Keith Worden

Tristan Gowdridge, Nikolaos Devilis, Keith Worden

This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of data, from the homology of a simplicial complex, calculated over a range of values. The required background theory and a method of computing persistent homology is presented here, with applications specific to structural health monitoring. These results allow for topological inference and the ability to deduce features in higher-dimensional data, that might otherwise be overlooked. A simplicial complex is constructed for data for a given distance parameter. This complex encodes information about the local proximity of data points. A singular homology value can be calculated from this simplicial complex. Extending this idea, the distance parameter is given for a range of values, and the homology is calculated over this range. The persistent homology is a representation of how the homological features of the data persist over this interval. The result is characteristic to the data. A method that allows for the comparison of the persistent homology for different data sets is also discussed.

Jack Poole, Paul Gardner, Nikolaos Dervilis et al.

The practical application of structural health monitoring (SHM) is often limited by the availability of labelled data. Transfer learning - specifically in the form of domain adaptation (DA) - gives rise to the possibility of leveraging information from a population of physical or numerical structures, by inferring a mapping that aligns the feature spaces. Typical DA methods rely on nonparametric distance metrics, which require sufficient data to perform density estimation. In addition, these methods can be prone to performance degradation under class imbalance. To address these issues, statistic alignment (SA) is discussed, with a demonstration of how these methods can be made robust to class imbalance, including a special case of class imbalance called a partial DA scenario. SA is demonstrated to facilitate damage localisation with no target labels in a numerical case study, outperforming other state-of-the-art DA methods. It is then shown to be capable of aligning the feature spaces of a real heterogeneous population, the Z24 and KW51 bridges, with only 220 samples used from the KW51 bridge. Finally, in scenarios where more complex mappings are required for knowledge transfer, SA is shown to be a vital pre-processing tool, increasing the performance of established DA methods.

Tristan Gowdridge, Nikolaos Dervilis, Keith Worden

Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be introduced. The main tool within topological data analysis (TDA) is persistent homology. Persistent homology is a method of quantifying the shape of data over a range of length scales. The required background and a method of computing persistent homology is briefly discussed in this work. Ideas from topological data analysis are then used for nonlinear dynamics to analyse some common attractors, by calculating their embedding dimension, and then to assess their general topologies. A method will also be proposed, that uses topological data analysis to determine the optimal delay for a time-delay embedding. TDA will also be applied to a Z24 Bridge case study in structural health monitoring, where it will be used to scrutinise different data partitions, classified by the conditions at which the data were collected. A metric, from topological data analysis, is used to compare data between the partitions. The results presented demonstrate that the presence of damage alters the manifold shape more significantly than the effects present from temperature.

Aidan J. Hughes, Jack Poole, Nikolaos Dervilis et al.

Population-based structural health monitoring (PBSHM), seeks to address some of the limitations associated with data scarcity that arise in traditional SHM. A tenet of the population-based approach to SHM is that information can be shared between sufficiently-similar structures in order to improve predictive models. Transfer learning techniques, such as domain adaptation, have been shown to be a highly-useful technology for sharing information between structures when developing statistical classifiers for PBSHM. Nonetheless, transfer-learning techniques are not without their pitfalls. In some circumstances, for example if the data distributions associated with the structures within a population are dissimilar, applying transfer-learning methods can be detrimental to classification performance -- this phenomenon is known as negative transfer. Given the potentially-severe consequences of negative transfer, it is prudent for engineers to ask the question `when, what, and how should one transfer between structures?'. The current paper aims to demonstrate a transfer-strategy decision process for a classification task for a population of simulated structures in the context of a representative SHM maintenance problem, supported by domain adaptation. The transfer decision framework is based upon the concept of expected value of information transfer. In order to compute the expected value of information transfer, predictions must be made regarding the classification (and decision performance) in the target domain following information transfer. In order to forecast the outcome of transfers, a probabilistic regression is used here to predict classification performance from a proxy for structural similarity based on the modal assurance criterion.

