Keith Worden

G. Tsialiamanis, D. J. Wagg, P. A. Gardner et al.

In the current work, a problem-splitting approach and a scheme motivated by transfer learning is applied to a structural health monitoring problem. The specific problem in this case is that of localising damage on an aircraft wing. The original experiment is described, together with the initial approach, in which a neural network was trained to localise damage. The results were not ideal, partly because of a scarcity of training data, and partly because of the difficulty in resolving two of the damage cases. In the current paper, the problem is split into two sub-problems and an increase in classification accuracy is obtained. The sub-problems are obtained by separating out the most difficult-to-classify damage cases. A second approach to the problem is considered by adopting ideas from transfer learning (usually applied in much deeper) networks to see if a network trained on the simpler damage cases can help with feature extraction in the more difficult cases. The transfer of a fixed trained batch of layers between the networks is found to improve classification by making the classes more separable in the feature space and to speed up convergence.

G. Tsialiamanis, D. J. Wagg, N. Dervilis et al.

A framework is proposed for generative models as a basis for digital twins or mirrors of structures. The proposal is based on the premise that deterministic models cannot account for the uncertainty present in most structural modelling applications. Two different types of generative models are considered here. The first is a physics-based model based on the stochastic finite element (SFE) method, which is widely used when modelling structures that have material and loading uncertainties imposed. Such models can be calibrated according to data from the structure and would be expected to outperform any other model if the modelling accurately captures the true underlying physics of the structure. The potential use of SFE models as digital mirrors is illustrated via application to a linear structure with stochastic material properties. For situations where the physical formulation of such models does not suffice, a data-driven framework is proposed, using machine learning and conditional generative adversarial networks (cGANs). The latter algorithm is used to learn the distribution of the quantity of interest in a structure with material nonlinearities and uncertainties. For the examples considered in this work, the data-driven cGANs model outperform the physics-based approach. Finally, an example is shown where the two methods are coupled such that a hybrid model approach is demonstrated.

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.

G. Tsialiamanis, M. D. Champneys, N. Dervilis et al.

Linear modal analysis is a useful and effective tool for the design and analysis of structures. However, a comprehensive basis for nonlinear modal analysis remains to be developed. In the current work, a machine learning scheme is proposed with a view to performing nonlinear modal analysis. The scheme is focussed on defining a one-to-one mapping from a latent `modal' space to the natural coordinate space, whilst also imposing orthogonality of the mode shapes. The mapping is achieved via the use of the recently-developed cycle-consistent generative adversarial network (cycle-GAN) and an assembly of neural networks targeted on maintaining the desired orthogonality. The method is tested on simulated data from structures with cubic nonlinearities and different numbers of degrees of freedom, and also on data from an experimental three-degree-of-freedom set-up with a column-bumper nonlinearity. The results reveal the method's efficiency in separating the `modes'. The method also provides a nonlinear superposition function, which in most cases has very good accuracy.

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.

T. A. Dardeno, K. Worden, N. Dervilis et al.

For situations that may benefit from information sharing among datasets, e.g., population-based SHM of similar structures, the hierarchical Bayesian approach provides a useful modelling structure. Hierarchical Bayesian models learn statistical distributions at the population (or parent) and the domain levels simultaneously, to bolster statistical strength among the parameters. As a result, variance is reduced among the parameter estimates, particularly when data are limited. In this paper, a combined probabilistic FRF model is developed for a small population of nominally-identical helicopter blades, using a hierarchical Bayesian structure, to support information transfer in the context of sparse data. The modelling approach is also demonstrated in a traditional SHM context, for a single helicopter blade exposed to varying temperatures, to show how the inclusion of physics-based knowledge can improve generalisation beyond the training data, in the context of scarce data. These models address critical challenges in SHM, by accommodating benign variations that present as differences in the underlying dynamics, while also considering (and utilising), the similarities among the domains.

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.

