# Data-Efficient Neural Network for Track Profile Modelling in Cold Spray Additive Manufacturing
## 2 Materials and Methods
### Data-Efficient Artificial Neural Network Model Design and Training  
To demonstrate the effectiveness of leveraging a previous modelling attempt or existing model, a mathematical Gaussian function model, previously proposed in [24] and expressed in Equation 1, was selected and built with its free coefficients, A and \(\sigma\), being predicted using a quadratic regression model as applied in [22]. \[y=\frac{A}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{\pi}{\sigma})^{2}}, \tag{1}\]  
This selection was due to the mathematical model framework being capable of predicting an asymmetric single-track profile at off-normal angles and being often used in cold spray and CSAM [24,25,36]. For preparing the training dataset of outputs for the quadratic regression model, appropriate free coefficients were found through a Gaussian function equation curve fitting to each single-track profile shown in Figures S1-S3 of the \begin{table}
\begin{tabular}{c c c c} \hline
**Level** & **Spray Angle (\({}^{\circ}\))** & **Traverse Speed (mm/s)** & **Standoff Distance (mm)** \\ \hline
1 & 45 & 25 & 30 \\
2 & 60 & 100 & 40 \\
3 & 75 & 200 & 50 \\
4 & 90 & - & - \\ \hline \end{tabular}
\end{table} 
Table 1: The levels of process variables in the experimental design matrix for the preparation of single-track profiles used for the training of the proposed data-efficient neural network model. Supplementary Materials. The Curve Fitting Tool in MATLAB version R2018a was used with the trust-region-reflective algorithm and nonlinear least square method. The resulting free coefficients are summarised in Table S3 of the Supplementary Materials. With the experimental process parameters listed in Table 1 as inputs, the quadratic model was developed with the QR decomposition algorithm using the iterative reweighted least square method in Statistics and Machine Learning Toolbox, MATLAB version R2018a. The proposed DANN modelling framework is shown in Figure 1, using a static ANN model for the geometric prediction of a single-track profile in CSAM. The DANN was developed to predict a polar length at a polar angle from the Tool Centre Point (TCP). A data-driven model for predicting a complete single-track profile can be developed by sampling a sufficient number of geometric points from the fabricated single-track profiles, as demonstrated by the area validation method in our previous work [31]. The polar length was sampled at 2.72\({}^{\circ}\) intervals around the TCP, resulting in 67 points from each single-track profile. Two data-efficient techniques were adopted in the proposed modelling framework, leveraging the Gaussian function model as an existing model, discussed above, to develop an ANN model. In Technique 1, the polar length approximated from the Gaussian function model was used as another input variable in addition to spray angle, traverse speed, standoff distance and polar angle. This technique explicitly leverages the partial domain knowledge of cold spray deposition, represented by the previously proposed Gaussian function model. At the same time, the DANN model learnt to compensate for the discrepancies between this knowledge and the true CSAM process, as successfully observed in other physical science fields [41; 42]. Technique 2 was the augmentation of training data with a virtual input-output subset generated by the Gaussian function model. Therefore, the overall training dataset consisted of an empirical dataset prepared using  
Figure 1: The proposed data-efficient artificial neural network modelling framework for the prediction of a single-track profile in cold spray additive manufacturing. The CSAM process part represents a digital version of the experimental equipment and setup described in Section 2.1.  
the DOE method in Section 2.1 and a virtual dataset created using the Gaussian function model.