Tina A Dardeno, Lawrence A Bull, Robin S Mills et al.

Structural health monitoring (SHM) has been an active research area for the last three decades, and has accumulated a number of critical advances over that period, as can be seen in the literature. However, SHM is still facing challenges because of the paucity of damage-state data, operational and environmental fluctuations, repeatability issues, and changes in boundary conditions. These issues present as inconsistencies in the captured features and can have a huge impact on the practical implementation, but more critically, on the generalisation of the technology. Population-based SHM has been designed to address some of these concerns by modelling and transferring missing information using data collected from groups of similar structures. In this work, vibration data were collected from four healthy, nominally-identical, full-scale composite helicopter blades. Manufacturing differences (e.g., slight differences in geometry and/or material properties), among the blades presented as variability in their structural dynamics, which can be very problematic for SHM based on machine learning from vibration data. This work aims to address this variability by defining a general model for the frequency response functions of the blades, called a form, using mixtures of Gaussian processes.

Daniel R. Clarkson, Lawrence A. Bull, Chandula T. Wickramarachchi et al.

Regression is a fundamental prediction task common in data-centric engineering applications that involves learning mappings between continuous variables. In many engineering applications (e.g.\ structural health monitoring), feature-label pairs used to learn such mappings are of limited availability which hinders the effectiveness of traditional supervised machine learning approaches. The current paper proposes a methodology for overcoming the issue of data scarcity by combining active learning with hierarchical Bayesian modelling. Active learning is an approach for preferentially acquiring feature-label pairs in a resource-efficient manner. In particular, the current work adopts a risk-informed approach that leverages contextual information associated with regression-based engineering decision-making tasks (e.g.\ inspection and maintenance). Hierarchical Bayesian modelling allow multiple related regression tasks to be learned over a population, capturing local and global effects. The information sharing facilitated by this modelling approach means that information acquired for one engineering system can improve predictive performance across the population. The proposed methodology is demonstrated using an experimental case study. Specifically, multiple regressions are performed over a population of machining tools, where the quantity of interest is the surface roughness of the workpieces. An inspection and maintenance decision process is defined using these regression tasks which is in turn used to construct the active-learning algorithm. The novel methodology proposed is benchmarked against an uninformed approach to label acquisition and independent modelling of the regression tasks. It is shown that the proposed approach has superior performance in terms of expected cost -- maintaining predictive performance while reducing the number of inspections required.

Simon M. Brealy, Aidan J. Hughes, Tina A. Dardeno et al.

Population-based structural health monitoring (PBSHM), aims to share information between members of a population. An offshore wind (OW) farm could be considered as a population of nominally-identical wind-turbine structures. However, benign variations exist among members, such as geometry, sea-bed conditions and temperature differences. These factors could influence structural properties and therefore the dynamic response, making it more difficult to detect structural problems via traditional SHM techniques. This paper explores the use of a Bayesian hierarchical model as a means of multitask learning, to infer foundation stiffness distribution parameters at both population and local levels. To do this, observations of natural frequency from populations of structures were first generated from both numerical and experimental models. These observations were then used in a partially-pooled Bayesian hierarchical model in tandem with surrogate FE models of the structures to infer foundation stiffness parameters. Finally, it is demonstrated how the learned parameters may be used as a basis to perform more robust anomaly detection (as compared to a no-pooling approach) e.g. as a result of scour.

Aidan J. Hughes, Giulia Delo, Jack Poole et al.

In traditional approaches to structural health monitoring, challenges often arise associated with the availability of labelled data. Population-based structural health monitoring seeks to overcomes these challenges by leveraging data/information from similar structures via technologies such as transfer learning. The current paper demonstrate a methodology for quantifying the value of information transfer in the context of operation and maintenance decision-making. This demonstration, based on a population of laboratory-scale aircraft models, highlights the steps required to evaluate the expected value of information transfer including similarity assessment and prediction of transfer efficacy. Once evaluated for a given population, the value of information transfer can be used to optimise transfer-learning strategies for newly-acquired target domains.

