Three-Dimensional Modeling and Analysis of Directed Energy Deposition Melt Pools Based on Physical Information Neural Networks

Abstract

In Directed Energy Deposition (DED), modeling the molten pool temperature field is crucial for precise temperature control, process optimization, and quality improvement. However, conventional numerical methods suffer from limitations such as high computational costs and poor transferability. This study proposes a physics-informed neural network with dynamic learning rate (DLR-PINN) model, which integrates transfer learning to enable rapid prediction of 3D temperature fields and dimensions of molten pools across process parameters. Its validity is verified by a finite element method (FEM) calibrated via single-track DED experiments. Results show that DLR-PINN exhibits superior convergence and stability compared to traditional PINN. Combined with transfer learning, training efficiency is significantly enhanced, with a single prediction taking only 10 s. Using the FEM as the benchmark, it achieves a mean absolute percentage error (MAPE) of 0.53% for temperature prediction, and MAPE of 3.69%, 2.48%, and 6.96% for molten pool dimension predictions, respectively. Sensitivity analysis of process parameters reveals that scanning speed has a significantly greater regulatory effect on molten pool characteristics than laser power. Additionally, the temperature field of the flat-top heat source is more uniform than that of the Gaussian heat source, which is more conducive to improving printing quality and efficiency.

1. Introduction

Directed Energy Deposition (DED)technology is extensively utilized in the fabrication of large-scale complex components owing to its high flexibility, superior material utilization rate, and absence of printing size constraints [1]. However, during the layer-by-layer deposition process, materials undergo rapid cyclic heating-cooling thermal cycles. These thermal histories induce spatial heterogeneity in grain morphology and precipitate distribution [2,3], leading to pronounced mechanical anisotropy and functional gradient distributions. For instance, significant temperature gradients can generate residual stresses that cause material warpage and cracking [4]. Although experimental measurements can directly acquire thermal history information, their high cost and limited applicability hinder their widespread use in process optimization. The finite element method (FEM), as a mature numerical computation method, provides an effective theoretical approach for establishing three-dimensional temperature field models of the DED process. For instance, Kiran et al. [5] constructed a thermomechanical model using FEM, which successfully predicted the evolution of temperature and residual stress during single-track and multi-track deposition, with the predicted results showing good agreement with experimental data. Shu et al. [6] investigated the prediction of temperature fields around laser-induced molten pools on metal substrates. They proposed that for reliable calculation of temperature histories, accurate characterization of the laser beam’s power density distribution on the substrate and consideration of convective heat transfer effects are essential. Liu et al. [7] developed a powder-scale three-dimensional multiphysics model, calculated the variations in the DED process across different time and space, and analyzed the influence of process parameters on molten pool dynamics, geometric characteristics, and temperature changes. Although numerical methods can establish effective temperature field models, as the complexity of the problem increases, extensive mesh processing and complex spatiotemporal mapping increase computational costs and reduce the response speed of data analysis, especially for high-fidelity models [8]. In addition, numerical models are usually constructed for specific working conditions, with limited generalization ability across different process parameters or geometric configurations and a lack of universality, often requiring tedious remodeling and calculation for specific problems. Therefore, based on a deep understanding of the core significance of temperature field modeling, developing new temperature field models with both computational efficiency and real-time response is particularly urgent and important.

With the rapid development of artificial intelligence technology, machine learning (ML) methods have shown great potential in the process analysis of metal additive manufacturing (AM). They can efficiently process high-dimensional and heterogeneous data, establish complex nonlinear relationships between inputs and outputs from such data, and thereby provide guidance for process optimization, quality control, and system improvement [9,10,11]. For example, Ren et al. [12] proposed a coupled recurrent neural network (RNN) and deep neural network (DNN) model, which successfully predicted the temperature field distribution under different scanning strategies. Compared with finite element results, the prediction accuracy reached over 95%. Zhang et al. [13] used molten pool temperature data collected by sensors to train two ML models, XGBoost and LSTM, for predicting the molten pool temperature field in DED, and analyzed their impacts on the microstructure, porosity, and mechanical properties of the components. Experimental results showed that both models could predict the molten pool temperature with high accuracy. Chen et al. [14] utilized data generated by high-fidelity numerical models to train a model for predicting the isotherms of the temperature field, and subsequently reconstructed the temperature field throughout the entire process by integrating interpolation methods. The prediction results exhibited good consistency with the simulation data.

However, data-driven models also have obvious limitations. On the one hand, the performance of these models is highly dependent on large volumes of high-quality, accurately labeled training datasets. Particularly in the AM field, due to the complexity of experimental processes and the high cost associated with data acquisition, there is a scarcity of available data for model training. On the other hand, the inherent “black-box” nature of ML results in a lack of transparency and interpretability in the model training process, making it challenging to uncover and incorporate the underlying physical laws within the field [15]. Moreover, such inherent uncertainty may even cause the model to learn input-output mappings that violate fundamental physical principles, thereby yielding physically unreasonable predictions [16]. To address the above issues, a new research paradigm—Physics Informed Machine Learning (PIML)—has gradually emerged [17,18,19]. This method improves the transparency and interpretability of models by integrating prior physical information (such as physically intuitive parameters and physical partial differential equations (PDEs)) with ML models. In particular, the Physics-Informed Neural Network (PINN) model [19], first proposed by Raissi et al. [20], is characterized by a core mechanism in which the governing PDEs describing specific physical systems, along with their boundary conditions, are incorporated as soft constraints and systematically embedded into the formulation of the neural network’s loss function. This thereby constitutes a general-purpose PDE solver, whose effectiveness has been validated in solving both forward and inverse problems of nonlinear PDEs. Leveraging its remarkable potential in addressing complex nonlinear problems, the PINN model has been swiftly adopted in numerous scientific and engineering domains, such as computational fluid dynamics [21,22,23], heat transfer [24], and power system analysis [25]. These successful applications offer a novel paradigm for tackling modeling and simulation challenges in complex physical systems.

In recent years, PINN has made a series of important progress in the field of modeling metal AM processes. Zhu et al. [26] successfully applied the PINN model to predict the molten pool temperature and velocity fields during metal AM processes. Xie et al. [27] used the PINN model to predict the temperature fields of single-layer and multi-layer DED, and effectively improved the model’s prediction accuracy by introducing approximately 20% labeled data. However, this work did not consider the influence of the substrate temperature field. Liao et al. [28] successfully predicted the thermophysical parameters such as specific heat capacity and heat transfer coefficient of materials by inverse solution with the help of PINN, which demonstrated the potential application of PINN in the inverse solution process. Li et al. [29] effectively alleviated the common abnormal gradient issues in deep networks by introducing a residual network structure into PINN combined with transfer learning. Nevertheless, current research on PINN-based modeling of metal AM processes still faces the following key challenges and limitations: First, the material models are overly simplistic, generally assuming thermophysical parameters as constants and ignoring the nonlinear variation in materials’ thermophysical parameters with temperature; Second, the complex loss functions composed of PDEs result in highly non-convex optimization surfaces, which are prone to falling into local minima, leading to difficulties in model convergence; Third, research has mainly focused on the prediction and verification of single physical quantities such as temperature fields, and has not yet deeply explored how to utilize PINN’s prediction results for practical process parameter optimization, analysis of defect formation mechanisms, or evaluation of mechanical properties; thus, the application value of the model needs to be further explored.

