## 3 Empirical Data Driven Methods for Production Process Modelling  
Production process modelling enables the development of stable and repeatable process models, which are of crucial relevance for manufacturing companies. Production processes can be viewed from different perspectives ranging from the overall production process chain including raw material delivery, manufacturing steps and the final stock exploitation with the final product to simulations of the microscopic material behaviour during a manufacturing step [46]. This section focuses on the modelling of manufacturing processes for control purposes. The following sections consider high-level perspectives accounting for multiple manufacturing machines and systems. In the domain of manufacturing processes, as for example additive manufacturing, models can be classified in analytical, numerical or empirical approaches [47]. Analytical models consist of the mathematical analysis of a physical problem described by partial differential equations (PDE) and given boundary conditions. Given the assumptions made, the obtained solution accurately represents the reflected domain. However, the process to determine analytical solutions is not trivial, and for complex processes, advanced numerical methods and demanding computational efforts are necessary to solve the considered physical equations. Numerical methods are typically finite element method (FEM), boundary element method (BEM), finite difference method (FDM) or finite volume method (FVM). The solving of PDE is an essential element for multi-physics simulations enabling the simulation of complex manufacturing processes. Offline process simulations allow to perform parameter or design engineering without performing costly experiments and are therefore widely spread in R&D departments. An important limitation of these methods is the computational burden for complex processes including multiple physical phenomena. Furthermore, the computational effort scales drastically with increasing simulation domain sizes. Besides computational drawbacks, setting up a model is not trivial and requires highly skilled employees to define model parameters. The model definition and its assumptions are crucial to obtain representative results for a predefined process window. In certain applications, it can be more suitable to create empirical models by performing a series of experiments and analysing the observations. Thus, relations between process inputs and obtained measurements can be captured without knowing the detail about the physical phenomena. This method is in particular suitable for low-cost experiments, where numerical or analytical models would require more resources. To approximate the captured observations in terms of an exploitable model, various approaches exist. One of the most straightforward methods in statistical analysis is linear regression, where a line is fitted to capture the relation between two variables. For more complex non-linear applications, machine learning based methods as neural networks (NN) have experienced a widespread adoption in recent years [48; 49; 50; 51; 52; 53]. Due to the increased availability of computational resources, the increasing development of machine learning libraries and the possibility to approximate highly non-linear relationships lead to many applications in industry, as for example machining operations and additive manufacturing [54]. The approximation capacities can be customized by varying NN hyperparameters as the number of hidden layers or the number of neurons of each layer. A further advantage of the NN-based model is the possibility to apply the model for similar processes. Therefore, only few additional data points are required to fine tune the already trained model by adapting the model to the changed conditions [55]. However, the main drawbacks of NN are the required amount of data to train the network and the reliability of the predictions. The structure of NNs makes it difficult to interpret and predict the behaviour, in particular for unseen data points.