Machine learning for molten pool dynamics prediction in laser manufacturing

LeSong1

WenboZhou3

XuyangChen2✉Emailchenxy.2018@tsinghua.org.cn

LinlinJia4

ZhiyongHuang1

BaoruiDu2

1School of ComputingNational University of Singapore117576SingaporeSingaporeSingapore

2Institute of Engineering ThermophysicsChinese Academy of Sciences100190BeijingBeijingChina

3Institute of Systems ScienceNational University of Singapore119615SingaporeSingaporeSingapore

4iCoSys InstituteUniversity of Applied Sciences and Arts Western Switzerland100190BeijingBeijingChina

Abstract

This study investigates the influence of line energy density, power and scanning speed on the molten pool geometry and maximum solute concentration obtained under steady-state conditions in hybrid laser–MIG welding of aluminum alloys. A comprehensive numerical model incorporating multiple reflections, Fresnel absorption, and laser–arc coupling was established to simulate the thermo-fluid behavior during the welding process. Key characteristics, including molten pool width, height, depth, and magnesium concentration, were quantitatively analyzed. To enhance predictive capability and improve process optimization efficiency, the Support Vector Regression (SVR) method was introduced to develop a machine learning model. The results demonstrate that SVR effectively captures the nonlinear relationship between line energy density, power, scanning speed and molten pool responses, achieving high predictive accuracy across multiple target variables and providing a reliable basis for process parameter optimization and weld quality improvement.

Introduction

Hybrid laser–MIG welding combines the deep penetration of lasers with the filler feed capability of gas-metal-arc weldingCasalino2010. For aluminum alloys, this technology offers high deposition rates and good joint quality, yet the molten pool remains a complex, multiscale thermo-fluid systemZhong2025,moradi2013investigation,miao2017comparative. Understanding its dynamics is essential for microstructure control and defect mitigation, but existing research leaves key gapsWang2022,PARK2025104750,Kopp2022.

Early attempts to predict weld geometry using machine learning include the following examples: Zhao et al.ZHAO2023181 developed a physics-informed deep learning model for bidirectional predictions of molten pool dimensions and process parameters in laser powder bed fusion, achieving high accuracy. Rahman et al.Rahman2024 used machine learning to predict melt-pool geometry, achieving accurate results with Extra Trees and Gaussian process regression models. Wang et al.Cai2019 modeled the relationship between the principal components of high-speed weld images and weld width using back-propagation neural networks (BPNN). Jamnikar et al.jamnikar2021 used a convolutional neural network to map molten pool images and thermographic data to bead geometry and microstructural properties in wire-feed laser additive manufacturing. Hu et al.Hu2024 developed a physics-informed model to predict molten pool depth in fiber laser cladding, improving prediction accuracy significantly. Kumar et al.Kumar2025 developed a deep learning model to predict melt pool geometry and defects in laser powder bed fusion, outperforming traditional methods. Deng et al.DENG2025826 developed a dimensionless learning network (DimensionlessNet) based on Buckingham’s Pi theorem to predict molten pool morphology in laser welding.These studies demonstrate the potential of machine learning.

Overall, prior research has primarily focused on single-source laser welding, simplified heat source models, or involved high computational costs. Most importantly, few studies have explored the combined effects of line energy density (laser power divided by scanning speed)UNNI2023108042, laser power, and scanning speed on molten pool geometry and solute distribution in hybrid laser–MIG welding. Additionally, existing machine-learning models are rarely trained using simulation data that account for multiple reflections, Fresnel absorption, and laser–arc coupling, often limiting their predictions to a single response variable.

To address these gaps, the present study integrates finite volume simulations with support vector regression to predict molten pool geometry (width, height, and depth) and maximum solute concentration under steady-state conditions. Our numerical model incorporates the governing equations of continuity, momentum, energy, and species transport, representing the key physical processesMuch2024 involved. The model is validated against literature and experimental data. Simulation results over 209 combinations of laser power and scanning speed are used to train SVR models that capture nonlinear relationships between process parameters and molten pool responses. The proposed framework delivers accurate predictions while reducing computation time from hours to seconds, enabling efficient parameter optimisation and providing actionable insights for hybrid laser–MIG welding of aluminium alloys.

Mathematical model

The material of the base metal is A7N01 aluminum alloy, and the material of the filler wire is ER5356 aluminum alloy\cite{zhang2016softening}. Main components of the base metal and filler wire are aluminum, magnesium and zinc, and the content of other elements is quite little. Therefore, the composition of filler wire is assumed as 95 percent aluminum and 5 percent magnesium, while the composition of base metal is assumed as 95.5 percent aluminum and 4.5 percent zinc, and other elements are neglected. When the base metal begins to melt, the liquid flow in the molten pool is assumed to be Newtonian, incompressible and laminar\cite{wei2016origin}.

