Green Physics-Informed Machine Learning Models For Structural Health Monitoring

Abstract: 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.

• The paper develops a Gaussian-process framework for comparing model greyness, data coverage, runtime, and emissions in structural health monitoring.
• Physics-informed models consistently reduced the data needed to reach NMSE below 10, but carbon savings occurred only when data reductions outweighed added hyperparameter costs.
• The strongest savings came from the more constrained Grey-2 model, cutting emissions by 73.7% on the toy case and 89.7% on upsampled GARTEUR data, while some cases increased emissions.
• The findings show that physics-informed ML is not automatically greener and is most promising for larger datasets where Gaussian-process training costs grow rapidly.
• Find recent papers about green physics-informed machine learning for structural health monitoring.

Motivation and scope

Machine learning (ML) adoption in structural engineering is expected to grow substantially, and with it the energy demand and carbon footprint of training and running models. While the computer science community has developed an extensive literature on sustainable ML — covering hardware choice, data centre power usage effectiveness (PUE), carbon intensity of electricity, and runtime (2604.27638) — work within engineering has largely treated ML as a tool for achieving sustainability goals rather than examining the sustainability of the models themselves. The paper by Bradley and Cross addresses this gap by asking a specific question: how does the environmental impact of a model change as physical knowledge is embedded into it?

The authors frame this through physics-informed machine learning (PIML), or "grey-box" modelling, in which physical insight is incorporated into otherwise data-driven learners. Their working hypothesis is that reduced emissions are achievable when embedded physics permits a reduction in training data without significantly increasing model complexity. Notably, they claim this is the first work to introduce physics knowledge into a model explicitly with the intention of reducing its environmental footprint.

Methodology

The study uses Gaussian process (GP) regression throughout, chosen deliberately not for computational burden but because GPs provide a framework in which physics can be embedded in multiple ways without altering the model architecture, enabling direct comparison of emissions across levels of "greyness". Three models are compared:

• Black-1: standard GP regression with constant mean and squared exponential (SE) kernel; four hyperparameters.
• Grey-1: SE kernel multiplied by a periodic kernel encoding knowledge of surface periodicity, with the period bounded within 10% of the true value; six hyperparameters.
• Grey-2: identical to Grey-1 but with the period fixed at the true value ("whiter"); five hyperparameters.

All models are trained via marginal likelihood maximisation using ADAM (1000 iterations) in GPyTorch v1.14.2, and emissions are estimated with CodeCarbon on a Dell Inspiron 5680 (Intel Core i7-8700 CPU, 32 GB RAM). Because CodeCarbon could not access live component telemetry, emissions were estimated from the CPU's thermal design power rather than measured directly — a limitation acknowledged by the authors.

A key methodological contribution is the framing of model comparison around task-specific error targets. The authors argue that engineers can often specify how good "good enough" is, allowing explicit trade-offs between model error and computational burden. Models were therefore compared at the minimum training data coverage required to achieve NMSE below 10 across nine optimisation runs (three repeats at three start points).

Two test cases are used: a synthetic "toy-box" surface $y = \sqrt{|x_2|}\sin(4x_1)$, and experimental data from the GARTEUR SM-AG19 benchmark aircraft wing, where the task is to predict acceleration response spatially and temporally in the first mode (~10 Hz), with the periodic kernel acting as a proxy for knowledge of the modal response. Two GARTEUR layouts were tested: the original sparse accelerometer layout and an upsampled variant doubling the data density per coverage interval.

Results

The central finding is that required training data coverage decreases monotonically with increasing embedded physics, but this does not consistently translate into carbon savings. On the toy-box case, both effects align favourably:

Model	Coverage	Emissions (gCO₂e)
Black-1	80%	0.471
Grey-1	60%	0.419 (−10.9%)
Grey-2	20%	0.124 (−73.7%)

On the upsampled GARTEUR data, Grey-2 again delivers large savings (0.209 gCO₂e vs 2.04 gCO₂e for Black-1, −89.7%), but Grey-1 increases emissions by 65.8% relative to Black-1 despite requiring 25% less coverage. On the original GARTEUR layout, both grey-box models increase emissions (+62.2% and +31.9%) even though coverage requirements fall. This is a genuinely contradictory result relative to the authors' hypothesis, and the paper reports it plainly.

Discussion

Because CodeCarbon used fixed constants for carbon intensity, TDP, and RAM power, emission differences reduce to runtime differences. GP regression runtime scales approximately as $\mathcal{O}(N^3) + \mathcal{O}(HN^2)$, where $N$ is training set size and $H$ the number of optimised hyperparameters. At small $N$, the hyperparameter term dominates, which explains why the six-hyperparameter Grey-1 has the longest runtime at any given coverage, followed by Grey-2 and Black-1. Physics-informed models only become cheaper when the reduction in $N$ overcomes the increase in $H$ — which occurred in the toy-box case and the upsampled GARTEUR case for Grey-2, but not elsewhere.

Two implications follow directly from these results. First, adding physics to a model can increase runtime when no training data reduction is achieved, so PIML is not automatically greener. Second, since the $\mathcal{O}(N^3)$ term grows with data size, the runtime advantage of reduced training data should widen substantially at larger $N$, suggesting that carbon savings from physics-informed GPs are more likely at industry-scale datasets than in the lightweight examples studied here. The upsampled GARTEUR results support this: runtimes diverge sharply between models in the higher-coverage regime.

Limitations and open questions

The authors are explicit that this is a preliminary investigation with deliberately simple test cases. Several limitations bear directly on the strength of the conclusions:

• Scale: both hyperparameter counts and dataset sizes are small, so absolute footprints (fractions of a gram to a few grams CO₂e) are negligible; the concept remains unvalidated at scale.
• Emissions estimation: CodeCarbon relied on TDP-based estimation with fixed carbon intensity constants rather than live measurement, so reported values reflect runtime differences only.
• Proxy physics: the periodic kernel serves as a stand-in for genuine physical insight; whether real engineering knowledge yields the same trade-off structure is unresolved.
• Unquantified trade-off: even for this simple benchmark, the relationship between hyperparameter count and training data size is not quantified analytically.

Open questions left by the paper include whether the observed trend holds for more complex GPs, for neural networks and PINNs (whose linear scaling laws with both hyperparameters and data may change the trade-off qualitatively), and what the critical data size is above which physics-informed GPs reliably reduce emissions.

Conclusion

This paper establishes a framework for evaluating ML sustainability in structural health monitoring through the lens of embedded physical knowledge. Its empirical findings are mixed but informative: increased physics consistently reduces the training data needed to meet a performance threshold, yet carbon savings materialise only when that data reduction outweighs the added cost of optimising additional hyperparameters. The evidence suggests GP-based PIML is more likely to reduce emissions at large training set sizes, motivating scale-up studies and extension to other model families before firm conclusions can be drawn.