G. Tsialiamanis, N. Dervilis, D. J. Wagg et al.

Machine learning has affected the way in which many phenomena for various domains are modelled, one of these domains being that of structural dynamics. However, because machine-learning algorithms are problem-specific, they often fail to perform efficiently in cases of data scarcity. To deal with such issues, combination of physics-based approaches and machine learning algorithms have been developed. Although such methods are effective, they also require the analyser's understanding of the underlying physics of the problem. The current work is aimed at motivating the use of models which learn such relationships from a population of phenomena, whose underlying physics are similar. The development of such models is motivated by the way that physics-based models, and more specifically finite element models, work. Such models are considered transferrable, explainable and trustworthy, attributes which are not trivially imposed or achieved for machine-learning models. For this reason, machine-learning approaches are less trusted by industry and often considered more difficult to form validated models. To achieve such data-driven models, a population-based scheme is followed here and two different machine-learning algorithms from the meta-learning domain are used. The two algorithms are the model-agnostic meta-learning (MAML) algorithm and the conditional neural processes (CNP) model. The algorithms seem to perform as intended and outperform a traditional machine-learning algorithm at approximating the quantities of interest. Moreover, they exhibit behaviour similar to traditional machine learning algorithms (e.g. neural networks or Gaussian processes), concerning their performance as a function of the available structures in the training population.

G. Tsialiamanis, C. Mylonas, E. Chatzi et al.

Attempts have been made recently in the field of population-based structural health monitoring (PBSHM), to transfer knowledge between SHM models of different structures. The attempts have been focussed on homogeneous and heterogeneous populations. A more general approach to transferring knowledge between structures, is by considering all plausible structures as points on a multidimensional base manifold and building a fibre bundle. The idea is quite powerful, since, a mapping between points in the base manifold and their fibres, the potential states of any arbitrary structure, can be learnt. A smaller scale problem, but still useful, is that of learning a specific point of every fibre, i.e. that corresponding to the undamaged state of structures within a population. Under the framework of PBSHM, a data-driven approach to the aforementioned problem is developed. Structures are converted into graphs and inference is attempted within a population, using a graph neural network (GNN) algorithm. The algorithm solves a major problem existing in such applications. Structures comprise different sizes and are defined as abstract objects, thus attempting to perform inference within a heterogeneous population is not trivial. The proposed approach is tested in a simulated population of trusses. The goal of the application is to predict the first natural frequency of trusses of different sizes, across different environmental temperatures and having different bar member types. After training the GNN using part of the total population, it was tested on trusses that were not included in the training dataset. Results show that the accuracy of the regression is satisfactory even in structures with higher number of nodes and members than those used to train it.

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.

G. Tsialiamanis, D. J. Wagg, N. Dervilis et al.

A powerful approach, and one of the most common ones in structural health monitoring (SHM), is to use data-driven models to make predictions and inferences about structures and their condition. Such methods almost exclusively rely on the quality of the data. Within the SHM discipline, data do not always suffice to build models with satisfactory accuracy for given tasks. Even worse, data may be completely missing from one's dataset, regarding the behaviour of a structure under different environmental conditions. In the current work, with a view to confronting such issues, the generation of artificial data using a variation of the generative adversarial network (GAN) algorithm, is used. The aforementioned variation is that of the conditional GAN or cGAN. The algorithm is not only used to generate artificial data, but also to learn transformations of manifolds according to some known parameters. Assuming that the structure's response is represented by points in a manifold, part of the space will be formed due to variations in external conditions affecting the structure. This idea proves efficient in SHM, as it is exploited to generate structural data for specific values of environmental coefficients. The scheme is applied here on a simulated structure which operates under different temperature and humidity conditions. The cGAN is trained on data for some discrete values of the temperature within some range, and is able to generate data for every temperature in this range with satisfactory accuracy. The novelty, compared to classic regression in similar problems, is that the cGAN allows unknown environmental parameters to affect the structure and can generate whole manifolds of data for every value of the known parameters, while the unknown ones vary within the generated manifolds.