Aidan J. Hughes, Jack Poole, Nikolaos Dervilis et al.

Decision-support for the operation and maintenance of structures provides significant motivation for the development and implementation of structural health monitoring (SHM) systems. Unfortunately, the limited availability of labelled training data hinders the development of the statistical models on which these decision-support systems rely. Population-based SHM seeks to mitigate the impact of data scarcity by using transfer learning techniques to share information between individual structures within a population. The current paper proposes a decision framework for selecting transfer strategies based upon a novel concept -- the expected value of information transfer -- such that negative transfer is avoided. By avoiding negative transfer, and by optimising information transfer strategies using the transfer-decision framework, one can reduce the costs associated with operating and maintaining structures, and improve safety.

Daniel R. Clarkson, Lawrence A. Bull, Tina A. Dardeno et al.

At present, most surface-quality prediction methods can only perform single-task prediction which results in under-utilised datasets, repetitive work and increased experimental costs. To counter this, the authors propose a Bayesian hierarchical model to predict surface-roughness measurements for a turning machining process. The hierarchical model is compared to multiple independent Bayesian linear regression models to showcase the benefits of partial pooling in a machining setting with respect to prediction accuracy and uncertainty quantification.

Christopher A. Lindley, Scott Beamish, Rob Dwyer-Joyce et al.

It has been shown that ultrasonic techniques work well for online measuring of circumferential oil film thickness profile in journal bearings; unfortunately, they can be limited by their measuring range and unable to capture details of the film all around the bearing circumference. Attempts to model the film thickness over the full range of the bearing rely on deterministic approaches, which assume the observations to be true with absolute certainty. Unaccounted uncertainties of the film thickness may lead to a cascade of inaccurate predictions for subsequent calculations of hydrodynamic parameters. In the present work, a probabilistic framework is proposed to model the film thickness with Gaussian Processes. The results are then used to estimate the location of the bearing shaft under various operational conditions. A further step in the process involves using the newly-constructed dataset to generate likelihood maps displaying the probable location of the shaft centre, given the bearing rotational speed and applied static load. The results offer the possibility to visualise the confidence of the predictions and allow the true location to be found within an area of high probability within the bearing's bore.

Simon M. Brealy, Lawrence A. Bull, Daniel S. Brennan et al.

Population-based Structural Health Monitoring (PBSHM) aims to share information between similar machines or structures. This paper takes a population-level perspective, exploring the use of additive Gaussian processes to reveal variations in turbine-specific and farm-level power models over a collected wind farm dataset. The predictions illustrate patterns in wind farm power generation, which follow intuition and should enable more informed control and decision-making.

Tristan Gowdridge, Elizabeth Cross, Nikolaos Dervilis et al.

The paper studies the topological changes from before and after cointegration, for the natural frequencies of the Z24 Bridge. The second natural frequency is known to be nonlinear in temperature, and this will serve as the main focal point of this work. Cointegration is a method of normalising time series data with respect to one another - often strongly-correlated time series. Cointegration is used in this paper to remove effects from Environmental and Operational Variations, by cointegrating the first four natural frequencies for the Z24 Bridge data. The temperature effects on the natural frequency data are clearly visible within the data, and it is desirable, for the purposes of structural health monitoring, that these effects are removed. The univariate time series are embedded in higher-dimensional space, such that interesting topologies are formed. Topological data analysis is used to analyse the raw time series, and the cointegrated equivalents. A standard topological data analysis pipeline is enacted, where simplicial complexes are constructed from the embedded point clouds. Topological properties are then calculated from the simplicial complexes; such as the persistent homology. The persistent homology is then analysed, to determine the topological structure of all the time series.

George Tsialiamanis, Keith Worden, Nikolaos Dervilis et al.