Material compositions of A7N01 aluminum and ER5356 welding wire1.

begin{table}[t]\caption{Material compositions of A7N01 aluminum and ER5356 welding wire}\label{Material compositions of A7N01 aluminum and ER5356 welding wire}\begin{tabular}{llllllllll}\\ \hline \\Materials & Mg & Zn & Fe & Mn & Si & Cr & Ti & Cu & Al \\A7N01 & 1.0-2.0 & 4.0-5.0 & 0.35 & 0.2-0.7 &

$(<)$

0.30 &

$(<)$

0.30 &

$(<)$

0.20 &

$(<)$

0.20 & Bal. \\ER5356 & 4.5-5.5 &

$(<)$

0.1 &

$(<)$

0.4 & 0.05-0.2 &

$(<)$

0.25 & 0.05-0.2 & 0.06-0.2 &

$(<)$

0.10 & Bal. \\\end{tabular}\end{table}

Governing equations

The nomenclature is provided2.

begin{table}[t]\caption{Nomenclature}\label{nomenclature}\begin{tabular}{llll}\\ \hline \\

$(g)$

& Acceleration due to gravity &

$(I)$

& Welding current \\

$(t)$

& Time &

$(h_{c})$

& Heat transfer coefficient \\

$(x_{i})$

& Distance along

$(i)$

directions &

$(t_{b x})$

& Tangential unit vector parallel to the x-z plane \\

$(u_{j})$

& Liquid velocity along

$(j)$

direction &

$(t_{\text {by }})$

& Tangential unit vector parallel to the y -z plane \\

$(P)$

& Pressure &

$(n_{b})$

& Outward normal vector \\V & Scanning speed &

$(f)$

& Frequency of droplet transfer \\

$(f_{l})$

& Liquid fraction &

$(\rho_{w})$

& Density of welding wire \\

$(C_{p})$

& Specific heat &

$(C_{w})$

& Specific heat capacity of welding wire \\

$(\Delta H)$

& Latent heat content &

$(d_{w})$

& Wire diameter \\

$(F)$

& Volume force &

$(U_{w})$

& Wire feed speed \\

$(F_{L x})$

& Electromagnetic force along x direction &

$(T)$

& Temperature \\

$(F_{L y})$

& Electromagnetic force along y direction &

$(T_{w})$

& Droplet temperature \\

$(F_{L z})$

& Electromagnetic force along z direction &

$(T_{l})$

& Liquidus temperature \\

$(h)$

& Sensible heat enthalpy &

$(T_{\text {ref }})$

& Reference temperature \\

$(k)$

& Thermal conductivity &

$(T_{a})$

& Ambient temperature \\C & Concentration & & \\

$(C_{l})$

& Concentration in liquid phase & \multicolumn{2}{|l|}{Greek symbols} \\

$(C_{s})$

& Concentration in solid phase &

$(\rho)$

& Density \\

$(C_{\text {ref }})$

& Reference Concentration &

$(\mu)$

& Viscosity \\

$(D)$

& Diffusion coefficient of element &

$(\mu_{m})$

& magnetic permeability \\

$(Q_{\text {laser }})$

& Body heat source of laser &

$(\beta)$

& Coefficient of volume expansion \\

$(Q_{\text {arc }})$

& Body heat source of arc &

$(\gamma)$

& Surface tension \\

$(Q_{d})$

& Sensible heat from droplets &

$(\frac{d \gamma}{d T})$

& Temperature coefficient of surface tension \\

$(q_{\text {laser }})$

& Surface heat source of laser &

$(\sigma)$

& Stefan-boltzmann constant \\

$(r)$

& Radius &

$(\sigma_{j})$

& Radius of the arc pressure \\

$(H)$

& Thickness of workpiece &

$(\varepsilon)$

& Surface emissivity \\L & Characteristic length & & \\\end{tabular}\end{table}

The continuity, momentum, energy and species equations are used as follows\cite{chen2020numerical}\cite{rao2013modelling}\cite{wei2015evolution}.

