Physics-derived covariance functions for machine learning in structural dynamics ⁎

This paper attempts to bridge the gap between standard engineering practice and machine learning when modelling stochastic processes. For a number of physical processes of interest, derivation of the (auto)covariance is achievable. This paper suggests their use as priors in a standard Gaussian process regression as a means of enhancing predictive capability in situations where they are reflective of the process of interest. A covariance function of a linear oscillator under random load is derived and used in a regression context to predict the displacements of a vibratory system. A simulation case study is used to demonstrate the enhancement over a standard Gaussian process regression model. ⁎ The authors would like to acknowledge the support of the EPSRC, particularly through grant reference number EP/S001565/

Convolution models for output only linear structural system identification and the problem of identifiability

This paper investigates the use of the Gaussian Process Convolution Model (GPCM) as an output only system identification tool for structural systems. The form of the model assumes a priori that the observed data arise as the result of a convolution between an unknown linear filter and an unobserved white noise process, where each of these are modelled as a GP. The GPCM infers both the linear time filter (which is the impulse response function, i.e. Green's function, of the system) and driving white noise process in a Bayesian probabilistic fashion with an approximate variational posterior over both signals. It will be shown that although the model structure is intuitive and sensible priors are applied, the GPCM falls short in recovering the linear impulse response of interest response due to the problem of identifiability. This is an interesting result indicating that physically informed kernel structures alone are not enough to recover the true impulse response in similar non-parametric probabilistic models. Despite this, the avenue of research remains highly promising, and several ideas are proposed to improve the model as a system identification tool

On the application of Gaussian process latent force models for joint input-state-parameter estimation : With a view to Bayesian operational identification

The problem of identifying dynamic structural systems is of key interest to modern engineering practice and is often a first step in an analysis chain, such as validation of computer models or structural health monitoring. While this topic has been well covered for tests conducted in a laboratory setting, identification of full-scale structures in place remains challenging. Additionally, during in service assessment, it is often not possible to measure the loading that a given structure is subjected to; this could be due to practical limitations or cost. Current solutions to this problem revolve around assumptions regarding the nature of the load a structure is subject to; almost exclusively this is assumed to be a white Gaussian noise. However, in many cases this assumption is insufficient and can lead to biased results in system identification. This current work presents a model which attempts the system identification task (in terms of the parametric estimation) in conjunction with estimation of the inputs to the system and the latent states --- the displacements and velocities of the system. Within this paper, a Bayesian framework is presented for rigorous uncertainty quantification over both the system parameters and the unknown input signal. A Gaussian process latent force model allows a flexible Bayesian prior to be placed over the unknown forcing signal, which in conjunction with the state-space representation, allows fully Bayesian inference over the complete dynamic system and the unknown inputs

Constraining Gaussian processes for physics-informed acoustic emission mapping

The automated localisation of damage in structures is a challenging but critical ingredient in the path towards predictive or condition-based maintenance of high value structures. The use of acoustic emission time of arrival mapping is a promising approach to this challenge, but is severely hindered by the need to collect a dense set of artificial acoustic emission measurements across the structure, resulting in a lengthy and often impractical data acquisition process. In this paper, we consider the use of physics-informed Gaussian processes for learning these maps to alleviate this problem. In the approach, the Gaussian process is constrained to the physical domain such that information relating to the geometry and boundary conditions of the structure are embedded directly into the learning process, returning a model that guarantees that any predictions made satisfy physically-consistent behaviour at the boundary. A number of scenarios that arise when training measurement acquisition is limited, including where training data are sparse, and also of limited coverage over the structure of interest. Using a complex plate-like structure as an experimental case study, we show that our approach significantly reduces the burden of data collection, where it is seen that incorporation of boundary condition knowledge significantly improves predictive accuracy as training observations are reduced, particularly when training measurements are not available across all parts of the structure

Distributions of fatigue damage from data-driven strain prediction using Gaussian process regression

