML pipeline, thereby fostering growing interest in its application to PHM. This work investigates how PIML is being leveraged in the context of PHM through a systematic literature review of 212 studies. The review introduces a four-class classification scheme, consisting of observational bias, inductive bias, learning bias, and hybrid approaches, and further categorizes studies by PHM task. Across all four classes, the reviewed studies consistently demonstrate improved predictive performance over conventional baselines across a broad range of assets, although the literature is heavily skewed toward lithium-ion batteries and bearings, and dominated by problem-specific solutions. Overall, the review indicates that physics-informed approaches already provide tangible benefits, whereas claims of improvements concerning some of the aforementioned limitations lack sufficient supporting evidence. Future research should prioritize transferable design patterns, benchmarks comparing integration strategies, and uncertainty-aware models that are lightweight and robust enough for online deployment in real-world settings.

Physics-informed machine learning (PIML) integrates mechanistic knowledge, typically in the form of partial differential equations (PDE), into data-driven models. Despite strong empirical performance, its statistical generalisation properties remain poorly understood, particularly in the regression setting with unbounded losses. Existing analyses rely on approximation or stability arguments and do not fully capture how physical structure influences generalisation from finite data. In this work, we develop a PAC-Bayesian framework for PIML that provides high-probability generalisation guarantees in the presence of unbounded losses. We adopt a multi-task perspective that jointly treats data fidelity, PDE residuals, initial and boundary conditions, avoiding the looseness induced by standard union-bound approaches. Our analysis leverages the structure of physics-informed objectives to derive novel bounds where the complexity scales with input-gradient norms of the losses, revealing a direct link between physical regularity and generalisation. We instantiate this framework under Sobolev and Poincaré-type assumptions, yielding two classes of bounds that trade off statistical complexity and smoothness in different regimes. Building on these results, we propose a self-bounding-aware learning algorithm that directly optimises tractable surrogates of the derived bounds, along with a practical procedure to estimate the associated constants in realistic settings. Empirical evaluations on standard PDE benchmarks demonstrate that our bounds are non-vacuous, significantly tighter than union-bound baselines, and can be effectively minimised during training. Overall, our results provide a principled statistical foundation for the generalisation of physics-informed models.