Physics-Based Data Augmentation Enables Accurate Machine Learning Prediction of Melt Pool Geometry

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The proposed physics-augmented learning framework enables accurate melt pool geometry prediction in L-PBF under sparse data conditions, supporting intelligent process optimization and quality control in metal additive manufacturing.

Abstract

Accurate melt pool geometry prediction is essential for ensuring quality and reliability in Laser Powder Bed Fusion (L-PBF). However, small experimental datasets and limited physical interpretability often restrict the effectiveness of traditional machine learning (ML) models. This study proposes a hybrid framework that integrates an explicit thermal model with ML algorithms to improve prediction under sparse data conditions. The explicit model—calibrated for variable penetration depth and absorptivity—generates synthetic melt pool data, augmenting 36 experimental samples across conduction, transition, and keyhole regimes for 316 L stainless steel. Three ML methods—Multilayer Perceptron (MLP), Random Forest, and XGBoost—are trained using fivefold cross-validation. The hybrid approach significantly improves prediction accuracy, especially in unstable transition regions (D/W ≈ 0.5–1.2), where morphological fluctuations hinder experimental sampling. The best-performing model (MLP) achieves R² > 0.98, with notable reductions in MAE and RMSE. The results highlight the benefit of incorporating physically consistent, nonlinearly distributed synthetic data to enhance generalization and robustness. This physics-augmented learning strategy not only demonstrates scientific novelty by integrating mechanistic modeling into data-driven learning, but also provides a scalable solution for intelligent process optimization, in situ monitoring, and digital twin development in metal additive manufacturing.

1. Introduction

Laser Powder Bed Fusion (L-PBF) is one of the most prominent technologies currently used in Additive Manufacturing (AM) methods for fabricating metal parts, providing unprecedented advantages for manufacturing complex-shaped metal components and dominating the metal additive manufacturing market with applications in various industries such as aerospace, automotive, pharmaceuticals, and even the energy industry [1,2,3,4,5].

In L-PBF, a laser beam selectively melts metal powders layer by layer to form melt pools, which solidify and stack into near-net-shaped parts. The quality and performance of the final product are affected by multiple factors, including raw powder properties, process parameters, and post-processing treatments [6,7,8]. The melt pool—defined as the localized region of melt powder along the laser scanning path—is central to the process. Its thermal behavior and geometric characteristics strongly influence the solidification dynamics, defect formation, and ultimately the microstructure and mechanical performance of the fabricated part [9]. However, melt pool instability can lead to serious issues such as increased porosity, poor surface finish, and mechanical degradation [10,11,12]. To mitigate such defects, many studies have explored the influence of process parameters—particularly laser power, scanning speed, and energy density—on melt pool geometry and quality [13,14,15]. While higher laser power tends to increase melt pool size, excessive energy input can lead to keyhole formation and instability. By contrast, insufficient energy may result in lack-of-fusion defects due to incomplete melting. Therefore, understanding and predicting melt pool characteristics under different processing conditions is essential for process control and defect mitigation [16].

Traditional simulation tools, such as finite element modeling (FEM), have been extensively used to capture melt pool evolution and thermal history [17,18,19]. Recent reviews [20] have highlighted how FEM approaches remain the primary tool for multiscale thermo-mechanical analysis in L-PBF, despite challenges in computational cost, parameter calibration, and scalability. While FEM provides valuable physical insight, its high resource demands and reliance on precise material models limit its applicability for real-time or data-intensive scenarios.

To overcome these limitations, recent studies have explored data-driven approaches, particularly machine learning (ML), which can capture complex nonlinear relationships and generalize across materials and systems [21]. ML has found success in AM applications such as defect detection, topology optimization, process monitoring, and post-processing [22,23,24,25,26]. For instance, Ankit et al. [27] used ML to predict porosity based on chemical composition with over 90% accuracy. Similarly, Anwaruddin et al. [28] applied deep learning to predict porosity and density in L-PBF-processed parts. While ML has demonstrated predictive power in AM, its application to melt pool geometry prediction remains limited [29,30,31]. A major challenge lies in data scarcity. High-fidelity experimental data in metal AM are difficult and expensive to obtain. Additionally, ML models often lack physical interpretability and may fail to generalize under sparse or noisy conditions [32,33]. Combining physics-based modeling with machine learning has emerged as a promising solution. This hybrid approach improves model robustness, reduces dependency on large datasets, and enhances interpretability [34]. One effective strategy is physics-informed data augmentation. Mondal et al. [35] combined an analytical heat transfer model with Gaussian process regression to predict melt pool geometry under limited data conditions. Zhu et al. [36] employed physics-informed neural networks to simulate temperature and flow fields in L-PBF, achieving physically consistent predictions even with sparse data. Sharma et al. [37] further enhanced multiphysics modeling by incorporating gas–melt pool interactions into a deep learning framework, showing that physics-guided learning reduces data requirements while improving prediction transparency. Ritto et al. [38] integrated stochastic physical modeling with ML to detect multiple damage scenarios in digital twins. Ren et al. [39] used FEA and ML to identify thermal patterns in directed energy deposition (DED). These studies highlight the potential of hybrid modeling in AM, particularly for low-volume, highly customized production. These hybrid strategies bridge the gap between computational modeling and experimental validation, providing more robust and interpretable models.