Continuity Equations:

$\frac{\partial \rho}{\partial t}+\frac{\partial\left(\rho u_i\right)}{\partial x_i}=0$

eq1

Momentum Equations:

begin{equation}\begin{aligned}\frac{\partial\left(\rho u_j\right)}{\partial t}+\frac{\partial\left(\rho u_i u_j\right)}{\partial x_i} & =\frac{\partial}{\partial x_i}\left(\mu \frac{\partial u_j}{\partial x_i}\right)+(-\frac{\partial P}{\partial x_i})+\frac{\partial}{\partial x_j}\left(\mu \frac{\partial u_j}{\partial x_i}\right)-V \frac{\partial\left(\rho u_i\right)}{\partial x} \\ & \quad -C\left(\frac{\left(1-f_l\right)^2}{f_l^3+B}\right) u_i+F \end{aligned}\label{eq2}\end{equation}Energy Equations:

begin{equation}\begin{aligned}\frac{\partial(\rho h)}{\partial t}+\frac{\partial\left(\rho u_i h\right)}{\partial x_i} & =\frac{\partial}{\partial x_i}\left(\frac{k}{C_p} \frac{\partial h}{\partial x_i}\right)-\frac{\partial(\rho \Delta H)}{\partial t}-\frac{\partial\left(\rho u_i \Delta H\right)}{\partial x_i}-V \frac{\partial(\rho h)}{\partial x}\& \quad-V \frac{\partial(\rho \Delta H)}{\partial x}+Q_{\text {laser }}\end{aligned}\label{eq3}\end{equation}

Species Equation:

begin{equation}\begin{aligned}\frac{\partial(\rho C)}{\partial t}+\frac{\partial\left(\rho u_i C\right)}{\partial x_i} & =\frac{\partial}{\partial x_i}\left(\rho D \frac{\partial C}{\partial x_i}\right)+\frac{\partial}{\partial x_i}\left(\rho D \frac{\partial\left(C_l-C\right)}{\partial x_i}\right)\& \quad-\frac{\partial}{\partial x_i}\left(\rho f_s\left(C_l-C_s\right) u_i\right)\end{aligned}\label{eq4}\end{equation}

The porous media term in the right-hand side of Equation (2) was represented on the basis of Carman-Kozeny equation.

$F$

is the volume force, including electromagnetic force and buoyancy caused by concentration and temperature variations. The electromagnetic force can be expressed as:

$F_{L x}=-\frac{\mu_{m} I^{2}}{4 \pi^{2} \sigma_{j}^{2} r} \exp \left(-\frac{r^{2}}{2 \sigma_{j}^{2}}\right)\left[1-\exp \left(-\frac{r^{2}}{2 \sigma_{j}^{2}}\right)\right]\left(1-\frac{z}{H}\right)^{2} \frac{x}{r} \tag{5}$

$F_{L y}=-\frac{\mu_{m} I^{2}}{4 \pi^{2} \sigma_{j}^{2} r} \exp \left(-\frac{r^{2}}{2 \sigma_{j}^{2}}\right)\left[1-\exp \left(-\frac{r^{2}}{2 \sigma_{j}^{2}}\right)\right]\left(1-\frac{z}{H}\right)^{2} \frac{y}{r} \tag{6}$

$F_{L z}=-\frac{\mu_{m} I^{2}}{4 \pi^{2} \sigma_{j}^{2} r} \exp \left(-\frac{r^{2}}{2 \sigma_{j}^{2}}\right)\left[1-\exp \left(-\frac{r^{2}}{2 \sigma_{j}^{2}}\right)\right]\left(1-\frac{z}{H}\right) \tag{7}$

The expression of buoyancy is as follows:

$F_{b}=\rho g \beta\left(T-T_{\text {ref }}\right)+\rho g \beta\left(C-C_{\text {ref }}\right) \tag{8}$

The last two terms in Eq. (4) represent the solute partitioning at the liquid-solid interface [20]. The formulation of nominal concentration [26], density, thermal conductivity, heat capacity, and mass diffusion coefficient for mushy zone can be summarized as follows.

$C=f_{s} C_{s}+f_{l} C_{l} \tag{9}$

$\rho=f_{s} \rho_{s}+f_{l} \rho_{l} \tag{10}$

$k=\left(\frac{g_{s}}{k_{s}}+\frac{g_{l}}{k_{l}}\right)^{-1} \tag{11}$

$c=f_{s} c_{s}+f_{l} c_{l} \tag{12}$

$D=f_{l} D_{l} \tag{13}$

Boundary conditions

The boundary condition for energy can be given as

$-k \nabla T \cdot \boldsymbol{n}_{b}=q_{\text {laser }}+q_{\text {arc }}+\mathrm{Q}_{\mathrm{d}}(x, y)-\sigma \varepsilon\left(T^{4}-T_{a}^{4}\right)-h_{c}\left(T-T_{a}\right) \tag{14}$

The detailed description of heat source model of laser and arc, and free surface evolution of the molten pool can be found in the literature.