To address the aforementioned issues, this study aims to develop and validate a PINN with dynamic learning rate (DLR-PINN) model for predicting the 3D molten pool temperature field and size during the DED process. In the existing literature, fixed thermophysical parameters and fixed learning rate settings are commonly adopted. When dealing with strong nonlinear problems caused by thermophysical parameters and complex loss functions, bottlenecks such as slow convergence speed and insufficient stability are frequently encountered, and no effective adaptive optimization scheme has been formed yet. The novelty of this study lies in utilizing a dynamic learning rate adjustment mechanism to targetly alleviate difficulties in model convergence by real-time adaptation to nonlinear characteristics during training. Results show that the model significantly improves convergence performance. Meanwhile, the trained model is further applied to process analysis, filling the research gap in temperature prediction under different heat source scenarios and providing more reliable theoretical and technical support for related engineering applications. Specifically, first, molten pool temperature data were collected through DED experiments to calibrate the finite element model (FEM) of the temperature field. Concurrently, region-specific differential point sampling was performed on the solution domain to construct a training dataset and train the DLR-PINN model. Subsequently, the calibrated FEM was used as a benchmark to validate the physical rationality of DLR-PINN, and combined with transfer learning, the rapid prediction of 3D molten pool temperature fields and sizes under different processes was realized. Finally, the application of this model was extended to predict 3D temperature fields under different heat sources. Results indicate that the improved PINN model can not only accurately and rapidly predict the 3D temperature field and geometric dimensions of the molten pool under different process conditions and heat source scenarios but also provide a versatile and real-time alternative modeling approach for process monitoring and optimization.

The remainder of this study is organized as follows: Section 2, “Methods”, will briefly elaborate on the PDEs involved in establishing the temperature field model, as well as the basic principles of the PINN model. Section 3, “Experiments and Validation”, calibrates and validates the finite element model through experimental methods. In Section 4, “Discussion and Analysis”, the trained model is used to predict the molten pool temperature field and verify its accuracy. Using transfer learning methods, the 3D molten pool temperature field and variation laws of molten pool dimensions under different combinations of process parameters are predicted and explored, with in-depth analysis of the influence of processes on molten pool characteristics. Meanwhile, the improved PINN model is employed to predict and analyze the temperature field under flat-top heat sources. Section 5, “Conclusions”, summarizes this work.

2. Methodology

2.1. Governing Equation

Based on Fourier’s law of heat conduction and the law of energy conservation, the heat storage capacity per unit volume in three-dimensional space equals the algebraic sum of the heat generated per unit volume and the heat inflow rate. This heat conduction process can be described by the following governing equation [30]:

(1)$\rho C_P{\displaystyle \frac{\partial T}{\partial t}}=k({\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}})+Q$

In the equation, T(x, y, z, t) represents the temperature at different positions (x, y, z) and different times t; $\rho$ denotes the material density; θ is the specific heat capacity; and k stands for the thermal conductivity. Considering that these parameters vary with the thermal conductivity and temperature, they are expressed as functions of temperature via interpolation methods based on the material property values listed in Table 1, so as to define the material parameter values at different temperatures. Q is the energy per unit volume generated by the laser beam.

Before the laser is turned on (i.e., at t = 0 s), the surface temperature of the material is assumed to be equal to the ambient temperature $T_{amb}=293.15 \mathrm{K}$:

(2)$T(x,y,z,0)=T_{amb},x\in \Omega$

During the movement of the laser heat source along the scanning direction, the heat generated by the laser beam exchanges heat with the surrounding materials and air through convection and radiation. Therefore, in the setting of boundary conditions, the heat exchange on the other five surfaces except the bottom surface can be described by Equation (3). For the bottom surface, it is assumed to be thermally insulated, meaning no heat exchange occurs.

(3)$-k{\displaystyle \frac{\partial T}{\partial \overrightarrow{n}}}=h_c\left(T-T_{amb}\right)+\sigma \epsilon \left(T^4-T_{amb}^4\right)$

where $h_c$ denotes the convective heat transfer coefficient (set to 40 W/(m²·K) determined through joint calibration using thermocouple and infrared camera measurements; $\sigma$ represents the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W/(m²·K⁴)); $\epsilon$ is the thermal emissivity of the material, ranging from 0 to 1, and is set to 0.7 herein [31].

For mesh generation in the numerical modeling process, we use the sweeping function of COMSOL 6.2 (a technique for generating high-quality structured meshes, which generates prismatic or hexahedral elements by sweeping the 2D mesh of the source surface to the target surface along a specified direction (e.g., axis)). High-quality structured meshes are generated as follows: a finer mesh is adopted for the deposition layer (the region close to the heat source) to ensure the accuracy of temperature field calculation near the heat source; a relatively sparse mesh is used for the substrate layer (the region far from the heat source) to reduce the total number of meshes and lower the computational cost, thereby achieving a balance between accuracy and efficiency.

During the interaction of the laser beam, powder, and material, thermal energy not only diffuses radially along the substrate but also penetrates along the depth direction with gradual attenuation. Thus, employing a volumetric heat source model that accounts for both radial and depth distributions better aligns with the actual scenario and more accurately reflects the thermal energy diffusion behavior within the substrate [32]. To precisely characterize the attenuation properties of laser beam energy in the depth direction, this study adopts a depth-attenuated Gaussian volumetric heat source model [33]. Owing to its depth-dependent attenuation characteristics, this model can effectively characterize the diffusion and transfer of heat at varying depths within the substrate, thereby accurately capturing the heat distribution features inside the material. Its energy distribution is expressed as:

(4)$q(x,y,z,t)={\displaystyle \frac{2\eta P}{\pi r^{2}d}}\mathit{exp}\left(-2{\displaystyle \frac{(x-(x_0+vt){)}^2+(y-y_0{)}^2}{r^2}}\right)\mathit{exp}\left(-{\displaystyle \frac{|z-z_0|}{d}}\right)$

where d denotes the laser penetration depth, i.e., the total penetration depth of the laser in both the deposited layer and substrate; $\eta$ is the material’s absorptivity of laser energy, set to 0.3 herein [31]; $r$ represents the spot radius of the laser beam projected onto the substrate, with a value of 1.5 mm; P is the laser power; v stands for the scanning speed of the laser along the deposition direction; $(x_0,y_0,z_0)$ is the initial position of the laser beam at t = 0 s; and $(x,y,z)$ is the actual position of the laser beam at different times.

Table 1

Temperature-dependent material parameters of Ti-6Al-4V [31,34].

TemperatureT (°C)	Densityρ (kg/m3)	Thermal Conductivityk (W/m∙°C)	Specific Heat CapacityCp (J/kg∙°C)
20	4420	7	546
205	4395	8.75	584
500	4350	12.6	651
995	4282	22.7	753
1100	4267	19.3	641
1200	4252	21	660
1600	4198	25.8	732
1650	3886	83.5	831
2000	3818	83.5	831

2.2. Neural Network Modeling of Physical Information

2.2.1. Data Sampling

Before model training, point sampling of the solution domain is required. To balance computational cost and accuracy, a region-specific differentiated point sampling strategy was adopted: dense sampling was performed in regions near the heat source on the upper surface of the substrate (where temperature gradients are large and physical behaviors are complex), with sampling intervals of 0.4 mm, 0.4 mm, and 0.2 mm in the x, y, and z directions, respectively; in regions far from the heat source, sparser sampling was applied, with intervals of 1 mm, 1 mm, and 0.8 mm (Figure 1a). Given the high energy density and complex thermal behaviors of the deposited layer, uniform dense sampling was conducted thereon to ensure the model’s capability to capture relevant features. Furthermore, to avoid wasting computational resources on undeposited regions, the sampling process was dynamically synchronized with the deposition process—specifically, sampling was only performed on currently completed deposited regions (Figure 1b). This strategy, analogous to the element activation technique in finite element analysis, effectively avoids redundant sampling, reducing the number of sampling points by approximately 10% and thereby lowering computational costs significantly (Table 2).