In this work, an explicit-model-augmented machine learning framework is presented for predicting melt pool geometry in L-PBF. A physics-based analytical model is first calibrated using experimental data to simulate melt pool width and depth. This model is then used to generate physically consistent synthetic data to augment a limited dataset of 36 parameter combinations, covering conduction, transition, and keyhole mode. Three ML algorithms—multilayer perceptron (MLP), random forest, and XGBoost—are trained using fivefold cross-validation. The results show that the hybrid model improves predictive accuracy, reduces dependency on large datasets, and enhances physical interpretability. This study provides an efficient and generalizable approach for intelligent process optimization in metal additive manufacturing.

2. Materials and Methods

2.1. Experimental Materials and Design

The material used in this study was atomized 316 L stainless steel powder provided by BLT Corporation, whose detailed chemical composition is listed in

Table 1. The chemical composition of the 316 L powder (wt%).

Elements	Fe	C	Cr	Ni	Mo	Mn	Si
316 L	67.98	0.0086	17.11	11.01	2.42	0.81	0.66

. The powder exhibits high sphericity, excellent flowability, and uniform spreading, making it suitable for laser powder bed fusion (L-PBF) processing. To remove surface moisture, the powder was sieved using a 250-mesh standard sieve and dried in a vacuum oven at 80 °C for 4 h prior to use.

L-PBF fabrication was conducted on an HBD-80 machine equipped with a Yb fiber laser (wavelength 1040 nm, maximum power 200 W), with an effective build size of Ø120 mm × 80 mm. Throughout the printing process, the build chamber was shielded with argon gas to maintain an oxygen concentration below 1000 ppm and suppress oxidation.

To avoid unstable or low-energy regimes that lead to process defects, a total of 36 sets of process parameter combinations were systematically designed using different levels of laser power and scanning speed to cover a wide range of energy input conditions. The process parameters are shown in

Table 2. L-PBF experimental process parameters.

Process Parameters (Symbol, Unit)	Values
Laser power (P, W)	100, 120, 140, 160, 180, 200
Scan speed (V, mm/s)	400, 500, 600, 700, 800, 900, 1000, 1100
Laser spot diameter ($d_0$, mm)	0.06
Hatch spacing (d, mm)	0.5
Layer thickness (H, mm)	0.03

. As illustrated in Figure 1, under each process condition, 15 single-track melt pool samples were fabricated using L-PBF, and the cross-sectional dimensions were measured and averaged to minimize experimental uncertainties.

2.2. 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 Parand 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. Equations for modeling melt pool geometry.

Task (Label)	Equation
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	$W=B\times {10}^6\times P^{0.59}{V}^{-0.36}{\rho }^{-0.39}{C}_p^{-0.39}{k}^{-0.21}{(T_m-T_0)}^{-0.60}$

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; ρ 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.

.

The explicit model improved in the article is based on the evaluation of Parand 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 (w₀), 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.

$$y=w_0{P}^{w_1}{V}^{w_2}{\rho }^{w_3}{C_P}^{w_4}{k}^{w_5}{(T_m-T_0)}^{w_6}$$

(1)

where,

$$\begin{array}{l}(a)w_1+w_3+w_5=0\\ (b)2w_1+w_2-3w_3+2w_4+w_5=1\\ (c)-3w_1-w_2-2w_4-3w_5=0\\ (d)-w_4-w_5+w_6=0\end{array}$$