Fatigue is a leading cause of structural failure; however, monitoring and prediction of damage accumulation remains an open problem, particularly in complex environments where maintaining sensing equipment is challenging. As a result, there is a growing interest in virtual loads monitoring, or inferential sensing, particularly for predicting strain in areas of interest using machine learning methods. This paper pursues a probabilistic approach, relying on a Gaussian process (GP) regression, to produce both strain predictions and a predictive distribution of the accumulated fatigue damage in a given time period. Here, the fatigue distribution is achieved via propagation of successive draws from the posterior GP through a rainflow count. The establishment of such a distribution crucially accounts for uncertainty in the predictive model and will form a valuable element in any probabilistic risk assessment. For demonstration of the method, distributions for predicted fatigue damage in an aircraft wing are produced across 84 flights. The distributions provide a robust measure of predicted damage accumulation and model uncertainty

Control of flexible structures using model predictive control and gaussian processes

There is a recognised need to address issues of vibration control by making use of recent developments in data-driven modelling. The present study considers the difficulties imposed by the limitations of the actuator in the range of active vibration control. The paper proposes and examines a data-based Gaussian process (GP) model of a proof mass actuator in a flexible structural framework, aiming to improve control performance. This requires incorporating an inverse GP of static nonlinearity within the Wiener-Hammerstein model. The model starts with designing model predictive control (MPC) for a cantilever beam, in which the aim is to identify the optimal control force. Utilising the GP is the second step towards quantifying the uncertainty and limitation of the proof mass actuator by designing an inverse GP for the static nonlinearity. This quantification forwards to an MPC controller using a steady-state target optimisation tracking approach, in which this controller provides the optimal voltage required to eliminate vibration within the controller's limitations. The numerical outcome shows that the proposed scheme was capable of supplying the necessary voltage, which eliminated the structure's vibration within an actuator's limits. The results of this work encourage additional research into the developed strategy, particularly in the context of experimental real-time implementation

Bayesian joint input-state estimation for nonlinear systems

This work suggests a solution for joint input-state estimation for nonlinear systems. The task is to recover the internal states of a nonlinear oscillator, the displacement and velocity of the system, and the unmeasured external forces applied. To do this, a Gaussian process latent force model is developed for nonlinear systems. The model places a Gaussian process prior over the unknown input forces for the system, converts this into a state-space form and then augments the nonlinear system with these additional hidden states. To perform inference over this nonlinear state-space model a particle Gibbs approach is used combining a “Particle Gibbs with Ancestor Sampling” Markov kernel for the states and a Metropolis-Hastings update for the hyperparameters of the Gaussian process. This approach is shown to be effective in a numerical case study on a Duffing oscillator where the internal states and the unknown forcing are recovered, each with a normalised mean-squared error less than 0.5%. It is also shown how this Bayesian approach allows uncertainty quantification of the estimates of the states and inputs which can be invaluable in further engineering analyses

A Bayesian methodology for localising acoustic emission sources in complex structures

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

Learning model discrepancy: A Gaussian process and sampling-based approach

Predicting events in the real world with a computer model (simulator) is challenging. Every simulator, to varying extents, has model discrepancy, a mismatch between real world observations and the simulator (given the ‘true’ parameters are known). Model discrepancy occurs for various reasons, including simplified or missing physics in the simulator, numerical approximations that are required to compute the simulator outputs, and the fact that assumptions in the simulator are not generally applicable to all real world contexts. The existence of model discrepancy is problematic for the engineer as performing calibration of the simulator will lead to biased parameter estimates, and the resulting simulator is unlikely to accurately predict (or even be valid for) various contexts of interest. This paper proposes an approach for inferring model discrepancy that overcomes non-identifiability problems associated with jointly inferring the simulator parameters along with the model discrepancy. Instead, the proposed procedure seeks to identify model discrepancy given some parameter distribution, which could come from a ‘likelihood-free’ approach that considers the presence of model discrepancy during calibration, such as Bayesian history matching. In this case, model discrepancy is inferred whilst marginalising out the uncertain simulator outputs via a sampling-based approach, therefore better reflecting the ‘true’ uncertainty associated with the model discrepancy. Verification of the approach is performed before a demonstration on an experiential case study, comprising a representative five storey building structure