Damage prognosis is, arguably, one of the most difficult tasks of structural health monitoring (SHM). To address common problems of damage prognosis, a population-based SHM (PBSHM) approach is adopted in the current work. In this approach the prognosis problem is considered as an information-sharing problem where data from past structures are exploited to make more accurate inferences regarding currently-degrading structures. For a given population, there may exist restrictions on the resources available to conduct monitoring; thus, the current work studies the problem of allocating such resources within a population of degrading structures with a view to maximising the damage-prognosis accuracy. The challenges of the current framework are mainly associated with the inference of outliers on the level of damage evolution, given partial data from the damage-evolution phenomenon. The current approach considers an initial population of structures for which damage evolution is extensively observed. Subsequently, a second population of structures with evolving damage is considered for which two monitoring systems are available, a low-availability and high-fidelity (low-uncertainty) one, and a widely-available and low-fidelity (high-uncertainty) one. The task of the current work is to follow an active-learning approach to identify the structures to which the high-fidelity system should be assigned in order to enhance the predictive capabilities of the machine-learning model throughout the population.

Aidan J. Hughes, Keith Worden, Nikolaos Dervilis et al.

Classification models are a key component of structural digital twin technologies used for supporting asset management decision-making. An important consideration when developing classification models is the dimensionality of the input, or feature space, used. If the dimensionality is too high, then the `curse of dimensionality' may rear its ugly head; manifesting as reduced predictive performance. To mitigate such effects, practitioners can employ dimensionality reduction techniques. The current paper formulates a decision-theoretic approach to dimensionality reduction for structural asset management. In this approach, the aim is to keep incurred misclassification costs to a minimum, as the dimensionality is reduced and discriminatory information may be lost. This formulation is constructed as an eigenvalue problem, with separabilities between classes weighted according to the cost of misclassifying them when considered in the context of a decision process. The approach is demonstrated using a synthetic case study.

Aidan J. Hughes, Lawrence A. Bull, Paul Gardner et al.

Risk-based active learning is an approach to developing statistical classifiers for online decision-support. In this approach, data-label querying is guided according to the expected value of perfect information for incipient data points. For SHM applications, the value of information is evaluated with respect to a maintenance decision process, and the data-label querying corresponds to the inspection of a structure to determine its health state. Sampling bias is a known issue within active-learning paradigms; this occurs when an active learning process over- or undersamples specific regions of a feature-space, thereby resulting in a training set that is not representative of the underlying distribution. This bias ultimately degrades decision-making performance, and as a consequence, results in unnecessary costs incurred. The current paper outlines a risk-based approach to active learning that utilises a semi-supervised Gaussian mixture model. The semi-supervised approach counteracts sampling bias by incorporating pseudo-labels for unlabelled data via an EM algorithm. The approach is demonstrated on a numerical example representative of the decision processes found in SHM.

Tina A. Dardeno, Lawrence A. Bull, Nikolaos Dervilis et al.

Reductions in natural frequency are often used as a damage indicator for structural health monitoring (SHM) purposes. However, fluctuations in operational and environmental conditions, changes in boundary conditions, and slight differences among nominally-identical structures can also affect stiffness, producing frequency changes that mimic or mask damage. This variability has limited the practical implementation and generalisation of SHM technologies. The aim of this work is to investigate the effects of normal variation, and to identify methods that account for the resulting uncertainty. This work considers vibration data collected from a set of four healthy full-scale composite helicopter blades. The blades were nominally-identical but distinct, and slight differences in material properties and geometry among the blades caused significant variability in the frequency response functions, which presented as four separate trajectories across the input space. In this paper, an overlapping mixture of Gaussian processes (OMGP), was used to generate labels and quantify the uncertainty of normal-condition frequency response data from the helicopter blades. Using a population-based approach, the OMGP model provided a generic representation, called a form, to characterise the normal condition of the blades. Additional simulated data were then compared against the form and evaluated for damage using a marginal-likelihood novelty index.

Aidan J. Hughes, Paul Gardner, Lawrence A. Bull et al.

