## 2 Materials and Methods
### Direct, Inverse Neural Network-Based Material Parameter Identification Process
#### 2.2.3 Bayesian Neural Networks
The negative sign is used because minimization is better to apply in computation and minimizing negative log-likelihood coincides with maximizing the likelihood. To obtain the MP estimates and the standard deviation, we sampled 500 times from the distribution object and calculated the statistics based thereon. In contrast, epistemic uncertainties can be reduced by adding more data. For implementation into the network, instead of deterministic weights or bias values, weight distributions are learned during the training process (see Figure 5). The preliminary distribution is called the prior, which is then adjusted during the learning process, yielding the posterior distribution. A variety of usable distributions exist, although the choice of a specific prior distribution is not straightforward [17]. The use of a normal distribution with zero mean, a standard deviation of one, and a diagonal covariance is often a suitable default setting [17], which is why this was also used within this work. For the implementation, the TFP DenseVariational layer (DVL) was applied, which allows performing complex calculations, such as the computation of the evidence's lower bound (ELBO). For further details on the mathematical backgrounds and software implementations, we refer to [16; 17] and the TensorFlow Probability package. As a consequence of implementing epistemic uncertainties into the network, it returns a different output for each prediction run, since a new set of weights are sampled from the distributions to build the BNN and predict an output. Hence, in the present study, we ran each prediction 100 times to determine the mean values of the outputs and corresponding standard deviations, whereas the latter should decrease when using more data. Due to the large input neuron number of 2400 for the time series, the additional parameters of the network (distribution parameters and covariance matrix) resulted in a very high number of network parameters to be trained, even with a small number of hidden layer neurons. This led to a very long training time or even to untrainable network architectures due to limited hardware resources. An applicable alternative is to utilize only a few or just the last layer as a stochastic one, which may be considered as learning a point estimate transformation, followed by a shallow BNN [17]. For the fully stochastic networks used, the number of points of the SOC was reduced to decrease the input data point amount and thus the network parameter number. In this work, different variants and settings for the BNN were used, which are listed in the Appendix in Table 14.  
Figure 5 illustrates a BNN based on an MLP, which can account for both aleatoric and epistemic uncertainties. BNNs have been previously been used for direct inverse MPI for a simple material model by [PERSON] et al. [18], who demonstrated some advantages in the context of MPI. In this work, the BNNs were applied to the proposed complex material models MAT_187_SAMP-1 and GISSMO, and we evaluated the effects of different settings on the prediction accuracy in comparison to other network types. Advantages for the subsequent numerical simulation are also shown, by which the effects of uncertainties can thus be evaluated. Furthermore, it is shown how well the underlying relationship could be learned and how much prediction repetitions or inherent uncertainties affect the prediction result.