## 3 Empirical Data Driven Methods for Production Process Modelling
Recent developments combine the advantages of physics-based simulations and neural networks by including the physical knowledge in the training of the neural network [56; 57; 58; 59]. In contrast to conventional neural networks, physics-informed neural networks contain additional loss terms. Besides the prediction error, terms satisfying the partial differential equation of the simulated domain penalize solutions disagreeing with the physical equations. As a result, less data are required to train and optimize the weights of the network, and predictions are more stable. Moreover, the additional loss terms shrink the solution space and prevent overfitting of the model. For data scare applications, this method is very promising but needs further research for industrial applications. In particular, the scalability needs to be investigated since the additional computations for each iteration and the coordinates dependent on PDE are potentially limiting factors for an adoption on larger part dimensions. With a similar objective, but without directly implementing the physics of the considered process, a recent method has been presented by [PERSON] et al. [60]. The authors presented the Fourier neural operator (FNO) to learn the underlying multiple partial differential equations (PDE) by mapping from a functional parametric dependence to the solution. The proposed architecture method appears to be significantly faster than traditional PDE solvers. To train the FNO, a sequence of timeframes of a physical phenomenon is given as input to predict the following time frames. The inputs are processed by transforming the input into the frequency space, filtering the higher frequency modes and transforming the filtered data back to the time space; furthermore, the inputs are multiplied with weights and summed up with the filtered results. Subsequently, the results are passed through an activation function, and the described layer is repeated multiple times. The advantages of the method are the mesh invariance and the capability of learning the relevant physics behind the process. Therefore, this method could be a promising approach for the modelling of manufacturing processes. First adoptions for solving the PDE for two-phase flow fields have been demonstrated by [PERSON] et al. [61]. However, the transfer from theoretical physical problems to empirical data is still missing. Besides AI-based algorithms inspired by the structure of NN, probabilistic approaches as Bayesian Optimization (BO) have also been proven to be suitable for data scare applications. Due to the probabilistic nature of BO, probabilities and uncertainties of the prediction are computed. These values enable the development of data sampling methods, minimizing the uncertainty of the predictions. As a result, less experimental data are required to obtain a model of the manufacturing process. [PERSON] et al. [62] successfully applied this method to optimize grinding parameters after a few trials for an unknown workpiece and tool combination. Table 2 summarizes the mentioned process modelling methods. However, a general challenge of process simulations remains the lack of generalizability of models and the increased computational burden for complex processes. Simulation domains are often tiny volumes and only represent a small element of the manufacturing step. Previously mentioned recent developments could be promising to handle both larger simulation domains and complex processes. Particularly, data-driven methods have recently shown great potential. Further research regarding the explainability and generalizability of data-driven methods must be carried out to enable an imminent transition to industry.