2.2.2. Dynamic Learning Rate-Based Physical Information Neural Networks

The PINN model integrates the powerful nonlinear mapping capability of artificial neural networks with the mathematical constraints of physical conservation laws. Its core innovation lies in embedding the dominant PDEs describing physical processes as a regularization term in the loss function during the training process, thereby realizing the modeling of the spatiotemporal evolution laws of physical fields. As shown in Figure 2, the PINN model consists of two components: forward propagation and backpropagation. In the forward propagation process, spatiotemporal coordinates (x, y, z, t) are input into the network as input vectors. In the hidden layers, data undergoes layer-by-layer processing through linear transformations between neurons in the multi-layer neural network (weighted calculation of weight matrix W and bias vector b) and nonlinear activation functions, constructing a complex mapping relationship from spatiotemporal coordinates to the physical field quantity of temperature T. This process can be formally expressed as:

(5)$A^0=(x,y,z,t{)}^{\mathsf{{\rm T}}}$

(6)$A^l=\sigma (W^l{A}^{l-1}+b^l),l=1,\dots ,l-1$

(7)$T=W^l{A}^{l-1}+b^l$

where A^l is the output vector of the l-th layer, A^(l−1) is the output vector of the (l−1) th layer and also the input of the l-th layer, b^l is the bias vector of the l-th layer, and W^l is the weight vector of neurons between layers. The $\sigma (·)$ activation function converts the linear expression between neurons into a nonlinear one to enhance the expressive ability and learning ability of the network.

Subsequently, the automatic differentiation (AD) technique is employed to perform differential operations on the temperature field T output by the network, calculating its partial derivatives with respect to spatiotemporal coordinates (e.g., ∂T/∂t, ∇T, etc.). Based on these derivative terms, combined with the dominant governing equation describing the heat conduction-convection process (Equation (1)), initial conditions (Equation (2)), and boundary conditions (Equation (3)), the physical constraint residuals for the governing equation R_pde, initial condition R_init, and boundary condition R_bc are constructed respectively:

(8)$\left\{\begin{array}{c}{R}_{pde}={\displaystyle \frac{\rho C_P\partial T}{\partial t}}-k\left({\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}\right)-Q\\ R_{bc}=k{\displaystyle \frac{\partial T}{\partial \overrightarrow{n}}}-h_c\left(T-T_{amb}\right)-\sigma \epsilon \left(T^4-T_{amb}^4\right)\\ R_{init}=T(x,y,z,0)-T_{amb}\end{array}\right.$

In the back propagation stage, the optimizer iteratively updates the network parameters through the gradient descent algorithm based on the gradient information generated from the final loss value (where θ = {W, b} represents trainable parameters) and the learning rate η (Equation (9)). The model continuously executes the closed-loop process of forward prediction, residual calculation, and parameter update until the maximum number of iterations is reached, and the training is terminated.

(9)${\theta }^{l+1}={\theta }^{l+1}-\eta \cdot {\nabla }_{\theta }L\left({\theta }^l\right)$

The loss function can be expressed as the mean squared error (MSE) of the above residual terms, and its mathematical formal expression is as follows:

(10)$\left\{\begin{array}{c}{L}_{pde}={\displaystyle \frac{1}{N_{pde}}}{\sum }_i^N{R}_{pde}^2\\ L_{pde}={\displaystyle \frac{1}{N_{pde}}}{\sum }_i^N{R}_{pde}^2\\ L_{bc}={\displaystyle \frac{1}{N_{bc}}}{\sum }_i^N{R}_{bc}^2\end{array}\right.$

In addition, a data residual loss term L_data is introduced to achieve dual-driven modeling by fusing simulation data with physical equation constraints. There are two main reasons for adding the data residual loss term: first, the DED process involves complex multi-physics field coupling, and the purely PDE-driven PINN may produce systematic errors due to equation simplification. The simulated temperature data T_obs can provide compensation information beyond physical mechanisms through L_data, significantly improving the prediction accuracy under complex working conditions; second, the multiple solution characteristics of the heat conduction equation may cause PINN to converge to an incorrect solution (such as a local minimum). The observed data, as a strong constraint, anchors the optimization direction, making the loss function form a more significant global minimum valley in the parameter space.

(11)$L_{data}={\displaystyle \frac{1}{N_{data}}}{\sum }_i^N|T_{pred}-T_{obs}{|}^2$

Herein, N_pde, N_bc, N_init, and N_data, respectively, represent the number of collocation points sampled in the governing equation domain, boundary condition domain, initial condition domain, and measured data domain. During the model training process, the temperature prediction value T gradually approximates the physical laws expressed in Equations (1) and (2) as the loss function decreases, and this convergence characteristic ensures the physical consistency between the predicted temperature field and the theoretical solution. The PINN constructed in this paper adopts a five-layer fully connected network architecture, where the input layer contains 4 neurons (spatiotemporal coordinates: x, y, z, t); there are 3 hidden layers, each with 64 neurons; the output layer is the physical quantity temperature T. The computer equipment utilized for model training is equipped with an NVIDIA GeForce RTX 3090 (Wuzhou, (Guangzhou, China)) graphics card, with the graphics card driver version 545.29.06 and CUDA toolkit version 12.3. The model was constructed using Python 3.7.16 based on the PyTorch 1.12.1 deep learning framework, with dependent libraries including NumPy 1.21.5, Torch 1.12.1, and SciPy 1.7.3, among others. Subsequent model-based temperature prediction and visualization were performed on a computer equipped with an NVIDIA GTX 1050 (Asus, (Taiwan, China)) graphics card, with the graphics card driver version 522.25 and CUDA toolkit version 11.8. The activation function provides nonlinear factors for the neural network, enhancing its ability to learn and handle complex tasks. Since the Tanh activation function is smooth and easy to differentiate, it converges more easily compared to the Sigmoid function, which helps accelerate the training process. In addition, it can alleviate the gradient vanishing problem, contributing to improving the stability of the model. Therefore, we choose Tanh to further enhance the expressive ability of the model. The Adam optimizer can handle sparse gradients and smooth data with low memory usage, so we select it to perform the network parameter optimization task.

2.3. Dynamically Adjusting Learning Rate Strategies

The learning rate, as a core hyperparameter in gradient descent and its derivative optimization algorithms, directly determines the update step size of model parameters during iteration, exerting a decisive influence on training convergence and efficiency. An excessively high learning rate may cause parameter updates to oscillate or even diverge, making it difficult to approach the global optimal solution. Conversely, an overly low learning rate significantly slows down the convergence speed and increases the risk of being trapped in local minima, ultimately restricting improvements in model performance. Currently, a fixed learning rate is mostly selected through trial-and-error methods [27], which not only increases the workload but also may reduce the training efficiency of the model. Additionally, this study considers the nonlinear relationship between material thermophysical parameters and temperature, which further enhances the model’s nonlinearity. To address these issues, this paper employs a dynamic learning rate adjustment strategy implemented via the Scheduler [35] module in the PyTorch deep learning framework, which can adaptively adjust the learning rate based on changes in the loss function during model training. Algorithm 1 illustrates the dynamic learning rate adjustment process, which continuously monitors changes in the loss value during training and compares them with the training loss from the previous cycle. If the loss decreases significantly, the current optimization direction is deemed effective, and the learning rate is maintained. If the loss does not improve, an adjustment mechanism is triggered. To avoid misjudgments caused by training fluctuations, the strategy introduces a patience period as a buffer threshold, allowing the model to experience temporary performance plateaus within a preset number of cycles. Only when no loss reduction is observed after exceeding the patience period will the model execute the learning rate adjustment strategy, thereby breaking through optimization stagnation points while ensuring training stability and guiding the model to continuously converge toward better solutions.