The heat distribution of droplet was approximated as a Gaussian:

$Q_{d}(x, y)=\frac{C_{w} \rho_{w} \frac{\pi}{4} d_{w}^{2} U_{w}\left(T_{w}-T_{l}\right)}{2 \pi r_{f}^{2} f} \exp \left(\frac{x^{2}+y^{2}}{2 r_{f}^{2}}\right) \tag{15}$

The boundary condition for momentum can be expressed as

$\mu \frac{\partial u}{\partial z} \cdot \boldsymbol{n}_{b}=f_{l} \frac{\partial \gamma}{\partial T}\left(\frac{d T}{d x} \cdot \boldsymbol{t}_{b x}\right)+f_{l} \frac{\partial \gamma}{\partial C}\left(\frac{d C}{d x} \cdot \boldsymbol{t}_{b x}\right) \tag{16}$

$\mu \frac{\partial \nu}{\partial z} \cdot \boldsymbol{n}_{b}=f_{l} \frac{\partial \gamma}{\partial T}\left(\frac{d T}{d y} \cdot \boldsymbol{t}_{b y}\right)+f_{l} \frac{\partial \gamma}{\partial C}\left(\frac{d C}{d y} \cdot \boldsymbol{t}_{b y}\right) \tag{17}$

$w \cdot \boldsymbol{n}_{b}=0 \tag{18}$

The Marangoni convection caused by temperature and concentration variation is represented at right-hand side in Eqs. (16) and (17).

Material properties of substrate and welding wire are as follows3.

begin{table}[t]\caption{Material properties}\label{Material properties}\begin{tabular}{lll}\\ \hline \\Property (Unit) & Substrate (A7N01) & Welding wire (ER5356) \\Solidus temperature(K) & 858 & 813 \\Liquidus temperature(K) & 923 & 908 \\Density of liquid metal (

$(\mathrm{kg} / \mathrm{m}^{3})$

) & 2700 & 2380 \\Enthalpy of solidus temperature (J/kg) &

$(7.6 \times 10^{5})$

$(7.4 \times 10^{5})$

\\Enthalpy of liquidus temperature (J/kg) &

$(1.1 \times 10^{6})$

$(1.03 \times 10^{6})$

\\Specific heat of liquid (J/(kg. K)) & 1200 & 1135 \\Specific heat of solid (J/(kg. K)) & 881 & 913 \\Thermal conductivity of liquid ((W/(

$(\text{m}\cdot\text{s})$

))) & 80 & 83 \\Thermal conductivity of solid ((W/(

$(\text{m}\cdot\text{s})$

))) & 101 & 108 \\Effective mass diffusivity (

$(\mathrm{m}^{2} / \mathrm{s})$

) &

$(7^{*} 10^{-7})$

$(7 * 10^{-7})$

\\Temperature coefficient of surface tension (

$(\mathrm{N} /(\mathrm{m} \cdot \mathrm{K}))$

) &

$(-1.55 \times 10^{-4})$

$(-1.7 \times 10^{-4})$

\\Viscosity of liquid ((kg/(

$(\text{m}\cdot\text{s})$

))) &

$(1.0 \times 10^{-3})$

\\\end{tabular}\end{table}

Processing parameters are as follows4.

begin{table}[t]\caption{Processing parameters}\label{Processing parameters}\begin{tabular}{ll}\\ \hline \\Parameter (Unit) & Value \\\hlineVoltage(V) & 21.3 \\Welding current(A) & 150 \\Effective laser radius (mm) & 0.15 \\Wire diameter (mm) & 1.2 \\Droplet temperature(K) & 1923 \\Wire feeding rate (m/min) & 9.3 \\Ambient temperature(K) & 298 \\Radius of droplet impact (mm) & 1.0 \\Distance between MIG torch and laser (mm) & 2 \\Frequency of droplet transition (Hz) & 317 \\Stefan--Boltzmann constant

$(\sigma)$

(W/(m

$(^{2})$

·K

$(^{4})$

)) &

$(5.670\times10^{-8})$

\\Surface emissivity & 0.47 \\\end{tabular}\end{table}

SVR modelling

Support Vector Regression (SVR) is a classic regression algorithm in machine learning, built on the theoretical framework of Support Vector Machines (SVM)\cite{article}. Unlike traditional regression methods, SVR fits data by finding an optimal hyperplane while allowing a certain margin of error (introduced via slack variables and a tolerance parameter), making it particularly effective for handling nonlinear and high-dimensional data\cite{xu2014application}PENG2023106812.

In nonlinear SVR, the regression function is expressed in its dual form as:

$f(x) = \sum_{i=1}^n (\alpha_i - \alpha_i^*) K(x_i, x) + b$

where:

$\alpha_i, \alpha_i^*$

are Lagrange multipliers introduced in the dual optimization problem. They represent the contribution of each training point to the regression function. Only points that lie outside the

$\epsilon$

-insensitive margin (support vectors) have non-zero multipliers.

$K(x_i, x)$

is the kernel function, enabling nonlinear mapping of input data into a higher-dimensional feature space.

$b$

is the bias term.

