# Gaussian Process Surrogates for Modeling Uncertainties in a Use Case of Forging Superalloys
## 1 Introduction  
Conducting experiments to better understand manufacturing processes is crucial, with real physical experiments being considered the gold standard. However, conducting real physical experiments for each new experimental setting is impractical because of expensive materials, production stoppages and labor hours for monitoring and evaluation. One good alternative is conducting experiments via simulation, where numerical methods-such as Finite Element Method (FEM)-present a well-observed method in the field of structural analysis. However, solving complex problems with FEM is time-consuming and computationally expensive. In order to reduce the computational effort, surrogate modeling may offer a promising solution [1]. Surrogate models are trained in a supervised manner and are designed to learn the function mapping between inputs and outputs. With a sufficient amount of training data with respect to the observed use case, a customized surrogate model is able to substitute for a FEM simulation up to a certain accuracy. When only specific dimensions with a controlled reduction in accuracy of a simulation result are desired, reduced-order surrogate modeling is an already known technique. Thus, reduced-order surrogate modeling aims to substitute the high-resolution simulation domain with some carefully selected dimensions of importance, e.g., selected displacement measures of a deformed part can be predicted by a reduced-order surrogate modeling with low computational effort, instead of performing a computationally intensive FEM simulation that predicts the displacement of each node representing the deformed part. Meanwhile, Gaussian process regression (GP) has been successfully used as a surrogate model in the past. In literature, GP regression is also called \"kriging\" after the statistician and mining engineer [PERSON] [2]. However, for consistency, we use only the term GP regression or plain GP in this paper. Regarding GP regression, one of the biggest advantages is that it predicts a distribution (described by mean and standard deviation) rather than just a point estimate. The predicted standard deviation can be seen as a quality criterion related to the corresponding predicted mean value. In the following, we will refer to that standard deviation of a prediction as epistemic uncertainty, i.e., how certain the model is with respect to its prediction. Considering real manufacturing processes, another source of uncertainty can be observed with regard to the lack of complete control over all influence parameters. These deviations occurring during repeated process iterations under the same conditions are referred to as aleatoric uncertainty. Recapitulating, we want to shed light on two types of uncertainties in surrogate modeling: (1) epistemic uncertainty referring to the lack of knowledge in respect to a simulation model and can be minimized by adding additional sources of information (with respect to machine learning models, it is mainly increasing the number of training instances at new locations in the feature space) and (2) aleatoric uncertainty referring to deviations of an observed manufacturing process itself, i.e., aleatoric uncertainty cannot be minimized even if more data is generated. Since epistemic and aleatoric uncertainties describe different properties, it seems natural to treat them separately when making predictions or optimization. However, it should be mentioned that in certain circumstances it may be useful to consider uncertainty as a whole rather than dividing it into aleatoric and epistemic uncertainty. In such cases, heteroskedastic GP regression represents a common approach for optimization with surrogate models [3; 4; 5]. In our problem definition, especially in solving inverse problems, we argue that the distinction of epistemic and aleatoric uncertainty shows clear advantages. There is a wealth of literature on surrogate models, reduced-order surrogate models, and optimization with GP regression.