As shown in Figure 3a, under a fixed learning rate, the model training loss fluctuates sharply with poor stability, and obvious abrupt loss changes occur, especially in the late stage of training. Since the stability of the loss function directly affects the convergence of the model, such severe fluctuations in loss may lead to deteriorated convergence or even divergence of the model. In contrast, after adopting the dynamic learning rate adjustment strategy, the descending trajectory of the model loss is significantly smoother and more stable, and the abnormal phenomenon of abrupt loss changes in the late training stage is effectively suppressed, ensuring that the model can steadily converge to the optimal solution. In addition, Figure 3b presents the training performance of the PINN model under the transfer learning paradigm. Transfer learning is an ML method whose core idea is to utilize the knowledge (such as model parameters, feature representations, training experience, etc.) obtained from a learned task (source task) to assist in solving another related new task (target task) that has limited data, high learning difficulty, or a different sample distribution, thereby improving the learning efficiency, convergence speed, or final performance of the target task. In this study, the model trained under the process parameter group of P = 1500 W and v = 10 mm/s is used as the “source model”. By transferring the parameters of the source model to the training model with new process parameters, the training of models under different combinations of process parameters is realized. It can be observed that compared with training from scratch, under the transfer learning framework, the model can reach the same loss level faster (i.e., with a faster convergence speed); within the same number of training epochs, the transfer learning model can achieve a lower final loss value (i.e., with higher model accuracy). With the support of transfer learning, the training time of the proposed model is only half of that of the finite element model.

Algorithm 1: Dynamic Learning Rate Adjustment During Training

Require: Initial learning rate α0; best loss value Lbest = ∞; patience counter p = 0; maximum patience threshold P = 50; improvement threshold δ = 0.001; decay factor γ = 0.05; maximum number of epochs N.1: Initialize model parameters; n←0;2: repeat3:   n←n + 1;4:   Forward propagation to compute predictions;5:   Calculate current loss Ln;6:   if Ln < Lbest × (1−δ) then:7:    Lbest←Ln;8:    p←0; Reset patience counter9:   else:10:    p←p + 1;11:    if p ≥ P then:12:    α0←α0 × γ; Apply learning rate decay13:    p←0; Reset patience counter14:    end if15:   end if16:   Update model parameters using optimizer with learning rate α0;17: until n > N;The optimized model parameters with adaptive learning rate.

3. Experimentation and Verification

To validate the improved PINN model proposed herein, a temperature field model was established using the commercial finite element software COMSOL Multiphysics 6.2, serving as the validation benchmark. To ensure the reliability of finite element results, the model was first calibrated and validated via experimental methods beforehand to align its outputs more closely with physical reality. Given the extensive applications of Ti-6Al-4V titanium alloy in aerospace, automotive, and biomedical engineering fields, attributed to its high strength, low density, superior corrosion resistance, and biocompatibility [36], the Ti-6Al-4V metal powder and substrate material were selected as the research objects for modeling and experiments in this study.

3.1. Experimental Data Collection

To calibrate and validate the rationality of the numerical model, this study employed both infrared thermography and contact-type thermocouples during the DED experiments to monitor the temperature field changes on the material surface and collect temperature data at key positions. Figure 4 presents the experimental system for temperature monitoring during the DED process along with its schematic diagram. The infrared camera used herein is the MCS640 manufactured by Lumasense Technologies (Milpitas, CA, USA), which features a resolution of 640 × 480 pixels and a maximum frame rate of 60 FPS. It records the temperature changes in the molten pool and deposition trajectory in real-time through an off-axis monitoring method. In addition, a thermocouple thermometer was utilized to monitor the temperature variation at a point 10 mm away from the deposition path. The DED system consists of an FANUC M-20iA/35M (Japan) six-axis robotic arm, a Raycus RFL-C6000X (Wuhuan, China) continuous-wave fiber laser (with a wavelength of 1080 nm, a fiber diameter of 100 μm, and a maximum power of 6 kW), a Huirui laser processing head (supporting spot diameter adjustment and coaxial powder feeding), 99.99% high-purity argon shielding gas (with a flow rate of 9 L/min and coaxial powder feeding), along with a fiber laser and a powder feeding system.

3.2. Calibration and Validation of Numerical Models

The temperature data collected in the above experiments were used to calibrate and validate the numerical model. To ensure the effectiveness of model calibration and validation, the spatial positions for collecting temperature curve data in numerical simulations strictly correspond to those of contact-type thermocouples and infrared cameras. First, the contact-type thermocouple measurement results were utilized to calibrate the numerical model. Since the measurement point is 10 mm away from the heat source trajectory, heat transfer requires time, resulting in a delay in thermocouple monitoring. To fully monitor the temperature changes at point A during the deposition process, the thermocouple measurement time was extended to 180 s. As the convective heat transfer coefficient mainly affects the cooling rate during the cooling process (a larger h_C leads to a higher cooling rate), this parameter was adjusted to gradually make the finite element curve consistent with the thermocouple data. Finally, h_c = 40 W/(m²⋅K) was determined, and the relative error between the thermocouple results and finite element results is within 5%, as shown in Figure 5a. During the deposition process, as the laser beam approaches point A, the temperature at point A rises rapidly. Since point A is not on the laser scanning trajectory, heat is mainly transferred to point A through thermal conduction, resulting in a certain delay in the temperature change at point A. Therefore, even after the entire deposition process ends, the temperature at point A continues to rise. Approximately 20 s later, the temperature at point A reaches its peak and then starts to decrease. The heat at this point begins to propagate to the surrounding materials and air in the form of thermal conduction and thermal radiation, and the temperature gradually returns to the ambient temperature.

To further verify the accuracy of the model, a quantitative comparison was conducted between the results of the calibrated finite element model and the infrared temperature measurement data (Figure 5b). It should be noted that since the minimum measurable temperature of the infrared camera is 800 °C (1073.15 K), temperature values around this threshold cannot be accurately captured. Figure 5b shows the comparison between the temperature measured by the infrared camera on the deposition track and the finite element results at t = 1 s. The minimum temperature captured by the infrared camera is 1121.25 K. Due to interference such as spatter during the cladding process, the camera captures small-range data fluctuations near the minimum measurable temperature, but this fluctuation area is relatively small and does not affect the temperature monitoring of the molten pool core area and adjacent regions. The results show that in the area above the solidus temperature (mainly the molten pool core area) and some adjacent regions, the finite element model results almost coincide with the infrared camera measurement results, with an average relative error of 1.2% and a maximum relative error of 3% in this area. Through the calibration and verification using thermocouples and infrared cameras, the rationality of the finite element model results is fully ensured, and at the same time, a reliable guarantee is provided for the subsequent verification of the PINN model.