In this study, the Radial Basis Function (RBF) kernel is employed, which is defined as:

$K(x_i, x) = \exp\left(-\gamma \|x_i - x\|^2\right)$

where

$\gamma$

is the kernel width parameter that controls the influence range of a training sample. A larger

$\gamma$

produces a more complex model (risk of overfitting), while a smaller

$\gamma$

yields a smoother model (risk of underfitting).

The primal optimization problem of SVR is formulated as:

$\min_{\mathbf{w}, b, \xi, \xi^*} \frac{1}{2} \|\mathbf{w}\|^2 + C \sum_{i=1}^n (\xi_i + \xi_i^*)$

subject to:

$y_i - f(x_i) \leq \epsilon + \xi_i$

$f(x_i) - y_i \leq \epsilon + \xi_i^*$

$\xi_i, \xi_i^* \geq 0$

where:

$\|\mathbf{w}\|^2$

is the regularization term controlling model complexity;

$\epsilon$

is the tolerance margin, within which errors are ignored;

$\xi_i, \xi_i^*$

are slack variables for deviations beyond

$\epsilon$

;

$C$

is the penalty parameter balancing model complexity and margin violations\cite{Rivas-Perea2013}.

To evaluate model performance, this study employs the coefficient of determination,

$R^2$

, defined as:

$R^2 = 1 - \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{\sum_{i=1}^n (y_i - \bar{y})^2}$

where:

$y_i$

is the true value,

$\hat{y}_i$

is the predicted value,

$\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i$

is the mean of the true values.

Here, the numerator

$\sum (y_i - \hat{y}_i)^2$

is the Residual Sum of Squares (RSS), which measures the discrepancy between predictions and observations, while the denominator

$\sum (y_i - \bar{y})^2$

is the Total Sum of Squares (TSS), reflecting the total variance of the data.

$R^2$

value close to 1 indicates excellent predictive performance,

$R^2 = 0$

means the model performs no better than predicting the mean, and

$R^2 < 0$

implies the model performs worse than the mean baseline\cite{Sekeroglu}.

In this work, SVR with the RBF kernel is employed to establish predictive models for the relationship between line energy density and melt pool characteristics. Input features and output variables are standardized, and hyperparameters

$C$

$\gamma$

, and

$\epsilon$

are optimized using grid search combined with cross-validation. The results demonstrate that the RBF-based SVR effectively captures the nonlinear dependence of molten pool responses on line energy density, achieving high predictive accuracy and providing a robust data-driven tool for welding process optimization. The overall framework is illustrated in Figure1.

Fig. 1

Overall framework of simulation –SVR hybrid modelling

Numerical approach

The computational domain is \SI{44}{\milli\meter}

$\times$

SI{10}{\milli\meter}

$\times$

SI{6}{\milli\meter}, and half of the workpiece is considered for high efficiency. A block-structured grid with refinement near the weld pool area is used to capture the intricate thermal and fluid flow phenomena accurately. The grid is divided into three x-zones, two y-zones, and two z-zones, resulting in a total of about \num{371200} elements for the entire domain. In the weld pool region, the smallest grid spacing is approximately \SIrange{0.12}{0.15}{\milli\meter}, while coarser spacing up to \SI{0.5}{\milli\meter} is applied farther away. To capture the rapid changes in temperature and fluid flow during welding, especially the molten pool dynamics, a time step of \SI{0.005}{\second} is used. The typical residual value for convergence in welding simulations is set at \num{1e-6} to determine the convergence of the solution in iterative calculations. During the welding process, the welding current is \SI{150}{\ampere}, the wire feed speed is \SI{0.15}{\meter\per\second} (

$\approx$

SI{9.0}{\meter\per\minute}), the distance between the laser and MIG torch is \SI{2}{\milli\meter}, and the inclination angle of the welding gun is \SI{22}{\degree}. The laser and MIG torch scan along a straight path. All computational programs used in this paper were developed in-house using the \texttt{C++} language.

Results and discussion

Simulation Results

The numerical simulations provide the foundation for machine learning training in this study. Due to the high cost of experiments, the scarcity of experimental data, and potential inaccuracies caused by measurement uncertainties (e.g., equipment misalignment), it is impractical to rely solely on experimental measurements to provide sufficient training samples. Therefore, simulation data generated by the finite volume method (FVM) were employed as the primary dataset.

Specifically, simulations were performed with laser power ranging from 1000 W to 2000 W (increment of 100 W) and scanning speed from 0.006 m/s to 0.024 m/s (increment of 0.001 m/s), total 209 combinations of laser power and scanning speed.For each operating condition, the transient evolution of the temperature field and velocity components (

$u, v, w$

) was computed at the discretized grid points (

$x, y, z$

). After reaching steady-state convergence, the molten pool geometry (width, height, and depth) and the maximum solute concentration (

$C_\text{max}$

) were extracted.