Gaining the ability to make informed decisions on operation and maintenance of structures provides motivation for the implementation of structural health monitoring (SHM) systems. However, descriptive labels for measured data corresponding to health-states of the monitored system are often unavailable. This issue limits the applicability of fully-supervised machine learning paradigms for the development of statistical classifiers to be used in decision-support in SHM systems. One approach to dealing with this problem is risk-based active learning. In such an approach, data-label querying is guided according to the expected value of perfect information for incipient data points. For risk-based active learning in SHM, the value of information is evaluated with respect to a maintenance decision process, and the data-label querying corresponds to the inspection of a structure to determine its health state. In the context of SHM, risk-based active learning has only been considered for generative classifiers. The current paper demonstrates several advantages of using an alternative type of classifier -- discriminative models. Using the Z24 Bridge dataset as a case study, it is shown that discriminative classifiers have benefits, in the context of SHM decision-support, including improved robustness to sampling bias, and reduced expenditure on structural inspections.

Nikolaos D. Tantaroudas, Jake Hollins, Konstantinos Agathos et al.

Finite element model updating is a mature discipline for linear structures, yet its extension to nonlinear regimes remains an open challenge. This paper presents a methodology that combines nonlinear model order reduction (NMOR) based on Taylor-series expansion of the equations of motion with the projection-basis adaptation scheme recently proposed by Hollins et al. [2026] for linear model updating. The structural equations of motion, augmented with proportional (Rayleigh) damping and polynomial stiffness nonlinearity, are recast as a first-order autonomous system whose Jacobian possesses complex eigenvectors forming a biorthogonal basis. Taylor operators of second and third order are derived for the nonlinear internal forces and projected onto the reduced eigenvector basis, yielding a low-dimensional nonlinear reduced-order model (ROM). The Cayley transform, generalised from the real orthogonal to the complex unitary group, parametrises the adaptation of the projection basis so that the ROM mode shapes optimally correlate with experimental measurements. The resulting nonlinear model-updating framework is applied to a representative wingbox panel model. Numerical studies demonstrate that the proposed approach captures amplitude-dependent natural frequencies and modal assurance criterion(MAC) values that a purely linear updating scheme cannot reproduce, while recovering the underlying stiffness parameters with improved accuracy.

Aidan J. Hughes, Paul Gardner, Keith Worden

The prospect of informed and optimal decision-making regarding the operation and maintenance (O&M) of structures provides impetus to the development of structural health monitoring (SHM) systems. A probabilistic risk-based framework for decision-making has already been proposed. However, in order to learn the statistical models necessary for decision-making, measured data from the structure of interest are required. Unfortunately, these data are seldom available across the range of environmental and operational conditions necessary to ensure good generalisation of the model. Recently, technologies have been developed that overcome this challenge, by extending SHM to populations of structures, such that valuable knowledge may be transferred between instances of structures that are sufficiently similar. This new approach is termed population-based structural heath monitoring (PBSHM). The current paper presents a formal representation of populations of structures, such that risk-based decision processes may be specified within them. The population-based representation is an extension to the hierarchical representation of a structure used within the probabilistic risk-based decision framework to define fault trees. The result is a series, consisting of systems of systems ranging from the individual component level up to an inventory of heterogeneous populations. The current paper considers an inventory of wind farms as a motivating example and highlights the inferences and decisions that can be made within the hierarchical representation.

Keith Worden, Tina A. Dardeno, Aidan J. Hughes et al.

Population-Based Structural Health Monitoring (PBSHM), aims to leverage information across populations of structures in order to enhance diagnostics on those with sparse data. The discipline of transfer learning provides the mechanism for this capability. One recent paper in PBSHM proposed a geometrical view in which the structures were represented as graphs in a metric "base space" with their data captured in the "total space" of a vector bundle above the graph space. This view was more suggestive than mathematically rigorous, although it did allow certain useful arguments. One bar to more rigorous analysis was the absence of a meaningful topology on the graph space, and thus no useful notion of continuity. The current paper aims to address this problem, by moving to parametric families of structures in the base space, essentially changing points in the graph space to open balls. This allows the definition of open sets in the fibre space and thus allows continuous variation between fibres. The new ideas motivate a new geometrical mechanism for transfer learning in data are transported from one fibre to an adjacent one; i.e., from one structure to another.