As shown in Figure 6, by observing the temperature distribution curve along the deposition trajectory, it can be found that the highest temperature point on the molten pool surface does not coincide with the position of the laser spot. This phenomenon is mainly attributed to the delay in temperature transfer during the heat conduction process: when the laser beam moves, heat continues to diffuse to the surrounding area through thermal conduction. Since the cooling rate of the material is lower than the heating rate, the temperature at this point does not drop immediately after the laser beam leaves; on the contrary, affected by the heat transferred from the front, the temperature at this point will continue to rise for a period of time, resulting in the highest temperature point appearing behind the laser spot. In addition, the hydrodynamics of the molten pool may also contribute to the offset of the surface’s highest temperature position. Specifically, the liquid metal flows to the surrounding under the action of surface tension, which may change the shape and position of the molten pool front, thereby affecting the location of the highest temperature point. Meanwhile, the convection inside the molten pool and the Marangoni convection caused by temperature gradients can also alter the flow and temperature distribution of the melt within the molten pool [37,38]. Furthermore, the recoil pressure generated by melt evaporation can also affect the shape and temperature distribution of the molten pool, leading to the offset of the highest temperature region relative to the center of the laser spot [39]. To sum up, the offset of the highest temperature position in the molten pool may be the result of the combined action of multiple factors such as heat conduction delay, molten pool hydrodynamics, internal convection, and melt evaporation.

4. Results and Discussion

In this section, the trained DLR-PINN model is utilized to predict the 3D temperature field and molten pool size during the single-track DED process. First, the temperature field under a single process parameter is predicted and compared with the finite element results for verification. Second, combined with transfer learning, the 3D temperature field and molten pool size under different combinations of process parameters are predicted, and a sensitivity analysis is conducted based on the prediction results.

4.1. Prediction of Three-Dimensional Temperature Field Based on Improved PINN

In this study, the trained model was first used to predict the 3D transient temperature field during the single-track DED process under the conditions of power P = 1500 W and scanning speed v = 10 mm/s. To ensure the reliability of the PINN model’s prediction results, the PINN predictions were compared and analyzed with the calculation results from the FEM established based on COMSOL Multiphysics. Figure 7 compares the three-dimensional temperature fields at t = 1 s and t = 2 s. Figure 7a displays the FEM results, where the temperature point cloud exhibits dense distribution near the heat source region but becomes sparse in distal regions. This spatial variation stems from the adaptive meshing strategy employed to concentrate computational resources within the heat-affected zone. In contrast, Figure 7b demonstrates that the DLR-PINN approach generates more uniformly distributed and informationally complete temperature point cloud data across the entire computational domain, including regions distant from the heat source. In terms of computational efficiency, traditional numerical models require approximately 40 min to complete computations. In contrast, the training of the DLR-PINN model integrated with transfer learning takes only approximately 20 min. Furthermore, on an NVIDIA GeForce GTX 1050 GPU, the DLR-PINN model completes the entire 3D temperature field prediction in approximately 10 s. For a further quantitative evaluation of accuracy, Figure 8 compares the temperature distributions along the scanning trajectory direction at the interface between the substrate and the deposited layer. The results show that at t = 1 s, the maximum temperature predicted by PINN is 2082.62 K, and the FEM calculation result is 2103.91 K; at t = 2 s, the maximum temperature predicted by DLR-PINN is 2102.85 K, and the FEM result is 2103.91 K. The two sets of temperature curves at different time instants are almost identical, with a relative error of approximately 1%, which fully verifies the rationality of the DLR-PINN model’s prediction results. In summary, compared with the traditional FEM, the DLR-PINN model can obtain 3D temperature field data with more uniform spatial distribution and more complete coverage at significantly higher computational efficiency while ensuring prediction accuracy comparable to FEM, especially in key regions near the heat source.

Furthermore, Figure 9 presents the temperature distribution results predicted by the improved PINN model at the interface between the substrate and the deposited layer at different time instants. It can be observed that as the heat source moves, the trailing edge of the molten pool exhibits a ‘comet-like’ morphology. This phenomenon arises because during the process from the start of laser scanning to the formation of a stable molten pool, the temperature at the center of the molten pool remains the highest, resulting in a lower surface tension in this region compared to other areas. Under the action of the surface tension gradient, the liquid metal in the molten pool flows from the high-temperature region to the surrounding low-temperature regions. Particularly at the rear end of the molten pool, after the material transforms from a powder state to a solid layer, its thermal conductivity increases, enabling rapid heat transfer, which in turn causes the melt at the rear end of the molten pool to cool and shrink rapidly. Meanwhile, since the cooling rate of the material is lower than the heating rate, the trailing edge of the molten pool gradually elongates as the beam moves, eventually forming a ‘comet-like’ shape. In addition, due to the difference in thermal conductivity between the front and rear ends of the molten pool, the temperature gradients at these two ends vary. It can be observed from Figure 9 that the temperature bands at the front end of the molten pool are denser, indicating a larger temperature gradient in this region. This also confirms that the prediction results of the PINN model are consistent with objective reality and can reflect the objective physical laws.

4.2. Prediction and Analysis of Melt Pool Temperature and Size Under Different Processes Based on Improved PINN Modeling

Laser power (P) and scanning speed (v), as core control parameters of the DED process, significantly affect the formation of the molten pool and the melting and solidification dynamics of materials, and ultimately determine the microstructure, mechanical properties, and surface morphology quality of the deposited layer. Based on the established improved PINN model framework, this study, combined with transfer learning methods, efficiently predicted the 3D transient temperature fields under multiple sets of key process parameter combinations (P = 1500 W, 1800 W, 2000 W; v = 6 mm/s, 8 mm/s, 10 mm/s). Based on the detailed temperature field data output by the PINN model, the geometric characteristics of the molten pool (such as length, width, and depth) under each parameter combination were further extracted and quantified. Finally, a comprehensive analysis of the 3D temperature distribution patterns and the evolution laws of molten pool sizes under different power and speed conditions was conducted.

4.2.1. Prediction of Molten Pool Temperature and Size at Different Laser Powers

To investigate the influence of laser power (P) on the thermal behavior of the molten pool, this study first utilized the trained model, effectively combined with transfer learning technology, to predict the 3D transient temperature field and molten pool geometry under constant scanning speed (v = 10 mm/s) and different laser powers (P = 1500 W, 1800 W, 2000 W).

Figure 10a specifically shows the temperature distribution curves corresponding to different powers along the scanning direction of the deposition trajectory (Path 1) predicted by the DLR-PINN model at t = 2 s. Figure 10b quantitatively compares the variation trends of key geometric dimensions (length, width, depth) of the molten pool predicted by the DLR-PINN model with power, and verifies them against the FEM calculation results. The analysis results indicate that with the increase in laser power, the peak temperature in the molten pool region increases significantly, and the size of the molten pool (including length, width, and depth) shows a systematic increasing trend. Figure 10a shows that under different power levels, the temperature distribution curves predicted by DLR-PINN are basically consistent with the FEM calculation results; Figure 10b further confirms the accuracy of DLR-PINN in predicting molten pool sizes. This fully confirms that the proposed DLR-PINN model, combined with the transfer learning strategy, is capable of accurately predicting the dynamic evolution of the molten pool temperature field and its geometric characteristics under varying laser power conditions. Furthermore, Figure 11 presents the temperature distribution along the longitudinal direction of the molten pool center. Compared with the temperature distribution on the surface deposition trajectory, the temperature gradient inside the molten pool is smaller than that in the surrounding area of the molten pool. This is because the exterior of the molten pool is in direct contact with the low-temperature substrate and air, and heat is rapidly dissipated through thermal conduction and convection, forming a steep temperature gradient. In contrast, within the molten pool, the combined effects—including the high thermal conductivity of liquid metal and Marangoni convection, which promotes heat redistribution inside the molten pool—result in a more uniform heat distribution with a significantly reduced gradient.