These simulations required substantial computational resources, with each case taking approximately ten hours to complete.

Due to limited space, we only show the cloud diagram of the melt pool dynamics when the scanning speed is 18mm/s and 6mm/s.as shown in figure 4. The mass transfer of Mg element in the hybrid laser-MIG welding process is shown in figure 7, which shows the concentration distribution of Mg at 0.5 s and 1.25 s, respectively. The concentration of magnesium is expressed by mass percentage.

Fig. 4

Temperature distribution at different times (a)t=1.25s,v=18mm/s (b)t=1.25s,v=6mm/s.

Fig. 7

Concentration distribution of magnesium at different times (a) 1.25s,v=18mm/s,(b) 1.25s,v=6mm/s.

Initially, due to the heating of laser and arc, the temperature of the heating area rises rapidly, the workpiece begins to melt and the molten pool begins to form. In the center of laser action, the penetration is increased significantly due to the multiple reflection and Fresnel absorption of keyhole. The temperature coefficient of surface tension is negative, which leads to the flow of liquid metal from the center of molten pool to the edge of molten pool. The liquid metal tends to flow downward and the outward flow is enhanced by hydrostatic pressure. Because of the high energy density of the laser beam, the temperature around the keyhole is higher than that of other regions, and the surface tension is small, then the liquid flows to the region. Due to the effect of electromagnetic force, the fluid in the molten pool tends to flow downward and forms a counter-clockwise eddy flow under the joint action of other driving forces. On the upper surface, the shape of the molten pool is elliptical, and the liquid tends to flow from the center to the edge. This complex flow pattern, influenced by surface tension, electromagnetic force, and hydrostatic pressure, is critical for understanding the weld pool dynamics.

begin{table}[t] \caption{Results of Characteristics of the molten pool at different scanning speeds} \label{sample-table} \begin{tabular}{llll} \multicolumn{2}{c}{\bf } & \multicolumn{2}{c}{\bf } \\ \hline \\ Scanning speed & 18mm/s & 12mm/s & 6mm/s \\ Max temperature & 3004K & 3088K & 3189K \\ Max velocity & 1.23m/s & 1.26m/s & 1.29m/s \\ length & 17.5mm & 16.0mm & 14.6mm \\ width & 5.3mm & 5.8mm & 6.3mm \\ depth & 1.2mm & 1.7mm\footnotemark[1] & 2.5mm \\ height & 1.7mm & 2.1mm & 2.8mm\footnotemark[2] \\ \end{tabular} \end{table}

SVR Prediction Results

Figures 8--11 present the prediction results of the Support Vector Regression (SVR) models trained on line energy density, power, and scanning speed as inputs to estimate the geometry of the molten pool (width, height, and depth) and the maximum solute concentration (

$C_\text{max}$

). The points in the figures are derived from the dataset, which is obtained through numerical simulations. The surfaces in the figures represent the SVR predicted surfaces at a fixed power value. The red dashed lines indicate the distances from the data points to the surface, which represent the prediction errors

Figure 12 (a-d) show the results of the Support Vector Regression (SVR) models, trained on line energy density, power, and scanning speed as inputs, for predicting the molten pool geometry (width, height, and depth) and the maximum solute concentration (

$C_\text{max}$

) on the test dataset. The x-axis represents each data point in the test set, and the y-axis shows the corresponding values. The black markers represent the values calculated by numerical simulation, while the colored markers represent the values predicted by the SVR model.

The accuracy for molten pool width is 97.0%, calculated as:

[\text{Accuracy} = \left( 1 - \frac{1}{n} \sum_{i=1}^{n} \left| \frac{y_{\text{pred},i} - y_{\text{true},i}}{y_{\text{true},i}} \right| \right) \times 100%\]

Similarly, the accuracy for molten pool height is 98.5%, the accuracy for molten pool depth is 76.2%, and the accuracy for maximum solute concentration (

$C_\text{max}$

) is 98.0%. These results indicate that the SVR model successfully captures the nonlinear growth trends for molten pool width, height, and

$C_\text{max}$

. The accuracy values demonstrate a strong agreement between the predicted values and the actual numerical simulation results, highlighting the effectiveness of the SVR model in modeling complex nonlinear relationships. Additionally, the molten pool depth prediction, while less accurate, still shows a reasonable alignment with the numerical data, suggesting the model can benefit from further refinement for more sensitive parameters.