Simon M. Brealy, Lawrence A. Bull, Pauline Beltrando et al.

Because of the global need to increase power production from renewable energy resources, developments in the online monitoring of the associated infrastructure is of interest to reduce operation and maintenance costs. However, challenges exist for data-driven approaches to this problem, such as incomplete or limited histories of labelled damage-state data, operational and environmental variability, or the desire for the quantification of uncertainty to support risk management. This work first introduces a probabilistic regression model for predicting wind-turbine power, which adjusts for wake effects learnt from data. Spatial correlations in the learned model parameters for different tasks (turbines) are then leveraged in a hierarchical Bayesian model (an approach to multi-task learning) to develop a "metamodel", which can be used to make power-predictions which adjust for turbine location - including on previously unobserved turbines not included in the training data. The results show that the metamodel is able to outperform a series of benchmark models, and demonstrates a novel strategy for making efficient use of data for inference in populations of structures, in particular where correlations exist in the variable(s) of interest (such as those from wind-turbine wake-effects).

Sarah Bee, Lawrence Bull, Nikolaos Dervilis et al.

A Nonlinear Auto-Regressive with eXogenous inputs (NARX) model can be used to describe time-varying processes; where the output depends on both previous outputs and current/previous external input variables. One limitation of NARX models is their propensity to overfit and result in poor generalisation for future predictions. The proposed method to help to overcome the issue of overfitting is a NARX model which predicts outputs at both the current time and several lead times into the future. This is a form of multi-task learner (MTL); whereby the lead time outputs will regularise the current time output. This work shows that for high noise level, MTL can be used to regularise NARX with a lower Normalised Mean Square Error (NMSE) compared to the NMSE of the independent learner counterpart.

Tina A. Dardeno, Lawrence A. Bull, Nikolaos Dervilis et al.

Despite recent advances in population-based structural health monitoring (PBSHM), knowledge transfer between highly-disparate structures (i.e., heterogeneous populations) remains a challenge. It has been proposed that heterogeneous transfer may be accomplished via intermediate structures that bridge the gap in information between the structures of interest. A key aspect of the technique is the idea that by varying parameters such as material properties and geometry, one structure can be continuously morphed into another. The current work demonstrates the development of these interpolating structures, via case studies involving the parameterisation of (and transfer between) a simple, simulated 'bridge' and 'aeroplane'. The facetious question 'When is a bridge not an aeroplane?' has been previously asked in the context of predicting positive transfer based on structural similarity. While the obvious answer to this question is 'Always,' the current work demonstrates that in some cases positive transfer can be achieved between highly-disparate systems.

Sarah Bee, Lawrence Bull, Nikolas Dervilis et al.

Multi-task neural networks learn tasks simultaneously to improve individual task performance. There are three mechanisms of multi-task learning (MTL) which are explored here for the context of structural health monitoring (SHM): (i) the natural occurrence of multiple tasks; (ii) using outputs as inputs (both linked to the recent research in population-based SHM (PBSHM)); and, (iii) additional loss functions to provide different insights. Each of these problem settings for MTL is detailed and an example is given.

Aidan J. Hughes, Lawrence A. Bull, Paul Gardner et al.