(2)

$$\begin{array}{l}\underset{W}{\min }{‖y-w_0{P}^{w_1}{V}^{w_2}{\rho }^{w_3}{C_P}^{w_4}{k}^{w_5}{(T_m-T_0)}^{w_6}‖}_2^2\\ s.t.:{(w_1+w_3+w_5)}^2+{(2w_1+w_2-3w_3+2w_4+w_5-1)}^2+\\ {(-3w_1-w_2-2w_4-3w_5)}^2+{(-w_4-w_5+w_6)}^2=0\end{array}$$

(3)

$$D=0.37\times {10}^6\times P^{0.51}{V}^{-0.46}{\rho }^{-0.46}{C}_p^{-0.46}{k}^{-0.06}{(T_m-T_0)}^{-0.51}$$

(4)

$$W=0.44\times {10}^6\times P^{0.59}{V}^{-0.36}{\rho }^{-0.39}{C}_p^{-0.39}{k}^{-0.21}{(T_m-T_0)}^{-0.60}$$

(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. In simulating the melt pool shape and dimensions, most studies treat this as a fixed value, and the deviation between simulated and actual melt pool shape and dimensions is significant.

2.3. Framework Overview and Motivation

To model the complex nonlinear relationship between process parameters and melt pool geometry, a machine learning (ML) framework is developed and trained on the augmented dataset comprising both experimental and synthetic samples. While the physics-based explicit model provides interpretable predictions of melt pool width and depth, its applicability is limited in high-dimensional spaces and in capturing multivariate couplings under varying processing conditions. Therefore, ML models are employed not to replace physical modeling, but to enhance its generalizability and flexibility by learning from both measured and simulated data. The melt pool dimensions—depth (D) and width (W)—are predicted using three representative ML algorithms: Multilayer Perceptron (MLP), Random Forest (RF), and Extreme Gradient Boosting (XGBoost). These algorithms were selected based on their ability to handle nonlinear, multivariable regression problems with small-to-moderate training datasets, and have been previously validated in metal additive manufacturing scenarios. Each model accepts laser power (P) and scan speed (V) as inputs and outputs two independent regression targets: melt pool width and depth. The dataset used for training consists of 36 experimental samples and 15 synthetic samples generated from the calibrated explicit model, yielding a total of 51 data points. To ensure the robustness of the models and reduce overfitting in the small-sample context, a fivefold cross-validation (5-CV) strategy is applied. Model performance is quantitatively evaluated using three widely adopted regression metrics: the coefficient of determination (R²), mean absolute error (MAE), and root mean square error (RMSE). These metrics are used to compare model accuracy on both the original dataset and augmented datasets. Hyperparameter tuning is performed using Bayesian optimization via the Tree-structured Parzen Estimator (TPE) algorithm, implemented in the HyperOpt package. Search spaces for each model’s key parameters (e.g., learning rate, number of estimators, max depth for XGBoost; number of layers and neurons for MLP) are defined based on preliminary grid tests, and the optimization objective is to maximize R² while minimizing MAE.

As shown in Figure 2, the proposed framework integrates physics-informed data generation with ML-based regression to achieve interpretable, accurate, and computationally efficient melt pool prediction. The enhanced model not only reduces dependency on expensive experimental data, but also enables reliable prediction over a wider process window, thereby offering practical value for real-time process design and control in L-PBF.

2.4. Machine Learning Modeling and Optimization

To achieve robust and accurate prediction of melt pool geometry, three supervised machine learning (ML) algorithms with complementary modeling characteristics were employed: (i) Multilayer Perceptron (MLP), (ii) Random Forest (RF), and (iii) eXtreme Gradient Boosting (XGBoost). These models were trained using the augmented dataset comprising 36 experimental samples and 15 physically consistent synthetic samples, and evaluated in terms of generalization and predictive accuracy.

MLP, a type of feedforward artificial neural network, consists of an input layer, one or more hidden layers, and an output layer. Each neuron computes a weighted sum of inputs and applies a nonlinear activation function (e.g., ReLU) to produce the output. Weight updates are achieved via backpropagation by minimizing the loss between the predicted and actual values, typically using a mean squared error (MSE) loss function. A schematic of the MLP architecture is shown in Figure 3a. To stabilize training and improve convergence, all input features were normalized. The network was trained using the Adam optimizer, a learning rate of 0.001, and a maximum of 1000 epochs. Several architectures with different hidden layer configurations were tested, and 5-fold cross-validation was used to evaluate performance. The best-performing architecture (Model 4), which comprises five layers of decreasing neuron sizes, is summarized in

Table 4. MLP hidden layer parameters.