To further reveal the influence mechanism of laser power P on the temperature distribution and size of the molten pool, Figure 12(b1–d3) respectively show the temperature distribution contour maps predicted by DLR-PINN on three key cross-sections (substrate-deposited layer interface, longitudinal section along the scanning trajectory, and molten pool cross-section) under different powers. In the figures, the molten pool contour lines are plotted according to the solidus temperature of Ti-6Al-4V alloy (~1878 K), and the quantitative results of geometric dimensions such as length, width, and depth of the corresponding molten pool are presented in Figure 10b. A comprehensive analysis of Figure 12(b1–d3) reveals that with the increase in laser power, the range of the heat-affected zone at the trailing edge of the molten pool widens significantly, and the overall geometric dimensions (length, width, depth) of the molten pool gradually increase (this size change trend is consistent with the quantitative results in Figure 10b. The dominant physical mechanism lies in the intense Marangoni convection within the molten pool: the surface tension gradient induced by the temperature gradient drives the molten metal to flow from the high-temperature central region (low surface tension) to the low-temperature edge region (high surface tension). The increase in power leads to a significant rise in energy density, causing more powder and substrate to melt, increasing the volume of liquid metal in the molten pool, and intensifying the temperature gradient. These factors collectively enhance the intensity of Marangoni convection, promoting more effective lateral spreading of molten metal to the edge of the molten pool, thereby significantly increasing the length and width of the molten pool; meanwhile, the enhanced energy input improves the thermal penetration capability of the molten metal, enabling the molten pool to further expand in the depth direction. Therefore, the increase in power, by strengthening Marangoni convection, is the fundamental reason for the expansion of the heat-affected zone of the molten pool and the systematic growth of three-dimensional geometric dimensions.

Increasing the width and depth of the molten pool is beneficial for enhancing the compactness between layers, reducing the probability of pores and lack-of-fusion defects during deposition, and decreasing surface roughness. Therefore, appropriately increasing the power helps improve deposition quality and enhance the mechanical properties of the formed parts [40]. However, excessively high laser power may intensify the evaporation of the molten pool and increase recoil pressure, thereby leading to more spatter and pore formation. Conversely, insufficient laser power results in inadequate energy input, forming irregular lack-of-fusion holes [41]. In summary, when selecting the laser power, it is necessary to balance the increase in molten pool size against the potential issues of evaporation, spatter, and pores to achieve the optimal deposition effect.

4.2.2. Effect of Scanning Speed on Temperature Distribution

Scanning speed is a key parameter affecting the solidification rate of the process, which directly influences the temperature field, cooling rate, and grain size of the molten pool, thereby affecting the microstructure and mechanical properties of the final formed part [42]. To further understand the impact of scanning speed on the temperature and size of the molten pool during the DED process, Figure 13a shows the temperature distribution curves predicted by the DLR-PINN model under different scanning speeds (v = 6 mm/s, 8 mm/s, 10 mm/s) along the scanning direction of the deposition trajectory (Path 1) at t = 2 s; Figure 13b quantitatively presents the variation trends of the molten pool’s length, width, and depth with the increase in scanning speed. Contrary to the influence law of laser power, the analysis results indicate that as the scanning speed increases, both the peak temperature and the overall geometric size of the molten pool show a decreasing trend. Similarly, Figure 14 displays the temperature distribution from the bottom of the substrate to the surface of the molten pool under different scanning speeds. From the bottom of the substrate to the surface of the molten pool, the temperature gradient inside the molten pool changes significantly. In contrast, in Figure 11, from the bottom of the substrate to the surface of the molten pool, the temperature gradient inside the molten pool changes relatively stably. It can be seen that the scanning speed has a more significant impact on the internal temperature gradient of the molten pool. This is because when the scanning speed increases, the action time of the heat source per unit area is shortened, the energy density decreases, the size of the molten pool (such as width and height) is significantly reduced, and heat is more concentrated in a narrow area, leading to a rapid increase in the temperature gradient [43]; at the same time, high-speed scanning accelerates the cooling rate of the molten pool, inhibits the temperature-equalizing effect of Marangoni convection, and further exacerbates the uneven distribution of heat. Therefore, changes in scanning speed will cause significant changes in the gradient.

Figure 15 further intuitively demonstrates the influence of scanning speed on molten pool size and heat-affected zone through temperature distribution contour maps and changes in molten pool contours on different cross-sections (substrate-deposited layer interface, longitudinal section along the scanning trajectory, and molten pool cross-section). Observing Figure 15(b1–b3), it can be found that as the scanning speed increases, the shape of the heat-affected zone gradually changes from wide and short to narrow and long. The reason for this change lies in the difference in laser-material interaction time. At low scanning speeds, the lower speed prolongs the action time of the laser in a unit area. Firstly, more energy input increases the molten pool temperature and the volume of liquid metal, and the enhanced temperature gradient drives stronger Marangoni convection. This convection allows molten metal sufficient time to flow and expand laterally towards the edge of the molten pool, thereby increasing the width of the molten pool; secondly, the continuous energy input improves the thermal penetration capability of the molten pool, leading to an increase in molten pool depth; furthermore, low scanning speed also reduces the solidification rate of the molten pool, allowing heat more time to conduct to the surrounding matrix, significantly expanding the range of the Heat Affected Zone (HAZ) and forming a wider temperature distribution zone. Conversely, at high scanning speeds, the action time of the laser in a unit area is greatly compressed. This results in a significant reduction in the total heat input absorbed by the material, and the rapid movement of the heat source causes a sharp increase in the cooling rate [44]. Meanwhile, the flow of liquid metal (including Marangoni convection) and heat conduction processes are severely limited by time. Under the combined effect of these factors, the molten pool rapidly solidifies and shrinks, with its length, width, and depth all significantly reduced; the limited energy input and short action time significantly narrow the range of the heat-affected zone, ultimately forming a narrow and long temperature heat-affected zone.

Although a lower scanning speed can increase the interaction time between the laser and the powder, promoting the sufficient melting of the powder and substrate, excessively low scanning speed will reduce the deposition efficiency. Conversely, an excessively high scanning speed will lead to a decrease in local energy density, which may cause defects such as a lack of fusion and spheroidization [2]. Therefore, in practical applications, it is necessary to reasonably select the scanning speed according to material properties and forming requirements to balance the forming efficiency of the molten pool and the quality of the formed part.

4.3. Sensitivity Analysis

The combination of different laser powers and scanning speeds essentially changes the energy density, which is a key parameter in the DED process. Energy density is closely related to laser power, scanning speed, and laser beam diameter, and it plays a crucial role in determining the thermal effects during the DED process. It affects the molten pool size, solidification rate, and ultimately the quality of the deposited material. The energy density can be described by the following formula [2]:

(12)$E={\displaystyle \frac{P}{vd}}$

In this study, we set the laser beam diameter d. By changing the power P and scanning speed v, we can adjust the energy density, thereby influencing the melting and solidification processes of the material. Table 3 below shows the energy density values calculated under different combinations of power and scanning speed, which are crucial for optimizing DED process parameters and predicting material responses.

To quantitatively evaluate the relative influence weights of laser power P and scanning speed v on the molten pool temperature distribution and geometric dimensions, this study further conducted a systematic sensitivity analysis. As shown in Figure 16, the statistical analysis results reveal significant correlations between the molten pool geometric dimensions (length, width, depth) and key process parameters: the molten pool size is positively correlated with laser power, while negatively correlated with scanning speed. It is worth noting that the change in scanning speed has a particularly significant impact on the molten pool, especially on the molten pool width, depth, and energy density (P/v). Specifically, under the condition of fixed laser power, reducing the scanning speed significantly promotes the expansion of the molten pool in the width and depth directions, while the impact on the length direction is relatively small. This phenomenon is highly consistent with the HAZ morphology changes observed in Figure 15: the reduction in scanning speed leads to a significant broadening of the HAZ range, and its spatial expansion mode is consistent with the growth trend of the molten pool in the width and depth directions, intuitively confirming the above correlation laws. In summary, the DLR-PINN model established in this study combined with the transfer learning strategy can not only quickly and accurately predict the 3D transient temperature field of the molten pool (the maximum mean absolute error for temperature prediction under different processes is 0.53%) and its 3D geometric dimensions (among them, the mean absolute errors for predicting the molten pool length, width, and depth are 3.69%, 2.48%, and 6.96%, respectively), but also the revealed quantitative correlation laws between process parameters and molten pool morphology are reasonable and reliable. These prediction results and physical insights provide critical data support and a theoretical basis for in-depth understanding of the thermophysical mechanisms underlying the DED process and optimizing the process parameter window (e.g., controlling the molten pool size and HAZ to regulate the microstructure).

