# Physics-Based Data Augmentation Enables Accurate Machine Learning Prediction of Melt Pool Geometry
## 2 Materials and Methods
### Explicit Model-Based Data Augmentation  
To compensate for the limited availability of experimental melt pool data and improve model generalization, this study integrates a physics-based explicit model to generate synthetic data. The analytical model adopted here builds upon the work of [PERSON] et al. [29], which demonstrated improved predictive accuracy over the classic Rosenthal model by incorporating empirical coefficients derived from extensive L-PBF experimental datasets. In this study, the melt pool geometry is characterized by its depth (\(D\)) and width (\(W\)), modeled as nonlinear functions of process parameters and thermophysical material properties. The governing equations are provided in Table 3.  
The explicit model improved in the article is based on the evaluation of [PERSON] et al. [29] from extensive experimental data processing, and significantly outperforms the Rosenthal melt pool geometry estimation. To accurately fit the problem across different measures and parameters, the nonlinear regression model is transformed into a constrained optimization problem, which is solved using the SciPy (v. 4.5.11) constrained optimization package and the trust domain method. In Equation (1), the model is structured as a product of power-law terms, where the constant variable is the multiplier (\(w_{0}\)), and each parameter has an exponential variable. Based on the constraints, Equations (2)-(5) are solved for the geometric estimation of the melt pool size. \[y=w_{0}P^{w_{1}}V^{w_{2}}\rho^{w_{3}}C_{p}{}^{w_{4}}k^{w_{5}}(T_{m}-T_{0})^{w_ {6}} \tag{1}\]  
where,  
\[ \begin{array}{l}(a)w_{1}+w_{3}+w_{5}=0\\ (b)2w_{1}+w_{2}-3w_{3}+2w_{4}+w_{5}=1\\ (c)-3w_{1}-w_{2}-2w_{4}-3w_{5}=0\\ (d)-w_{4}-w_{5}+w_{6}=0\end{array}  \tag{2}\]  
\[ \begin{array}{l}\min_{W}\
orm{y-w_{0}P^{w_{1}}V^{w_{2}}\rho^{w_{3}}C_{p}{}^{ w_{4}}k^{w_{5}}(T_{m}-T_{0})^{w_{6}}}\
orm{y-w_{0}P^{w_{1}}}_{2}^{2}\\ s.t.:(w_{1}+w_{3}+w_{5})^{2}+(2w_{1}+w_{2}-3w_{3}+2w_{4}+w_{5}-1)^{2}+\\ (-3w_{1}-w_{2}-2w_{4}-3w_{5})^{2}+(-w_{4}-w_{5}+w_{6})^{2}=0\\ D=0.37\times 10^{6}\times P^{0.51}V^{-0.46}\rho^{-0.46}C_{p}^{-0.46}k^{-0.06}(T_{m} -T_{0})^{-0.51}\end{array}  \tag{3}\]  
Figure 1: (**a**) Cross-sectional morphology of a single-track melt pool formed during L-PBF processing of 316 L stainless steel; (**b**) example of surface morphology of a melt pool; (**c**) 36 sets of test sample appearance based on L-PBF and 15 single-track melt pools per cube surface. \begin{table}
\begin{tabular}{c c} \hline \hline
**Task (Label)** & **Equation** \\ \hline Depth & \(D=A\times 10^{6}\times P^{0.51}V^{-0.46}\rho^{-0.46}C_{p}^{-0.46}k^{-0.06}(T_{m}-T _{0})^{-0.51}\) \\ Width & \(W=B\times 10^{6}\times P^{0.59}V^{-0.36}\rho^{-0.39}C_{p}^{-0.39}k^{-0.21}(T_{m} -T_{0})^{-0.60}\) \\ \hline \hline \end{tabular} where \(D\) is the depth of the melt pool; \(W\) is the width of the melt pool; \(P\) is the laser power; \(V\) is the scanning speed of the beam; \(C_{p}\) is the thermal conductivity of the alloy; \(k\) is the specific heat capacity of the alloy; \(p\) is the density of the alloy; \(T_{m}\) is the temperature; \(T_{0}\) is the temperature far away from the melt pool; and \(A\) and \(B\) are fitted constants calibrated using experimental data.  
\end{table} 
Table 3: Equations for modeling melt pool geometry. \[W=0.44\times 10^{6}\times P^{0.59}V^{-0.36}\rho^{-0.39}C_{p}^{-0.39}k^{-0.21}(T_{m}-T_{0 })^{-0.60} \tag{5}\]  
This study proposes a physics-informed machine learning framework, considering the laser penetration depth and absorption rate, thus changing the fixed coefficients in Equations (4) and (5) to variable calibration parameters \(A\) and \(B\), as shown in Table 3. It has been observed that there is an important relationship between laser penetration depth and absorption rate, and high energy input conditions, the complex physical reactions increase the depth of the melt pool [40], while a large number of small pore walls are generated that reflect the laser multiple times, thereby increasing the absorption rate.