Model Number	The Number of Neurons in the Hidden Layer
	First Layer	Second Layer	Third Layer	Forth Layer	Fifth Layer
Model 1	48	32	-	-	-
Model 2	48	32	16	-	-
Model 3	64	48	32	16	-
Model 4	64	48	32	16	8

.

Random Forest is an ensemble learning algorithm that combines multiple decision trees to enhance robustness and mitigate overfitting. Each tree is trained on a bootstrap sample, and final predictions are averaged. Hyperparameters including the number of estimators, maximum tree depth, and minimum sample splits were optimized using Bayesian optimization. The model structure is illustrated in Figure 3b, and optimal parameters are listed in

Table 5. Specific parameter selection of Random Forest and XGBoost models in Bayesian optimization.

Model	Hyperparameters	Before Data Enhancement	Data Enhancement
RF	n_estimators	372	148
	max_depth	6	191
	min_samples_split	2	2
	min_samples_leaf	1	1
XGBoost	n_estimators	1213	1066
	learning_rate	0.1	0.3
	max_depth	4	3
	min_child_weight	1	1
	Subsample	0.8	0.8

.

XGBoost is a gradient boosting framework optimized for speed and regularization. It builds additive regression trees sequentially, where each tree corrects the residual errors of the previous ensemble. Compared to standard Gradient Boosting Decision Tree (GBDT), XGBoost introduces L1/L2 regularization (α and λ) and shrinkage to control overfitting—features particularly important for small sample sizes. Hyperparameter tuning was also conducted via Bayesian optimization. The final parameter selection is detailed in Table 5.

All models underwent hyperparameter tuning using the Tree-structured Parzen Estimator (TPE) algorithm implemented in the HyperOpt Python3.11 library. The objective was to maximize the coefficient of determination (R²) and minimize error metrics. The search space for each algorithm was defined based on preliminary sensitivity analysis. To ensure robustness and mitigate overfitting under small sample conditions, 5-fold cross-validation was adopted throughout the optimization and validation process. The performance of each model was evaluated using three metrics:

• $R^2$ (coefficient of determination): measures how well predictions approximate actual values.
• MAE (mean absolute error): evaluates average absolute deviation.
• RMSE (root mean squared error): penalizes larger errors more severely.

The calculation of these three evaluation metrics is shown below:

$$R^2=1-{\displaystyle \sum _{i=1}^n{({\hat{Y}}_i-\overline{Y})}^2}/{\displaystyle \sum _{i=1}^n{(Y_i-\overline{Y})}^2}$$

(6)

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|{\hat{Y}}_i-{\overline{Y}}_i\right|}$$

(7)

$$RMSE=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{({\hat{Y}}_i-Y_i)}^2}}{n}}}$$

(8)

where Y is the actual value; $\overline{Y}$ the mean of the actual value; $\hat{Y}$ is the predicted value; and n is the number of test samples.

3. Results and Discussion

3.1. Analysis of Experimental Data

The melt pool depth-to-width ratio (D/W, denoted as N) serves as a critical dimensionless parameter for distinguishing melt pool modes. While prior studies have proposed conflicting threshold values for optimal N ranges—Guo et al. [41] suggested N > 0.5 indicates keyhole mode through numerical simulations, whereas Dilip et al. [42] experimentally determined N = 0.37–0.6 for effective substrate bonding—this study systematically captured a comprehensive dataset spanning N = 0.34–1.29, encompassing conduction, transition, and keyhole modes across the parameter matrix.

According to Figure 4, 36 sets of melt pool measurements were obtained experimentally. Figure 5a,b show their respective profiles under varying laser power at fixed scanning speeds, while Figure 5c,d illustrate the variation of melt pool width and depth as functions of laser power and scanning speed. It can be observed that both width and depth generally increase with rising laser power and decrease with increasing scan speed. Notably, the melt pool depth exhibits a larger dynamic range (24 μm to 173 μm) compared to width (72 μm to 143 μm), indicating its higher sensitivity to processing conditions. To account for experimental uncertainty, each data point in Figure 5c,d represents the mean value of 15 single-track measurements under identical processing conditions. The error bars denote one standard deviation (±1σ), calculated from these replicate measurements. These bars provide a quantitative indication of measurement repeatability and process stability. The increasing size of error bars at high laser power and low scan speeds suggests elevated thermal instability and inconsistent melt pool morphology in these regimes. Specifically, the melt pool width exhibits larger variability than depth under these conditions, implying greater susceptibility to fluctuation in lateral melt propagation.