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.

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.

G. Tsialiamanis, N. Dervilis, D. J. Wagg et al.

A major problem of machine-learning approaches in structural dynamics is the frequent lack of structural data. Inspired by the recently-emerging field of population-based structural health monitoring (PBSHM), and the use of transfer learning in this novel field, the current work attempts to create models that are able to transfer knowledge within populations of structures. The approach followed here is meta-learning, which is developed with a view to creating neural network models which are able to exploit knowledge from a population of various tasks to perform well in newly-presented tasks, with minimal training and a small number of data samples from the new task. Essentially, the method attempts to perform transfer learning in an automatic manner within the population of tasks. For the purposes of population-based structural modelling, the different tasks refer to different structures. The method is applied here to a population of simulated structures with a view to predicting their responses as a function of some environmental parameters. The meta-learning approach, which is used herein is the model-agnostic meta-learning (MAML) approach; it is compared to a traditional data-driven modelling approach, that of Gaussian processes, which is a quite effective alternative when few data samples are available for a problem. It is observed that the models trained using meta-learning approaches, are able to outperform conventional machine learning methods regarding inference about structures of the population, for which only a small number of samples are available. Moreover, the models prove to learn part of the physics of the problem, making them more robust than plain machine-learning algorithms. Another advantage of the methods is that the structures do not need to be parametrised in order for the knowledge transfer to be performed.

G. Tsialiamanis, D. Wagg, N. Dervilis et al.

A major problem of structural health monitoring (SHM) has been the prognosis of damage and the definition of the remaining useful life of a structure. Both tasks depend on many parameters, many of which are often uncertain. Many models have been developed for the aforementioned tasks but they have been either deterministic or stochastic with the ability to take into account only a restricted amount of past states of the structure. In the current work, a generative model is proposed in order to make predictions about the damage evolution of structures. The model is able to perform in a population-based SHM (PBSHM) framework, to take into account many past states of the damaged structure, to incorporate uncertainties in the modelling process and to generate potential damage evolution outcomes according to data acquired from a structure. The algorithm is tested on a simulated damage evolution example and the results reveal that it is able to provide quite confident predictions about the remaining useful life of structures within a population.

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.

S. C. Bee, E. Papatheou, M Haywood-Alexander et al.

There have been recent efforts to move to population-based structural health monitoring (PBSHM) systems. One area of PBSHM which has been recognised for potential development is the use of multi-task learning (MTL); algorithms which differ from traditional independent learning algorithms. Presented here is the use of the MTL, ''Joint Feature Selection with LASSO'', to provide automatic feature selection for a structural dataset. The classification task is to differentiate between the port and starboard side of a tailplane, for samples from two aircraft of the same model. The independent learner produced perfect F1 scores but had poor engineering insight; whereas the MTL results were interpretable, highlighting structural differences as opposed to differences in experimental set-up.

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.

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.

S. M. Smith, A. J. Hughes, T. 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 hierarchical Bayesian model to infer expected soil stiffness distributions at both population and local levels, as a basis to perform anomaly detection, in the form of scour, for new and existing turbines. To do this, observations of natural frequency will be generated as though they are from a small population of wind turbines. Differences between individual observations will be introduced by postulating distributions over the soil stiffness and measurement noise, as well as reducing soil depth (to represent scour), in the case of anomaly detection.

S C Bee, N Dervilis, K Worden et al.

Equation discovery provides a grey-box approach to system identification by uncovering governing dynamics directly from observed data. However, a persistent challenge lies in ensuring that identified models generalise across operating conditions rather than over-fitting to specific datasets. This work investigates this issue by applying a Bayesian relevance vector machine (RVM) within a multi-task learning (MTL) framework for simultaneous parameter identification across multiple datasets. In this formulation, responses from the same structure under different excitation levels are treated as related tasks that share model parameters but retain task-specific noise characteristics. A simulated single degree-of-freedom oscillator with linear and cubic stiffness provided the case study, with datasets generated under three excitation regimes. Standard single-task RVM models were able to reproduce system responses but often failed to recover the true governing terms when excitations insufficiently stimulated non-linear dynamics. By contrast, the MTL-RVM combined information across tasks, improving parameter recovery for weakly and moderately excited datasets, while maintaining strong performance under high excitation. These findings demonstrate that multi-task Bayesian inference can mitigate over-fitting and promote generalisation in equation discovery. The approach is particularly relevant to structural health monitoring, where varying load conditions reveal complementary aspects of system physics.