Classification models are a fundamental component of physical-asset management technologies such as structural health monitoring (SHM) systems and digital twins. Previous work introduced risk-based active learning, an online approach for the development of statistical classifiers that takes into account the decision-support context in which they are applied. Decision-making is considered by preferentially querying data labels according to expected value of perfect information (EVPI). Although several benefits are gained by adopting a risk-based active learning approach, including improved decision-making performance, the algorithms suffer from issues relating to sampling bias as a result of the guided querying process. This sampling bias ultimately manifests as a decline in decision-making performance during the later stages of active learning, which in turn corresponds to lost resource/utility. The current paper proposes two novel approaches to counteract the effects of sampling bias: semi-supervised learning, and discriminative classification models. These approaches are first visualised using a synthetic dataset, then subsequently applied to an experimental case study, specifically, the Z24 Bridge dataset. The semi-supervised learning approach is shown to have variable performance; with robustness to sampling bias dependent on the suitability of the generative distributions selected for the model with respect to each dataset. In contrast, the discriminative classifiers are shown to have excellent robustness to the effects of sampling bias. Moreover, it was found that the number of inspections made during a monitoring campaign, and therefore resource expenditure, could be reduced with the careful selection of the statistical classifiers used within a decision-supporting monitoring system.

Sikai Zhang, Tingna Wang, Keith Worden et al.

Feature selection refers to the process of selecting useful features for machine learning tasks, and it is also a key step for structural health monitoring (SHM). This paper proposes a fast feature selection algorithm by efficiently computing the sum of squared canonical correlation coefficients between monitored features and target variables of interest in greedy search. The proposed algorithm is applied to both synthetic and real datasets to illustrate its advantages in terms of computational speed, general classification and regression tasks, as well as damage-sensitive feature selection tasks. Furthermore, the performance of the proposed algorithm is evaluated under varying environmental conditions and on an edge computing device to investigate its applicability in real-world SHM scenarios. The results show that the proposed algorithm can successfully select useful features with extraordinarily fast computational speed, which implies that the proposed algorithm has great potential where features need to be selected and updated online frequently, or where devices have limited computing capability.

George Tsialiamanis, Charilaos Mylonas, Eleni Chatzi et al.

One of the requirements of the population-based approach to Structural Health Monitoring (SHM) proposed in the earlier papers in this sequence, is that structures be represented by points in an abstract space. Furthermore, these spaces should be metric spaces in a loose sense; i.e. there should be some measure of distance applicable to pairs of points; similar structures should then be close in the metric. However, this geometrical construction is not enough for the framing of problems in data-based SHM, as it leaves undefined the notion of feature spaces. Interpreting the feature values on a structure-by-structure basis as a type of field over the space of structures, it seems sensible to borrow an idea from modern theoretical physics, and define feature assignments as sections in a vector bundle over the structure space. With this idea in place, one can interpret the effect of environmental and operational variations as gauge degrees of freedom, as in modern gauge field theories. This paper will discuss the various geometrical structures required for an abstract theory of feature spaces in SHM, and will draw analogies with how these structures have shown their power in modern physics. In the second part of the paper, the problem of determining the normal condition cross section of a feature bundle is addressed. The solution is provided by the application of Graph Neural Networks (GNN), a versatile non-Euclidean machine learning algorithm which is not restricted to inputs and outputs from vector spaces. In particular, the algorithm is well suited to operating directly on the sort of graph structures which are an important part of the proposed framework for PBSHM. The solution of the normal section problem is demonstrated for a heterogeneous population of truss structures for which the feature of interest is the first natural frequency.

Lawrence A. Bull, Paul Gardner, Timothy J. Rogers et al.

In data-driven SHM, the signals recorded from systems in operation can be noisy and incomplete. Data corresponding to each of the operational, environmental, and damage states are rarely available a priori; furthermore, labelling to describe the measurements is often unavailable. In consequence, the algorithms used to implement SHM should be robust and adaptive, while accommodating for missing information in the training-data -- such that new information can be included if it becomes available. By reviewing novel techniques for statistical learning (introduced in previous work), it is argued that probabilistic algorithms offer a natural solution to the modelling of SHM data in practice. In three case-studies, probabilistic methods are adapted for applications to SHM signals -- including semi-supervised learning, active learning, and multi-task learning.

Kartik Chandrasekhar, Nevena Stevanovic, Elizabeth J. Cross et al.