To further validate the rationality of the melt pool modes classification and its relevance to build quality, representative cross-sectional morphologies, pore characteristics, and tensile test results were examined across conduction, transition, and keyhole conditions (Figure 6). The tensile tests were conducted at room temperature using a Shimadzu AG-IC 100 kN testing machine(Shimadzu Corporation; Kyoto, Japan). Specimens were prepared in accordance with ASTM E8/E8M-15 standards and tested at a strain rate of 3 mm/min [43]. Conduction-mode samples showed shallow melt pools with lack-of-fusion defects, while keyhole-mode cases exhibited deep, unstable geometries with gas-induced porosity. By contrast, the transition regime featured stable, cup-shaped melt pools with minimal porosity. This trend is reflected in tensile performance: transition-mode specimens achieved the highest UTS and YS values, while conduction and keyhole modes suffered strength degradation due to insufficient fusion and instability-induced defects, respectively. These observations provide supporting evidence that the D/W ratio correlates well with the process–structure–property relationship in L-PBF.

3.2. Calibration of the Explicit Model

This study enhances the original explicit model by replacing its fixed constant term with a calibratable variable coefficient. This dynamic coefficient is calibrated against experimental data to account for complex physical mechanisms, including dynamic laser penetration depth and absorption variations. Calibration was performed using 18 experimentally validated datasets covering laser powers of 100–200 W and scanning speeds of 700–900 mm/s—selected for their parametric completeness. For each constant scanning speed, linear regression analysis established relationships between laser power and melt pool dimensions (width/depth). After calibration, the explicit model achieved an average coefficient of determination ($R^2$) of 0.925 for melt pool width and 0.965 for depth, with corresponding mean absolute errors of 1.53% and 5.23%, respectively. The predictive effectiveness of the improved explicit model is illustrated in Figure 7. Based on the calibrated regression equations, constant values were computed for laser powers of 110 W, 130 W, 150 W, 170 W, and 190 W under scanning speeds of 700, 800, and 900 mm/s. These values were then substituted into the equations listed in Table 3 to generate 15 synthetic melt pool samples.

Owing to its foundation on large-scale datasets across diverse metallic materials, the improved explicit model exhibits broad applicability and significantly enhanced predictive accuracy when calibrated against experimental measurements. The spatial distribution of both experimental and simulated data is presented in Figure 8. Notably, the synthetic data complement the experimental dataset by extending coverage into process regimes that are experimentally difficult to access—particularly at the transition-to-keyhole boundary (D/W ≈ 0.6–1.2), where melt pool behavior is highly nonlinear and sensitive to process instabilities. These regions exhibit essential nonlinearities poorly represented in physical measurements. By encoding mechanisms like power-dependent absorption, the model generates physically consistent data with enhanced nonlinear features, enabling ML algorithms to capture behavioral transitions and inflection points critical for defect prediction [44,45].

As shown in Figure 9, the explicit model-generated data (yellow points) effectively supplement regions of the parameter space where experimental coverage is sparse, particularly near the conduction-to-keyhole transition. In both width and depth surfaces, these samples enhance the representation of nonlinear zones with steep geometric gradients, which are critical for learning melt pool morphological shifts. By enriching the dataset in these regions, the augmented data improve the model’s ability to generalize and capture key transition behaviors.

3.3. Model Training Results and Discussion

In this section, the predictive performance of the optimized machine learning (ML) models is assessed, with a focus on comparing melt pool width and depth predictions to evaluate the impact of data quality. The dataset simulated by the calibrated explicit model was integrated with the experimental data to form a hybrid training set. This augmented dataset was then used to retrain the ML models, and the results were benchmarked against models trained solely on experimental data. Details of the dataset composition are provided in

Table 6. Correspondence between process parameters and sample numbers.

Scanning Speed (mm/s)	Laser Power (W)
	100	110	120	130	140	150	160	170	180	190	200
400	S0	-	S9	-	-	-	-	-	-	-	-
500	S1	-	S10	-	S18	-	S27	-	-	-	-
600	S2	-	S11	-	S19	-	S28	-	S36	-	S45
700	S3	S6 *	S12	S15 *	S20	S24 *	S29	S33 *	S37	S42 *	S46
800	S4	S7 *	S13	S16 *	S21	S25 *	S30	S34 *	S38	S43 *	S47
900	S5	S8 *	S14	S17 *	S22	S26 *	S31	S35 *	S39	S44 *	S48
1000	-	-	-	-	S23	-	S32	-	S40	-	S49
1100	-	-	-	-	-	-	-	-	S41	-	S50

Where bold is the deleted sample, those with symbols ‘*’ are simulated data, and the rest are experimental data.

