288 K. Worden et al. carried out multiple times, one can define the training schedule (resp. testing schedule) as the set of schedules associated with acquiring data for training (resp. testing); the set being denoted by Dtr (resp. Dt ). (Of course, these sets are specific to a context and a schedule, but the notation will become too unwieldy if this is made explicit.) Now, a model of S for a context Cwill be defined as a mathematical functionMC which attempts to predict the behaviour of S for any schedule specific to the context C. Depending on the environmental and predictive variables, this may be a multi-scale and/or multi-physics model, and it will almost always be implemented in computer code in some appropriate language.6 Asimulationfor a context Cunder a schedule WC is then defined as, mC W(t) =MC[eC W(t) ≡WC] (34.2) Now, it is clear that one can obtain the simulation mC i (t) corresponding to a test TC i = { eC i ,rC i } (with i now a schedule label), so that one can attempt to assess the fidelity of the model by comparing its predictions to reality. Ametric on a given context Cwill be defined here simply as a function dC(x,y) such that dC(x,y) ≥0, with the zero only if x =y. (This is only one of the conditions for a true mathematical metric, but it will do here for now.) Finally, the main definitions of the paper are possible: Definition 2.1 ( -Mirror) Amodel MC for a given context Cis an -mirror if and only if dC(mC(t),rC(t) ≤ (34.3) for all scheduled tests in Dt . Definition 2.2 (Fitness-for-Purpose) Amodel MC is fit-for-purpose in a given context C iff it is an -mirror for C and ≤ T where T is a critical threshold based on engineering judgement and/or context requirements. 34.2.2 Hybrid Models and Uncertainty So far, only pure physics-based models have been considered; models sometimes termedwhite-box models. At the other end of the modelling spectrum are black-box models which are formed by taking a model basis with a universal approximation property, and tuning the parameters of the model to a set of observed data; examples of such models are artificial neural networks or support vector machines [8, 9]. One can also make use of hybrid or grey-box models, which combine some element specified by physics with an element of learning from data. Suppose that it is desirable or necessary to form or update a model based on data. The model will be established using data acquired from a training schedule Dtr and tested on data from a test schedule Dt . 7 The resulting model MhC(Dtr) is then an -mirror if it satisfies the conditions of Definition 2.1 onDt . There is no distinction here on how MhC(Dtr) is obtained. One might start with a white-box model and learn the parameters via system identification, or one might adopt a grey-box structure where a physics-based model is augmented with a nonparametric machine learner [10]. As the use of machine learning has been raised, it would seem to be an appropriate point to discuss uncertainty. This is because many modern machine learning algorithms are probabilistic and accommodate uncertainty directly. For example, Bayesian approaches to parameter estimation can characterise the entire density functions of parameters, rather than simply producing point estimates [11, 12]. Furthermore, nonparametric learners like Gaussian process regression can produce a natural confidence interval on predictions [13]. So, under the circumstances, one might allow the possibility that the model MhC(Dtr) is a function that returns a random variable, i.e. the simulation responses are stochastic processes, MCt =MhC[eC W(t)](Dtr) (34.4) 6In fact, it may be the case that different models are needed in order to completely cover the context of interest. For notational simplicity, it is assumed here that MC represents the set of relevant models, returning the values required by the overall context C; there is no overall loss of generality at this point. 7Following best practice in machine learning, different data sets are potentially required in order to fit parameters and establish hyperparameters [8]. In order to keep the notation simpler here and avoid confusion about the term ‘validation’, it is assumed that the modeller simply partitions Dtr appropriately.
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