Fig. 8

Width Prediction Surfaces under Varying Scanning Speeds and Energy Densities at Fixed Laser Powers (1000 –2000 W)

Fig. 9

Height Prediction Surfaces under Varying Scanning Speeds and Energy Densities at Fixed Laser Powers (1000 –2000 W)

Fig. 10

Depth Prediction Surfaces under Varying Scanning Speeds and Energy Densities at Fixed Laser Powers (1000 –2000 W)

Fig. 11

Maximum Solute Concentration Prediction Surfaces under Varying Scanning Speeds and Energy Densities at Fixed Laser Powers (1000 –2000 W)

Fig. 16

Comparison between SVR Predictions and Numerical Simulations on the Test Set for Width, Height, Depth, and

$(C_\text{max})$

Simulation Results vs SVR Prediction Results

Figures 17presents a comparison between the numerical simulation time and SVR prediction time on the test set. The simulation time is represented in blue, while the SVR prediction time is shown in red, with their average values indicated by horizontal dashed lines. The average simulation time (16.32 hours) is significantly higher than the average SVR prediction time (0.000351 hours), highlighting the efficiency of the SVR model in predicting time costs. There is considerable variation in the time costs for both methods, with SVR consistently predicting faster than numerical simulations.

Figure 18 shows the performance of the SVR model in predicting molten pool characteristics (Width, Height, Depth, and Maximum Solute Concentration,

$(C_{\text{max}})$

) by comparing the training and test R² scores. The graph shows that the SVR model performs exceptionally well in predicting molten pool width and height, with training R² values of 0.929 and 0.996, and test R² values of 0.916 and 0.975, respectively. For molten pool depth, the training R² is 0.841 and the test R² is 0.659. For maximum solute concentration (

$(C_{\text{max}})$

), the training R² is 0.820 and the test R² is 0.890. These results demonstrate that the SVR model shows strong predictive capabilities for all molten pool variables.

Fig. 18

SVR Model Performance: Training vs Test

$(R^2)$

Score

One of the major advantages of introducing machine learning lies in the significant reduction of computational cost. For each operating condition, a full numerical simulation based on the finite volume method requires approximately 10 hours of computation time on a standard workstation, as it resolves detailed transient thermal-fluid dynamics until steady-state convergence is reached. In contrast, once trained, the SVR model can provide predictions of molten pool geometry and maximum solute concentration for a new line energy density in only a few seconds.

This difference becomes particularly critical when a large number of operating conditions must be explored. For example, predicting 100 new parameter combinations would take nearly 1000 hours using numerical simulations, but less than two minutes with the SVR model. Such efficiency enables rapid process optimization: researchers no longer need to repeatedly perform time-consuming numerical simulations. Instead, the trained SVR can serve as a surrogate model that instantly provides accurate predictions for new inputs that have not been explicitly simulated but still fall within the reasonable range of the training data, thereby greatly accelerating design iterations and parameter optimization.

Experiment validation

As a representative case, we investigated the effect of scanning speed on molten pool size and composition distribution. A range of scanning speeds, including 1 mm/s, 3 mm/s, 5 mm/s, 7 mm/s, 11 mm/s, 13 mm/s, 15 mm/s, 17 mm/s, 19 mm/s, and 21 mm/s, was selected as the key parameter for this analysis. For each scanning speed, the molten pool dimensions and the distribution of key alloying elements, such as magnesium, were calculated using a numerical model. These results served as a benchmark for evaluating the machine learning predictions.

We utilized the Support Vector Regression (SVR) model to capture complex relationships between input parameters and output responses. The SVR model was trained on a subset of numerical simulation data to predict the molten pool size and composition distribution at different scanning speeds. To validate the reliability of the machine learning approach, we compared the SVR predictions with both the results obtained from the numerical model and experimental data. These comparisons, illustrated in the accompanying figure, demonstrate the high accuracy and generalization ability of the SVR model.

Fig. 23

Comparison of SVR predicted, calculated, and experimental value(a) Depth (b) Width (c) Height (d) Mg concentration.

The calculation results shown in the figure 23 provide a detailed comparison of the SVR (Support Vector Regression) model's predictions with numerical simulation results and experimental data. From the analysis, it can be observed that the minimum error between the SVR predictions and the numerical calculations is only 1.05 percent, while the maximum error is 5.6 percent. Similarly, the minimum error between the SVR predictions and the experimental results is 2.3 percent, and the maximum error is 8.5 percent. Both sets of errors remain consistently below 10 percent, demonstrating the reliability and accuracy of the SVR model.

The data comparison and analysis revealed that, for the maximum magnesium concentration in the molten pool, the average error between the SVR prediction results and the numerical calculations was 2.56 percent, while the average error between the SVR prediction results and the experimental data was 4.83 percent. Both error rates are consistently below 10 percent, demonstrating the reliability and accuracy of the SVR model.

Figure 23 shows that SVR predictions are closer to the experimental data compared to the numerical simulation , indicating better prediction accuracy for SVR in most conditions.