. To ensure statistical consistency during fivefold cross-validation, the S0 dataset was excluded, resulting in a total of 50 samples with balanced partitions for training and testing.

When trained on the original 35 experimental data points, the XGBoost model achieved the highest R² (0.864) for melt pool width prediction, while the MLP model yielded the lowest RMSE (5.877). For melt pool depth, the MLP model exhibited the best overall performance, attaining an R² of 0.977, MAE of 3.939, and RMSE of 5.197—substantially outperforming the other two models. The Random Forest model consistently underperformed in both tasks.

The three machine learning models selected and optimized in this paper have different prediction performances due to different mathematical principles, but it can be observed that the changes in the three indicators—R², MAE and RMSE—are broadly consistent. The MLP and XGBoost models show excellent prediction performance for melt pool width after data augmentation with the improved explicit model. Compared with the Random Forest model, which has an R² value of only 0.814, the prediction accuracies of these two models are similar and above 0.9. The MLP model has the highest prediction accuracies, with an R², MAE, and RMSE of 0.919, 3.535, and 5.202, respectively, and outperforms the other two models in all three scoring criteria. The difference in prediction performance among the three models is even more obvious in melt pool depth prediction: the R² value of the MLP model is as high as 0.984, which is much higher than that of the remaining two algorithms, and its MAE and RMSE are close to each other, more than twice lower than those of the remaining algorithms. Meanwhile, the XGBoost model is also very effective in predicting the depth of the melt pool, with an R², MAE, and RMSE of 0.922, 6.308, and 10.131, respectively. Overall, in terms of performance on the data-enhanced hybrid dataset, among the three models proposed in this paper, the MLP model has the best predictive results, followed by the XGBoost model, and the Random Forest model delivers a comparatively average performance.

Figure 10 and Figure 11 demonstrate that, despite the dataset expansion using the explicit model, prediction accuracy remained sensitive to data distribution. Across all models, melt pool depth was more accurately predicted than width, regardless of dataset type. This may be attributed to greater experimental variability in melt pool width. Elevated prediction errors were observed under extreme energy conditions (e.g., S9 and S17 for width; S27, S32, and S37 for depth), further highlighting the influence of data quality.

After data augmentation, all models showed marked improvement. The MLP model achieved the highest accuracy gains, with increases of up to 23.5% in R², and reductions of 27.1% in MAE and 17.8% in RMSE. These results confirm the efficacy of the proposed augmentation strategy. By incorporating physically consistent synthetic data, the framework not only enhances predictive accuracy but also improves the physical interpretability of the model. A detailed comparison is presented in Figure 12.

4. Conclusions

This study establishes a hybrid prediction framework that couples an improved explicit model with machine learning to predict melt pool geometry in L-PBF with high accuracy and enhanced interpretability. The explicit model was calibrated using experimental data and effectively simulates melt pool width and depth by accounting for variable absorptivity and penetration depth under different processing conditions. When integrated into a machine learning pipeline, the augmented dataset enables accurate interpolation across conduction, transition, and keyhole regimes.

Model evaluation reveals that MLP outperforms other algorithms, especially in predicting melt pool depth. Data augmentation leads to a maximum R² improvement of 23.5%, with reductions of up to 27.1% in MAE and 17.8% in RMSE. These results confirm that the predictive performance of ML models is highly dependent on dataset characteristics, particularly in high- or low-energy regimes where experimental data are scarce or unstable. Notably, the model demonstrates superior generalization in transition zones, where abrupt melt pool morphology changes typically elude experimental capture.

In summary, the explicit-model-augmented approach bridges the gap between physical interpretability and data-driven modeling. It enhances predictive robustness under small-sample constraints and offers a scalable strategy for real-time optimization in L-PBF. Despite its strong predictive performance, the model is subject to limitations related to dataset scope, lack of thermal field resolution, and material-specific calibration. These aspects can be addressed in future work by integrating physics-informed features and extending to multi-material datasets. This methodology provides a foundation for broader integration of hybrid modeling in metal additive manufacturing, supporting intelligent process control, in situ monitoring, and digital twin development.