# Surrogate Modelling Approach#

Part of a series: Uncertainty quantification in Implants Materials.

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## Surrogate Models#

It is necessary to repeatedly evaluate computational models when uncertainty is quantified or optimization is taken into account. This task cannot be accomplished by costly mathematical models. A surrogate model is constructed and evaluated instead of the original model to alleviate the computational burden. The surrogate models can be viewed as functional approximations that are built based on the original computational models; these models are more compact and faster to evaluate in a domain of interest defined by input parameters [Sobester *et al.*, 2008, Wang *et al.*, 2021, Kersaudy *et al.*, 2015, Zhang *et al.*, 2020]. Surrogate models are frequently used in data-driven appraches, in which data is utilized to reflect the behavior of a physical system. The need for such tool in these applications because the need for interpolation of the data from the deterministic simulations, e.g. finite element method (FEM) and finite volume methods (FVM). The main challenge most of those practical design problems share is that a single FEM simulation can take numerous hours or even days. The well-trained surrogate model will replace the original computational model for further UQ and sensitivity analysis, in which a large number of model evaluation is needed [Sudret *et al.*, 2017].

The choice of a surrogate model depends on the complexity of the system being modeled and the desired accuracy of the model. The more complex the system, the more complex the model will need to be in order to accurately approximate the behavior of the system. Additionally, the accuracy of the model will depend on the amount of data available and the accuracy of the data points. Finally, the type of model chosen will depend on the type of data available, such as whether the data points are linear or non-linear [Böttcher *et al.*, 2021, Ghanem *et al.*, 2017, Sobester *et al.*, 2008, Sudret *et al.*, 2017].
Surrogate models can be categorized according to their types, the data needed for their construction, or training strategies. Additionally, these models can be further classified based on their specific applications. In recent times, there has been a development of hybrid surrogate models, wherein multiple surrogate models are combined together [Böttcher *et al.*, 2021, Ghanem *et al.*, 2017, Sobester *et al.*, 2008, Hou and Behdinan, 2022].

## Constructing Surrogate Models#

In non-intrusive UQ approach, the computationally expensive model is treated as a black-box. The training data for the surrogate model are sampled from the input distribution in the parametric space of the input parameters. The sampled parameter combinations are evaluated using the expensive computational model (black-box) and the resulting observations are exploited to build the surrogate model on the whole domain of the input distribution [Al Baraghtheh *et al.*, 2023]. Figure Fig. 35. Note that, the sampling of the input variable range should be representative and capture all possible information about the original computational model [Wang *et al.*, 2016].

The training data, also known as numerical experiment design (DOE), is generated by evaluating the balck-box model output for a certain number of generated sample points of the input parameter vector. Here, the black-box form can be a physical or computer experiment; e.g. finite element code. The quantity of interest (QoI) simulated by the surrogate model, which for the degradation models could be volume loss, degradation rate or mass loss. The general mathematical presentation of the surrogate model is

where \(\mathcal{M}\) is the surrogate shape function that maps the black-box model for the input parameters \(f(\textbf{x})\), which is presented as a vector \(\mathcal{M}(\textbf{x})\) in \(\subset\mathbb{R}^M\), to a vector of QoI, which is denoted by \(\textbf{y}\).
The accuracy and reliability of surrogate models must be validated. Validation of the surrogate model predictions against the original complex model or experimental data [Sudret *et al.*, 2017, Zeller-Plumhoff *et al.*, 2022]. Therefore, it is essential to establish appropriate validation procedures and metrics to assess the performance of the surrogate models, for example using leave-out-error \(\varepsilon_{LOO}\) to estimate the performance of surrogate models, which is given by [Zeller-Plumhoff *et al.*, 2022]

## Surrogate Models In Application#

In the current case study of quantifying the uncertainty associated the degradation models of biodegradable implants, three different surrogate models are used; Kriging, Polynomial Chaos Expansion and Polynomial Chaos Kriging.

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## Contributors#

Jordan Gault, Berit Zeller-Plumhoff