# Data-Efficient Neural Network for Track Profile Modelling in Cold Spray Additive Manufacturing
## 3 Results
### Data-Efficient Artificial Neural Network Model Evaluation and Comparison  
The proposed DANN model, incorporating Techniques 1 and 2, was evaluated and compared with other modelling approaches, including a mathematical Gaussian function model [24] and a purely data-driven ANN model [31]. Here, the mathematical Gaussian function model was not the Gaussian function model used for the data-efficient ANN model as the existing model, but one with the optimal coefficients, listed in Table S4 of the Supplementary Materials, that were found through the curve-fitting method described in Section 2.2. The resulting model was referred to as the curve-fitted Gaussian function model and prepared to allow for the comparison of the proposed data-efficient ANN model against the best predictive performance that could be achieved using the previously proposed framework for mathematical Gaussian function modelling [24]. For the purely data-driven ANN model, the best performance was achieved with the architecture [4571], resulting in an MSE of 3.852 \(\times\) 10\({}^{-3}\). Furthermore, to investigate the effectiveness of each data-efficient technique, the data-efficient ANN models built using Technique 1 or Technique 2 solely were evaluated and compared. The prediction results of each model are summarised in Table 2 in absolute percent error and visually presented in Figure 3.  
Compared with the purely data-driven ANN model, the data-efficient ANN model with Technique 1 or 2 alone showed better predictive performance with lower MAPEs and MXAPEs. This result indicates that both Technique 1 and 2 effectively achieved data-efficient learning and development of a data-driven ANN model. Furthermore, Technique 1 was more effective than Technique 2, with a MAPE half that of Technique 2. This result might be attributed to Technique 1 being more direct in guiding the learning process of weights and biases through the approximated target output (or polar length) than augmentation of the training dataset. Figure 2: The results of the developed data-efficient neural network model with [51141] architecture: (a) normalised data-efficient ANN predictions vs. target outputs (or polar lengths) with a Mean Squared Error (MSE) of 1.032 \(\times\) 10\({}^{-4}\); (b) the training process of the developed data-efficient ANN model, showing no overfitting and underfitting. Hence, the proposed DANN model that combined the two data-efficient techniques achieved better predictive performance than the purely data-driven ANN model, showing that all of the prediction errors fell below the MAPE of the purely data-driven ANN model. The DANN model was also found to outperform the curved-fitted Gaussian function model with a lower MAPE and MXAPE. Notably, there was a lower number of predictions with large absolute percent errors (i.e., narrower upper quartile), as seen in Figure 3. This predictive capability became more significant when the entire single-track profile was predicted in the CSAM profiles, as presented in Figure 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Absolute** & \multicolumn{3}{c}{**Data-efficient ANN**} & \multicolumn{2}{c}{**Curve-Fitted**} & **Purely Data-Driven** \\ \cline{2-5}
**Error \%** & **Tech. 1** & **Tech. 2** & **Tech. 1 + 2** & **Gaussian** & **ANN** \\ \hline Mean & 2.060 & 4.040 & 1.230 & 1.873 & 7.174 \\ Minimum & 0.003 & 0.003 & 0.006 & 0.001 & 0.060 \\ Lower Q & 0.8147 & 1.113 & 0.3724 & 0.2682 & 2.510 \\ Median & 1.719 & 2.795 & 0.9081 & 0.8204 & 5.306 \\ Upper Q & 3.004 & 5.173 & 1.753 & 2.619 & 9.831 \\ Maximum & 9.685 & 20.78 & 5.748 & 11.83 & 33.26 \\ \hline R\({}^{2}\) & 0.9984 & 0.9964 & 0.9988 & 0.9931 & 0.9925 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the prediction results in absolute percent error for the testing single-track profiles in CSAM. The results are presented for: data-efficient ANN with the two techniques applied individually (Tech. 1 and Tech. 2), both applied (Tech.