## 4 Results
### Discussion  
We now proceed to summarize our results:  
* We consider the problem of predicting a discrete time curve that contains information about a test dummy during a crash event. We compare the DL methods proposed with two state-of-the-art methods, SST and ERC, in order to have a benchmark. The results in Tables 3 and 5 show that FFNNs have a clear predictive advantage in all the metrics analyzed. * It is known that FFNNs can be overconfident in their predictions [29]. In order to alleviate this problem, we also propose using BNNs, which not only give a prediction of the curve but also an uncertainty measurement. The main effect we observe is that BNNs cause a slight decrease in the performance metrics compared to FFNNs, as seen in Tables 7 and 8, but at the same time, they provide the information of which parts of the predicted curve are more certain. * DL approaches also excel in terms of computational power and time efficiency: while FE values were obtained using 45 processing hours in a 36 CPU cluster, SST, ERC, and DL methods were run on a standard computer (Processor Intel Core i7 3820, GPU Nvidia Titan V, 16 GB of RAM), with runtime in the range of minutes: ERC training took \(\sim\)100 s, while FFNN training took \(\sim\)7 s.  
Regarding potential limitations of our work, the dataset used is not too large, composed of 140 samples, due to the high computational load that entails generating one of these trajectories using FE simulations. Thus, it is important to ensure that the model is generalizing well; in order to do that, we follow standard practices by training using a set of samples and validating using a different set of samples, unseen during training. All the reported results are given using the validation trajectories. This is a limitation of our current work that can be addressed either by increasing the dataset size or by the use of techniques that specialize in cases of low sample size, i.e., few-shot learning techniques. We also highlight the importance of having an uncertainty prediction by using BNNs. As commented, NNs are known to be overconfident in their prediction [29], which can  
Figure 10: Comparison between predicted curves of the BNN and FFNN and the real curve chosen for two different test samples for the resultant head acceleration, where the average curve between the 100 predictions of the BNN is presented in red, the prediction of FFNN in green, the real values in black, and the BNN interval in blue. Note that the parts where the curve becomes less predictable have a higher uncertainty in the BNN (see Figure 2), indicating the regions where the prediction is less certain. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Metric** & **FFNN** & **BNN** \\ \hline \(R^{2}\) & 0.993 & 0.986 \\ MSE & 3.153\(\cdot 10^{-4}\) & 6.078\(\cdot 10^{-4}\) \\ MAE & 8.543\(\cdot 10^{-3}\) & 1.047\(\cdot 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Evaluation metrics comparison between FFNN and BNN for the mean value of 100 predictions for the Head Acceleration output curve. be mitigated by having not only an estimate of the trajectory, but also an estimation of its variance. Uncertainty has a key role in engineering [38], and in our problem, it means that a designer can obtain information regarding where the predictor is unsure about the trajectory, as well as information such as the maximum and minimum expected values of the trajectory (which in our cases are acceleration experimented by the test dummy). To sum up, we provide two different approaches: FFNNs provide a slightly better prediction, at the risk of overconfidence, whereas BNNs provide uncertainty estimates as well. We consider that both approaches are complementary and can be used to provide a very accurate first approximation for the prediction problem. Note that the low computational load of DL methods can be exploited also to make a first approximation of crash responses, which can later be refined (and validated) by making use of other tools, such as FE.