## 2 Materials and Methods
### Direct, Inverse Neural Network-Based Material Parameter Identification Process
#### 2.2.2 Convolutional Neural Networks  
CNNs are often applied in the context of machine image processing [53], where the model learns an internal representation of a two- or three-dimensional input. However, in recent years they have been successfully applied in various other research areas [54], such as language processing tasks [55] and time series analysis [56]. In the latter, usually, one-dimensional sequences of data constitute the input. The model learns to extract features from sequences of observations and how to map the internal features to the possible outputs. Convolutional neural networks consist of one or more convolutional layer, followed by a pooling layer, and these units may be repeated arbitrarily several times [49] (see Figure 4). The central processing step is the convolution, which can be understood as applying and sliding a filter of definable size over the receptive fields of each data point in the time series. The repetition of this linear transformation results in a feature map that indicates the intensity with which the feature encoded in the weights of the filter is present over the length of the time series. Then, the values of the feature map are passed through a non-linear activation function. Applying several filters, e.g., to learn multiple discriminative features, to a time series, results in as many feature maps as filters were applied. The values of the filter depend highly on the investigated dataset [49; 57]. Typically, a pooling layer is used after convolution, which filters out redundant information of the resulting time series by aggregating over a sliding window. Despite the data reduction, the performance of the CNN is usually not reduced, but rather, the computational efficiency is increased, enabling the solving of more complex tasks and also counteracting overlearning. Often, for time series data, additional batch normalization over each channel is applied to enable faster convergence and to keep the mean output close to zero and the output standard deviation near to one. Through a flattened layer, the resulting arrays of pooled feature maps are transformed into a one-dimensional vector. Subsequently, this is fed as input to a fully connected layer (FCL), which resembles the architecture of the MLPs. This can be used for classification or regression, where the number of neurons corresponds to the number of classes or (material) parameters to be determined. Consequently, both feature extraction and regression operations are fused into one process which can be optimized to maximize prediction accuracy. This is an advantage of 1D CNNs, which additionally have comparatively low computational complexity [20, 49, 57]. Hence, CNNs can learn directly from raw time series data and do not require domain expertise to manually engineer input features. The time series used in this work can be interpreted as one-dimensional grid structures, with data points having local dependencies through the continuously increasing abscissa values (e.g., displacements). The main advantage results from the grid-like structure of the feature maps, which allows one to consider local dependencies in the input in learning the activation patterns. Strong correlations of neighboring data points can be detected by the feature maps and used as activation patterns. The CNNs possess comparatively fewer learnable parameters than MLPs, for which a separate weight is assigned to each input value. The greatest influence on the number of learnable parameters is caused by the width and depth of the subsequent MLP. In contrast, the convolutional layers of the CNN have significantly fewer parameters due to the shared weights and the sparse connections between the neurons [49]. Within this work, various CNN architectures from the literature were implemented in the self-developed framework of direct inverse MPI, and their performances were investigated in the comparative study. Their internal denotation and the corresponding literature sources of the CNNs are listed in Table 4. The CNN proposed by [PERSON] et al.