On the Detection and Quantification of Nonlinearity via Statistics of the Gradients of a Black-Box Model

Detection and identification of nonlinearity is a task of high importance for structural dynamics. Detecting nonlinearity in a structure, which has been designed to operate in its linear region, might indicate the existence of damage. Therefore, it is important, even for safety reasons, to detect when a structure exhibits nonlinear behaviour. In the current work, a method to detect nonlinearity is proposed, based on the distribution of the gradients of a data-driven model, which is fitted on data acquired from the structure of interest. The data-driven model herein is a neural network. The selection of such a type of model was done in order to not allow the user to decide how linear or nonlinear the model shall be, but to let the training algorithm of the neural network shape the level of nonlinearity according to the training data. The neural network is trained to predict the accelerations of the structure for a time-instant using as inputs accelerations of previous time-instants, i.e. one-step-ahead predictions. Afterwards, the gradients of the output of the neural network with respect to its inputs are calculated. Given that the structure is linear, the distribution of the aforementioned gradients should be quite peaked, while in the case of a structure with nonlinearities, the distribution of the gradients shall be more spread and, potentially, multimodal. To test the above assumption, data from an experimental structure are considered. The structure is tested under different scenarios, some of which are linear and some nonlinear. The statistics of the distributions of the gradients for the different scenarios can be used to identify cases where nonlinearity is present. Moreover, via the proposed method one is able to quantify the nonlinearity by observing higher values of standard deviation of the distribution of the gradients for "more nonlinear" scenarios

A Meta-Learning Approach to Population-Based Modelling of Structures

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

On a meta-learning population-based approach to damage prognosis

The current work studies the application of population-based structural health monitoring (PBSHM) to the problem of damage prognosis. Two methods are proposed for population-informed damage prognosis and they are evaluated according to their performance using an experimental dataset. The first method is an attempt to define a functional subspace, which includes the potential behaviour of members of the population subjected to the phenomenon of damage evolution. The second approach is a meta-learning method, the deep kernel transfer (DKT) method, which seeks to exploit information from a population in order to enhance the predictive performance of a Gaussian process. The predictive capabilities of the two methods are tested in an experimental crack-growth problem. The results reveal that the two methods are properly informed by the population to make predictions about new structures and show potential in dealing with the problem of damage evolution, which is a problem of imbalanced and difficult-to-acquire data

A neat approach to structural health monitoring

In the current paper, an application of the neuroevolution of augmenting topologies (NEAT) algorithm is considered in a structural health monitoring (SHM) application. The algorithm is a variation of genetic algorithms, applied in neural networks, and has the goal of optimising both the topology and the weights and biases of a neural network model. The algorithm is applied here to an SHM problem instead of using feedforward neural networks. The algorithm is called to search for the best-fitting topology in the task, which would otherwise be sought through experimenting with the size and number of the layers of the neural network. Having used the algorithm, the accuracy is found to be close to the one achieved using classically trained neural networks. Another aspect of the application is that subnetworks were defined for every damage case of the problem, whose topologies are much simpler than a fully-connected feedforward neural network. These subnetworks define classification submodels that may be used in different combinations, building models for a subset of damage cases and input features

On an application of generative adversarial networks on remaining lifetime estimation

A major problem of structural health monitoring (SHM) has been the prognosis of damage and the prediction of the remaining useful life of a structure. Both tasks depend on multiple parameters, many of which are often uncertain. A wide range of 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 set 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

Towards a population-informed approach to the definition of data-driven models for structural dynamics

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 transferable, 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 two approaches have been developed to perform within a population of tasks and, herein, they are tested on a simulated dataset of a population of structures, with data available from a small subset of the population. Such situations are considered to be similar to having data available from existing structures or structures in a laboratory environment or even from a model and needing to model a new structure with only a few available data samples. 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, i.e. the more training structures, the better and more robustly the algorithms learn the underlying relationships

On generative models as the basis for digital twins

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 modeling 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 modeling 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 modeling 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 outperforms the physics-based approach. Finally, an example is shown where the two methods are coupled such that a hybrid model approach is demonstrated

An application of generative adversarial networks in structural health monitoring

In the current work, the use of generative adversarial networks (GANs) in a simulated structural health monitoring (SHM) application is studied. A specific type of GAN is considered, aiming at a disentangled representation of underlying features and clusters of data through some latent variables. This idea could prove useful in SHM, since explanation of how damage mechanisms or environmental conditions affect a structure may be exploited in order to monitor structures more effectively. In a simulated mass-spring example, different damage cases are introduced by reducing the stiffness of specific springs and different damage levels by applying different extents of stiffness reduction. The GAN implementation proves able to capture different damage cases through its categorical latent variables, as well as the damage extent within its continuous latent variables. The results demonstrate that the latent variables are indeed capturing the effect of damage in the structure and can be exploited for the purpose of condition assessment

On the dynamic properties of statistically-independent nonlinear normal modes

Much attention has been given to the study of nonlinear normal modes (NNMs), a nonlinear extension to the eminently useful framework for the analysis of linear dynamics provided by linear modal analysis (LMA). In the literature, several approaches have gained traction, with each able to preserve a subset of the useful properties of LMA. A recently-proposed framework (Worden and Green, 2017) casts nonlinear modal analysis as a problem in machine learning, viewing the NNM as directions in a latent modal coordinate space within which the modal dynamics are statistically uncorrelated. Thus far, the performance of this framework has been measured in a largely qualitative way. This paper presents, for the first time, an exploration into the underlying dynamics of the statistically-independent NNMs using techniques from nonlinear system identification (NLSI) and higher-order frequency-response functions (HFRFs). In this work, the statistically-uncorrelated NNMs are found for two simulated nonlinear cubic-stiffness systems using a recently-proposed neural-network based approach. NLSI models are fitted to both physical and modal displacements and the HFRFs of these models are compared to theoretical values. In particular, it is found for both systems that the modal decompositions here permit an independent single-input single-output (SISO) representation that can be projected back onto the original displacements with low error. It is also shown via the HFRFs that the underlying linear natural frequencies of the modal dynamics lie very close to the underlying linear natural frequencies of the nonlinear systems, indicating that a true nonlinear decomposition has been identified

Foundations of population-based SHM, Part IV : the geometry of spaces of structures and their feature spaces

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. One can then regard data normalisation procedures like cointegration as gauge-fixing operations. 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. Having motivated a number of problems in Population-Based SHM (PBSHM) in geometrical terms, it remains to show how these problems might be solved. 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. The GNN approach is shown to not only solve the normal section problem, but also to accommodate varying temperatures across the population; it thus provides a means of data normalisation