Conclusion

This study investigated the effect of line energy density, power and scanning speed on the steady-state molten pool characteristics in hybrid laser–MIG welding of aluminum alloys by combining high-fidelity numerical simulations with machine learning techniques. The key findings are summarized as follows:

A finite volume method (FVM)-based numerical model incorporating multiple laser reflections, Fresnel absorption, and laser –arc interactions was developed to simulate the thermo-fluid behavior of the molten pool. Under steady-state conditions, the molten pool geometry (width, height, and depth) and the maximum solute concentration (

$(C_\text{max})$

) were extracted total 209 combinations of laser power and scanning speed.

The results indicate that line energy density, power and scanning speed are critical parameters governing molten pool dynamics. The molten pool width, depth, and maximum solute concentration exhibit pronounced nonlinear relationships with line energy density, with varying response trends across different ranges. Such nonlinear effects directly influence weld quality and alloy composition distribution.

The Support Vector Regression (SVR) model with an RBF kernel achieved high predictive accuracy for molten pool width, height, and

$(C_\text{max})$

, with the average

$(R^2)$

values exceeding 0.8 for all four targets.

Once trained, the SVR model can rapidly predict molten pool characteristics for new line energy density values that were not explicitly simulated but fall within the training data range. Compared with numerical simulations, which require approximately 10 hours per case, SVR provides results within seconds, thus reducing computational costs by several orders of magnitude.

The integration of numerical simulation and machine learning establishes an efficient and reliable predictive framework. This approach enables systematic exploration of process parameters, accelerates design iterations, and supports process optimization in advanced manufacturing applications of aluminum alloy welding.

bibliography{sn-bibliography}

Author Contribution

L.S.: Conceptualization; Formal analysis; Investigation; Project administration; Writing - original draftW.Z.: Data curation; Formal analysis; Validation; Visualization; Writing - original draftX.C: Conceptualization; Funding acquisition; Methodology; Resources; Supervision; Writing - original draftL.J: Writing -review & editingZ.H: Writing -review & editingB.D: Writing -review & editing

Data Availability

The dataset can be obtained from the corresponding author.

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Hu, K. and Wang, Y. and Li, F. and others (2024) Thermal-fluid modeling and physics-informed machine learning for predicting molten pool depth in single-layer multi-track fiber laser cladding. International Journal of Advanced Manufacturing Technology 135: 3591--3613 https://doi.org/10.1007/s00170-024-14706-1

Kumar, A. and Shukla, M. (2025) Prediction of Melt Pool Geometry and Defects in Laser Additive Manufacturing of Inconel 718 Alloy: A Machine and Deep Learning Approach. JOM https://doi.org/10.1007/s11837-025-07918-7

Rahman, M. Shafiqur and Sattar, Naw Safrin and Ahmed, Radif Uddin and Ciaccio, Jonathan and Chakravarty, Uttam K. (2024) A Machine Learning Framework for Melt-Pool Geometry Prediction and Process Parameter Optimization in the Laser Powder-Bed Fusion Process. Journal of Engineering Materials and Technology 146(4): 041006 https://doi.org/10.1115/1.4065687, https://asmedigitalcollection.asme.org/materialstechnology/article-pdf/146/4/041006/7358810/mats\_146\_4\_041006.pdf, https://doi.org/10.1115/1.4065687, 0094-4289, This study presents a cost-effective and high-precision machine learning (ML) method for predicting the melt-pool geometry and optimizing the process parameters in the laser powder-bed fusion (LPBF) process with Ti-6Al-4V alloy. Unlike many ML models, the presented method incorporates five key features, including three process parameters (laser power, scanning speed, and spot size) and two material parameters (layer thickness and powder porosity). The target variables are the melt-pool width and depth that collectively define the melt-pool geometry and give insight into the melt-pool dynamics in LPBF. The dataset integrates information from an extensive literature survey, computational fluid dynamics (CFD) modeling, and laser melting experiments. Multiple ML regression methods are assessed to determine the best model to predict the melt-pool geometry. Tenfold cross-validation is applied to evaluate the model performance using five evaluation metrics. Several data pre-processing, augmentation, and feature engineering techniques are performed to improve the accuracy of the models. Results show that the “Extra Trees regression ” and “Gaussian process regression ” models yield the least errors for predicting melt-pool width and depth, respectively. The ML modeling results are compared with the experimental and CFD modeling results to validate the proposed ML models. The most influential parameter affecting the melt-pool geometry is also determined by the sensitivity analysis. The processing parameters are optimized using an iterative grid search method employing the trained ML models. The presented ML framework offers computational speed and simplicity, which can be implemented in other additive manufacturing techniques to comprehend the critical traits., 08

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