The application of reliable structural health monitoring (SHM) technologies to operational wind turbine blades is a challenging task, due to the uncertain nature of the environments they operate in. In this paper, a novel SHM methodology, which uses Gaussian Processes (GPs) is proposed. The methodology takes advantage of the fact that the blades on a turbine are nominally identical in structural properties and encounter the same environmental and operational variables (EOVs). The properties of interest are the first edgewise frequencies of the blades. The GPs are used to predict the edge frequencies of one blade given that of another, after these relationships between the pairs of blades have been learned when the blades are in a healthy state. In using this approach, the proposed SHM methodology is able to identify when the blades start behaving differently from one another over time. To validate the concept, the proposed SHM system is applied to real onshore wind turbine blade data, where some form of damage was known to have taken place. X-bar control chart analysis of the residual errors between the GP predictions and actual frequencies show that the system successfully identified early onset of damage as early as six months before it was identified and remedied.

Marcus Haywood-Alexander, Nikolaos Dervilis, Keith Worden et al.

The use of ultrasonic guided waves to probe the materials/structures for damage continues to increase in popularity for non-destructive evaluation (NDE) and structural health monitoring (SHM). The use of high-frequency waves such as these offers an advantage over low-frequency methods from their ability to detect damage on a smaller scale. However, in order to assess damage in a structure, and implement any NDE or SHM tool, knowledge of the behaviour of a guided wave throughout the material/structure is important (especially when designing sensor placement for SHM systems). Determining this behaviour is extremely diffcult in complex materials, such as fibre-matrix composites, where unique phenomena such as continuous mode conversion takes place. This paper introduces a novel method for modelling the feature-space of guided waves in a composite material. This technique is based on a data-driven model, where prior physical knowledge can be used to create structured machine learning tools; where constraints are applied to provide said structure. The method shown makes use of Gaussian processes, a full Bayesian analysis tool, and in this paper it is shown how physical knowledge of the guided waves can be utilised in modelling using an ML tool. This paper shows that through careful consideration when applying machine learning techniques, more robust models can be generated which offer advantages such as extrapolation ability and physical interpretation.

Matthew R. Jones, Tim J. Rogers, Keith Worden et al.

In the field of structural health monitoring (SHM), the acquisition of acoustic emissions to localise damage sources has emerged as a popular approach. Despite recent advances, the task of locating damage within composite materials and structures that contain non-trivial geometrical features, still poses a significant challenge. Within this paper, a Bayesian source localisation strategy that is robust to these complexities is presented. Under this new framework, a Gaussian process is first used to learn the relationship between source locations and the corresponding difference-in-time-of-arrival values for a number of sensor pairings. As an acoustic emission event with an unknown origin is observed, a mapping is then generated that quantifies the likelihood of the emission location across the surface of the structure. The new probabilistic mapping offers multiple benefits, leading to a localisation strategy that is more informative than deterministic predictions or single-point estimates with an associated confidence bound. The performance of the approach is investigated on a structure with numerous complex geometrical features and demonstrates a favourable performance in comparison to other similar localisation methods.

Chon Lok Lei, Sanmitra Ghosh, Dominic G. Whittaker et al.

Uncertainty quantification (UQ) is a vital step in using mathematical models and simulations to take decisions. The field of cardiac simulation has begun to explore and adopt UQ methods to characterise uncertainty in model inputs and how that propagates through to outputs or predictions. In this perspective piece we draw attention to an important and under-addressed source of uncertainty in our predictions -- that of uncertainty in the model structure or the equations themselves. The difference between imperfect models and reality is termed model discrepancy, and we are often uncertain as to the size and consequences of this discrepancy. Here we provide two examples of the consequences of discrepancy when calibrating models at the ion channel and action potential scales. Furthermore, we attempt to account for this discrepancy when calibrating and validating an ion channel model using different methods, based on modelling the discrepancy using Gaussian processes (GPs) and autoregressive-moving-average (ARMA) models, then highlight the advantages and shortcomings of each approach. Finally, suggestions and lines of enquiry for future work are provided.