# Data-Efficient Neural Network for Track Profile Modelling in Cold Spray Additive Manufacturing
###### Abstract
In CSAM, a mathematical Gaussian function model is often chosen due to the mass distribution of jetted powder being assumed to be of Gaussian function profile [24; 25]. Some other studies have utilised different mathematical function models, such as triangular [26; 27] and trapezoidal [28]. Due to the complex processes underlying each HPRAM, there is no agreement on the choice of a single mathematical function model with simplifying assumptions, and the suitable model often depends on process conditions and their combination [23], leading to limited prediction accuracy over a wide range of process conditions using only a single mathematical functional model. Data-driven modelling has emerged as an alternative approach due to its excellent non-linear mapping capability and increased accessibility of available software options [18; 29]. [PERSON] et al. developed an Artificial Neural Network (ANN) model to predict the height and width of a single-track bead profile in WAAM [30]. The results were compared with those of a quadratic regression model and showed that the ANN model outperformed in the prediction of key geometric features. However, data-driven modelling has previously been limited to only predicting the height and width in HPRAM, and unlike the mathematical modelling approach, it has not been adopted to describe an entire track profile. Thus, there has been no exploration of the technique beyond symmetric single-track profiles or predicting details in track profiles. Our previous study attempted to address this issue in CSAM, focusing on the ANN modelling of a single-track profile with high morphology at normal and off-normal spray angles [31]. Our results demonstrated the potential of a data-driven modelling approach for better prediction accuracy than a mathematical counterpart, i.e., the Gaussian function model. However, the limitation of a data-driven modelling approach was also observed in our previous study [31], namely, the necessity of a large amount of process training data to achieve a high prediction accuracy, which has also been identified recently in relevant manufacturing studies [32; 33]. This data scarcity issue is associated with high experimental costs and the lack of an automated measurement system in HPRAM. [PERSON] et al. applied a grey modelling technique for the first time in a thermal spray process in an attempt to overcome the issue, harnessing both mathematical (or white box) and data-driven (or black-box) modelling approaches [32]. Despite the reasonable prediction accuracy achieved in this study, the authors concluded that more complex and nonlinear phenomena existed and suggested further exploration of data-efficient modelling approaches to improve prediction accuracy. Therefore, this study focuses on the prediction of a single-track profile in CSAM, at both normal and off-normal spray angles, using a data-efficient ANN (DANN) approach to demonstrate that data-driven modelling can achieve better prediction accuracy than its mathematical counterpart that has already been adopted in CSAM. Inspired by the study by [PERSON] et al. [32], we leverage a mathematical function model as domain knowledge or the existing model at hand into the development of a DANN model. Specifically, a Gaussian function model, the model adopted elsewhere in CSAM studies, is selected with its coefficients computed by a quadratic regression model as applied in [22]. The significance of this study is four-fold: (1) the application of a data-driven modelling approach with a data-efficient focus in the prediction of a single-track profile in CSAM; (2) the comparative study among purely mathematical function, purely data-driven and data-efficient data-driven modelling approaches, in the context of HPRAM; (3) the demonstration that data-driven modelling can outperform more widely used mathematical modelling with appropriate data-efficient techniques in HPRAM; and (4) that existing models at hand can contribute to the development of a new data-driven model with better prediction accuracy without further experimentation.