4.4. Orthogonal Experiment

To verify the stability of the model, this study designed a series of orthogonal experiments, covering combinations of three groups of different laser powers (1500 W, 1800 W, 2000 W) and three groups of different scanning speeds (6 mm/s, 8 mm/s, 10 mm/s). The specific parameters are shown in Table 4. This table lists the maximum temperature error and mean absolute percentage error (MAPE) under different combinations of power and scanning speed. MAPE is a commonly used indicator to measure the prediction accuracy of the model, which is calculated by comparing the difference between the predicted value and the actual value. The formula is as follows:

(13)$MAPE={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|{\displaystyle \frac{y_{pre}-y_i}{y_i}}\right|\times 100\%$

where y_i is the actual observed value, $y_{pre}$ is the model’s predicted value, and N is the total number of samples. A lower MAPE value indicates a higher accuracy of the model’s prediction.

As can be seen from the data in Table 4, under different combinations of power and scanning speed, the maximum error of the temperature field ranges from 32.35 K to 80.68 K, and the MAPE value ranges from 0.29% to 0.53%. The above results indicate that the proposed model can maintain high stability and prediction accuracy even when the process parameters change.

4.5. Prediction and Analysis of Temperature Field Under Flat-Top Heat Source Based on DLR-PINN

Currently, most heat source models in the DED process employ the Gaussian heat source model. However, the energy distribution in the molten pool under a Gaussian heat source is characterized by high energy at the center and low energy at the edges, which tends to generate large temperature gradients and form an inhomogeneous temperature field. This, in turn, leads to high porosity, impairs the forming quality, and reduces mechanical properties. Furthermore, such an energy distribution lowers energy utilization efficiency, which is detrimental to improving printing efficiency [45]. Currently, most heat source models in the DED process employ the Gaussian heat source model. However, the energy distribution in the molten pool under a Gaussian heat source is characterized by high energy at the center and low energy at the edges, which tends to generate large temperature gradients and form an inhomogeneous temperature field. This, in turn, leads to high porosity, impairs the forming quality, and reduces mechanical properties. Furthermore, such an energy distribution lowers energy utilization efficiency, which is detrimental to improving printing efficiency [46]. Moreover, since DED typically operates in the heat conduction mode and the molten pool is relatively large, a uniform energy distribution is more readily achieved under the flat-top beam heat source. In this section, the improved PINN model is utilized to predict the temperature field under the flat-top beam heat source and compare it with that under the Gaussian beam. This aims to deepen the understanding of the mechanism by which different heat source models influence the DED process, thereby providing theoretical support for process optimization. The heat source model of the flat-top beam can be expressed as [47]:

(14)$I_f\left(x,y,t\right)={\displaystyle \frac{2\eta P}{\pi r_l^2}}\exp \left[-2{\left({\displaystyle \frac{{\left(x-(x_0+v\cdot t)\right)}^2+{(y_0-y)}^2}{{r_l}^2}}\right)}^{10}\right]$

Based on the heat source expression of the flat-top beam in Equation (14), this study predicted the molten pool temperature distribution under this heat source using the PINN model. Figure 17a,b present the temperature distribution maps from the PINN prediction and simulation, respectively, while Figure 17c shows the comparison of temperature distribution curves along the top deposition trajectory. It can be seen from the figures that the temperature distribution curve predicted by the PINN model is in good agreement with that from the simulation. This indicates that the PINN model can accurately predict the temperature distribution under the flat-top beam, and also verifies that the model still has good stability and accuracy under different working conditions and scenarios.

(15)$I_g\left(x,y,t\right)={\displaystyle \frac{2\eta P}{\pi r^2}}\exp \left[-2\left({\displaystyle \frac{{\left(x-(x_0+v\cdot t)\right)}^2+{(y-y_0)}^2}{r^2}}\right)\right]$

To further clarify the influence of the two heat sources on the molten pool temperature field, this study compared the molten pool temperature distributions under the flat-top beam and the Gaussian beam. As can be seen from Equation (14), the flat-top heat source model describes a planar energy distribution. For effective comparison, the Gaussian volumetric heat source model (Equation (4)) was also converted to a Gaussian surface heat source model (Equation (15)). This transformation only ignores the penetration effect of the heat source in the depth direction without changing the Gaussian characteristics of the heat source. Figure 18(a1–b2) show the temperature distributions under the two different heat sources, respectively. It can be observed that the molten pool area under the flat-top beam is larger, which can avoid excessive energy concentration and make the energy distribution in the molten pool more uniform. However, this uniformity is mainly manifested at the imaging position or near the focal point of the optical system [1] (as shown in Figure 18d) and is difficult to maintain during laser propagation. In contrast, the molten pool area under the Gaussian beam is smaller, with a more concentrated energy distribution. Figure 18c presents the temperature change in a point 0.2 mm below the surface of the deposited layer during the entire deposition process. It can be observed that between approximately 1.2 s and 2.0 s, the point is in the heating stage, with the temperature rising rapidly, and the heating rates under the two heat sources are comparable. Between 2.0 s and 2.5 s, in the cooling stage, the temperature of the point drops rapidly, and the cooling rate is lower than that in the heating stage. It is worth noting that the cooling rate of the Gaussian beam in the cooling stage is higher, which can lead to the formation of finer and unbalanced microstructures, increase residual stress, and thus cause cracks and deformations [2] and may also result in an increase in porosity [48]. Figure 18d shows the temperature distribution at different positions along the direction perpendicular to the scanning direction at the same moment. It can be seen that within the molten pool area, the temperature change per unit distance under the flat-top beam is smaller than that under the Gaussian beam, indicating that the temperature gradient in the molten pool area is smaller and the temperature distribution is more uniform. In summary, the larger molten pool area and more uniform temperature distribution under the flat-top beam are beneficial in improving inter-layer compactness, reducing pore defects, thereby improving the quality of formed parts [49]. Meanwhile, the larger molten pool also helps to enhance the laser printing efficiency [50].

5. Conclusions

This study primarily developed a PINN model with a dynamic learning rate, which was combined with transfer learning to predict the three-dimensional temperature field and molten pool size under varying power and scanning speed conditions. Based on the prediction results, we analyzed the effects of laser power and scanning speed on molten pool temperature and size, and quantitatively evaluated the significance of power P and scanning speed v with respect to molten pool temperature and size, respectively. The results are as follows:

(1). Aiming at the multi-scale physical characteristics in the DED process, a region-specific differentiated point sampling strategy is proposed. This strategy avoids redundant sampling while ensuring prediction accuracy, effectively reducing the training data by approximately 10%, and significantly lowering the computational costs of data generation and model training.