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).

C. A. Lindley, N. Dervilis, K. Worden

Whenever data-based systems are employed in engineering applications, defining an optimal statistical representation is subject to the problem of model selection. This paper focusses on how well models can generalise in Structural Health Monitoring (SHM). Although statistical model validation in this field is often performed heuristically, it is possible to estimate generalisation more rigorously using the bounds provided by Statistical Learning Theory (SLT). Therefore, this paper explores the selection process of a kernel smoother for modelling the impulse response of a linear oscillator from the perspective of SLT. It is demonstrated that incorporating domain knowledge into the regression problem yields a lower guaranteed risk, thereby enhancing generalisation.

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.

W. Lin, K. Worden, E. J. Cross

An important challenge faced by wind farm operators is to reduce operation and maintenance cost. Structural health monitoring provides a means of cost reduction through minimising unnecessary maintenance trips as well as prolonging turbine service life. Population-based structural health monitoring can further reduce the cost of health monitoring systems by implementing one system for multiple structures (i.e.~turbines). At the same time, shared data within a population of structures may improve the predictions of structural behaviour. To monitor turbine performance at a population/farm level, an important initial step is to construct a model that describes the behaviour of all turbines under normal conditions. This paper proposes a population-level model that explicitly captures the spatial and temporal correlations (between turbines) induced by the wake effect. The proposed model is a Gaussian process-based spatial autoregressive model, named here a GP-SPARX model. This approach is developed since (a) it reflects our physical understanding of the wake effect, and (b) it benefits from a stochastic data-based learner. A case study is provided to demonstrate the capability of the GP-SPARX model in capturing spatial and temporal variations as well as its potential applicability in a health monitoring system.

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.

L. A. Bull, P. A. Gardner, T. J. Rogers et al.

Power curves capture the relationship between wind speed and output power for a specific wind turbine. Accurate regression models of this function prove useful in monitoring, maintenance, design, and planning. In practice, however, the measurements do not always correspond to the ideal curve: power curtailments will appear as (additional) functional components. Such multivalued relationships cannot be modelled by conventional regression, and the associated data are usually removed during pre-processing. The current work suggests an alternative method to infer multivalued relationships in curtailed power data. Using a population-based approach, an overlapping mixture of probabilistic regression models is applied to signals recorded from turbines within an operational wind farm. The model is shown to provide an accurate representation of practical power data across the population.

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.

A. J. Hughes, L. A. Bull, P. Gardner et al.

A primary motivation for the development and implementation of structural health monitoring systems, is the prospect of gaining the ability to make informed decisions regarding the operation and maintenance of structures and infrastructure. Unfortunately, descriptive labels for measured data corresponding to health-state information for the structure of interest are seldom available prior to the implementation of a monitoring system. This issue limits the applicability of the traditional supervised and unsupervised approaches to machine learning in the development of statistical classifiers for decision-supporting SHM systems. The current paper presents a risk-based formulation of active learning, in which the querying of class-label information is guided by the expected value of said information for each incipient data point. When applied to structural health monitoring, the querying of class labels can be mapped onto the inspection of a structure of interest in order to determine its health state. In the current paper, the risk-based active learning process is explained and visualised via a representative numerical example and subsequently applied to the Z24 Bridge benchmark. The results of the case studies indicate that a decision-maker's performance can be improved via the risk-based active learning of a statistical classifier, such that the decision process itself is taken into account.

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.

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.