(2). Considering the nonlinearity between material thermophysical parameters and temperature increases the nonlinearity of the model. Therefore, a dynamic learning rate adjustment strategy is adopted, which significantly improves the convergence and training stability of the model. By combining transfer learning technology, the convergence speed of model training is remarkably enhanced. Under the hardware environment of the NVIDIA GTX 1050 GPU, the prediction of a single three-dimensional transient temperature field only takes about 10 s, providing a technical basis for real-time process monitoring and optimization.

(3). The proposed DLR-PINN model exhibits excellent performance in predicting key parameters of the molten pool. Compared with the finite element results at the same moment, the model can obtain a more complete three-dimensional temperature point cloud distribution, and the maximum mean absolute percentage error of temperature field prediction under different processes is 0.53%. The mean absolute percentage error for predicting the length, width, and depth of the molten pool is 3.69%, 2.48%, and 6.96%, respectively. Moreover, the prediction results of the model successfully reproduce the evolution law of the HAZ with process parameters (as shown in Figure 13), demonstrating that the model has strong physical consistency.

(4). Parameter sensitivity analysis shows that the maximum temperature and size of the molten pool increase significantly with the increase in laser power or the decrease in scanning speed. Among them, the scanning speed v is the primary influencing factor, and its regulatory weight on the molten pool characteristics is higher than that of the laser power.

(5). The developed PINN model can accurately predict the temperature distribution under the flat-top heat source. Compared with the Gaussian heat source, the temperature gradient around the molten pool under the flat-top heat source is smaller, and the temperature distribution is more uniform, which is beneficial to improving the forming quality.

The DED process involves multiple complex and coupled physical fields, including heat transfer, fluid flow, material phase transition, and stress-strain. PINN provides a new approach to addressing the challenges of multi-physics field coupling. However, current research mostly focuses on model development for a single field. Future work could further explore the application of PINNs in multi-physics field coupling in DED, such as thermo-mechanical coupling and fluid-solid coupling. Meanwhile, the more complex the physical field, the stronger the nonlinearity, making further optimization of network structures crucial—for instance, by introducing attention mechanisms, adaptive activation functions, and exploring hybrid architectures. Additionally, regarding the limitations of this study, further exploration could be conducted on the impacts of different sampling strategies and sampling densities on model training efficiency and prediction accuracy.

Author Contributions

Conceptualization, X.H. and Y.L.; methodology, X.H., Z.Q. and Z.P.; software, X.H.; validation, X.H. and Z.Q.; formal analysis, X.H. and Z.P.; investigation, X.H., Z.Q., X.G. and H.L.; resources, Y.L.; data curation, X.H., Z.Q. and Z.P.; writing—original draft preparation, X.H.; writing—review and editing, H.L. and Y.L.; visualization, X.H. and X.G.; supervision, Y.L.; project administration, Y.L.; funding acquisition, Y.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in the study are included in the article; further inquiries can be directed at the corresponding author.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 Sampling of the model training dataset: (a) schematic of the entire solution domain sampling; (b) dynamic synchronized sampling for the sedimentary layer. (Among these, the blue color denotes the sampling points on the substrate boundary; the green color denotes the sampling points in-side the substrate; the red color denotes the sampling points on the deposition layer).

Figure 2 Physics-informed neural network model with a dynamic learning rate.

Figure 3 Comparison of model validation loss under different scenarios: (a) loss under dynamic versus fixed learning rate; (b) model loss under migration learning.

Figure 4 DED process temperature monitoring and acquisition: (a) DED experimental equipment; (b) schematic of DED process temperature acquisition (Among these, the blue dotted line indicates the approximate monitoring range of the infrared camera).

Figure 5 Numerical model calibration and validation (a) numerical model calibration (b) model validation.

Figure 6 Temperature profiles along the deposition trajectory at t = 1 s, with a localized enlargement near the peak temperature provided in the lower right corner.

Figure 7 Comparison of 3D temperature field distribution at different moments (P = 1500 W, v = 10 mm/s): (a) 3D temperature point cloud calculated by FEM; (b) 3D temperature point cloud predicted by the improved PINN model.

Figure 8 Comparison of the temperature distribution of the substrate and deposition layer interface path 1.

Figure 9 Temperature distribution at different moments at the interface between substrate and deposited layer (z = 2.0 mm): (a) t = 1 s; (b) t = 2 s.

Figure 10 Effect of laser power on melt pool temperature and size (t = 2 s, v = 10 mm/s, P = 1500 W, 1800 W, 2000 W): (a) temperature distribution along the scanning trajectory; (b) melt pool size (length, width, depth).

Figure 11 Longitudinal temperature distribution in the center of the melt pool at different powers (z-axis indicates the distance from the bottom of the substrate to the surface of the deposited layer).

Figure 12 Multi-view temperature field display at different laser powers: (a1–a3) 3D temperature point cloud predicted by DLR-PINN (b1–b3) temperature distribution at the interface between the substrate and the deposition layer; (c1–c3) temperature distribution in the longitudinal profile along the melt pool trajectory; and (d1–d3) temperature distribution in the transverse cross-section of the melt pool along the perpendicular scanning direction.

Figure 13 Effect of scanning speed on melt pool temperature and size (t = 2 s, v = 10 mm/s, P = 1500 W, 1800 W, 2000 W): (a) Temperature distribution along the scanning trajectory; (b) melting pool size (length, width, depth).

Figure 14 Longitudinal temperature distribution in the center of the melt pool at different scanning speeds (z-axis indicates the distance from the bottom of the substrate to the surface of the deposited layer).

Figure 15 Multi-view temperature field visualization under different scanning speeds: (a1–a3) 3D temperature point cloud map predicted by DLR-PINN; (b1–b3) temperature distribution at the interface between the substrate and the deposition layer; (c1–c3) temperature distribution along the longitudinal section of the melt pool trajectory; (d1–d3) temperature distribution of the transverse cross-section of the melt pool along the direction perpendicular to the scanning direction.

Figure 16 Sensitivity analysis of melt pool characteristics and process parameters.

Figure 17 Comparison of PINN and simulation results for the temperature field distribution of the flat-top beam under (a) PINN, (b) simulation, and (c) regression of PINN on the deposition trajectory for the prediction of the temperature distribution.

Figure 18 Comparison of temperature distribution under Gaussian temperature heat source and flat-top heat source at P = 1500 W and v = 10 mm/s: (a1,a2) Gaussian; (b1,b2) flat-top. Among them, Figure (a1,b1) show the energy distributions under different heat sources at t = 0 s, where the black marked lines represent the melt pool contour lines. (c) Temperature distribution curves of a point on the deposited base layer at different moments. (d) Temperature distribution curves at different locations along the vertical scanning direction at upper t = 2 s.

Table 2

Comparison of efficiency under different sampling strategies.

Sampling Strategy	Number of Sampling Points per Unit Time	Training Time (50 Epochs)
Differential sampling	20,939	24.5 s
Uniform sampling	23,266	30.7 s

Table 3

Energy density at different laser powers and scanning speeds.

P (W)	V (mm/s)	E (J/mm2)
1500	10	50
1800	10	60
2000	10	66.67
1500	6	83.33
1500	8	62.5

Table 4

Nine sets of orthogonal experiments with different laser parameters were conducted by transfer learning, and the following table shows the maximum error (Max error) and mean percentage error (MAPE) between DLR-PINN and SIM at t = 2 s.

	P (W)	V (mm/s)	Max Error (K)	MAPE (%)
1	1500	6	51.21	0.31
2	1800	6	58.38	0.36
3	2000	6	62.25	0.38
4	1500	8	41.67	0.36
5	1800	8	59.15	0.37
6	2000	8	60.23	0.29
7	1500	10	32.35	0.35
8	1800	10	77	0.53
9	2000	10	80.68	0.53

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