Review of Recent Additive Manufacturing and Welding Research with Application of Physics-Informed Neural Networks

Taehwan Ko; Heuisu Kim; Yeoungcheol Shin; Dukyong Kim; Young Hoon Lee; Jinsu Hong; Seung Hwan Lee

doi:10.5781/JWJ.2024.42.4.3

Abstract

This review introduces recent research on applying physics-informed neural networks (PINNs) to additive manufacturing and welding. PINNs, which are artificial intelligence models, integrate governing equations containing physical information with artificial neural networks, enabling the modeling of complex physical phenomena at a lower computational cost than traditional numerical models. Although PINNs have been employed in a limited number of studies on welding processes, they have been extensively used in various studies within the field of additive manufacturing. This study reviews the theoretical background of PINNs to explore their effective application to welding processes, examining 12 research cases in additive manufacturing and two research cases in welding processes. The analysis included the structure of the PINN, governing equations, and prediction results of each study. Results indicate that PINNs provide faster computation speeds and higher prediction accuracies than numerical models. Moreover, they could perform analyses without additional training even when process parameters and materials changed. Additionally, PINNs have been effectively applied to predict the mechanical properties of the molten zone. Consequently, PINNs are anticipated to be actively applied in future research on welding process modeling and mechanical property prediction.

Key words: Physics-informed neural network, Additive manufacturing, Welding, Temperature field prediction, Melt pool behavior prediction, Mechanical property prediction

1. Introduction

Additive manufacturing and welding processes are multiphysics systems involving complex physical phenomena such as heat transfer and fluid dynamics during the melting and solidification of metal materials. Consequently, fluid and thermodynamics-based numerical models have been employed to observe these multi-physical phenomena¹^,²⁾. For instance, Sun et al.³⁾ simulated the temperature distribution and flow within a melt pool during direct energy deposition (DED) using numerical models. Similarly, Pamnani et al.⁴⁾ modeled the temperature distribution in shielded metal arc welding (SMAW) and activated gas tungsten arc welding (A- GTAW) processes using numerical models. However, these numerical models require separate analyses whenever the process conditions, such as process parameters or material properties, change. Additionally, the high-temperature gradients and cooling rates of both processes necessitate precise analysis, resulting in high computational costs⁵^,⁶⁾.

Physics-informed neural networks (PINNs), which combine physical-governing equations with artificial intelligence (AI) algorithms, have recently attracted considerable attention as alternatives to traditional numerical models. PINNs integrate governing equations into artificial neural networks, enhancing prediction accuracy and speed by leveraging data and physical information⁷⁾. According to prior research, PINNs can incorporate process conditions, such as process parameters and material properties, as input values, eliminating the need for separate analyses for different process conditions. Using automatic differentiation through the chain rule, a characteristic of neural network structures, enables the rapid and accurate computation of nonlinear partial differential equations, resulting in lower computational costs than traditional numerical models⁵⁾. Furthermore, PINNs can extend beyond numerical analysis to mechanical property prediction and in situ process monitoring applications by incorporating equations related to mechanical properties (e.g., Murakami equation) and process-monitoring criteria into the neural network. Hence, PINNs serve as versatile tools to predict the quality of products during the manufacturing process.

Recent trends reveal that while research on applying PINNs in welding processes remains sparse, PINNs have been successfully implemented in numerous additive manufacturing studies. For example, PINNs have been used to predict 3D temperature fields in DED processes, model molten pool behavior in laser powder bed fusion (LPBF) processes, and predict the fatigue life of components produced via LPBF. This study reviews 12 recent investigations that applied PINNs to additive manufacturing, summarizing their application methods, required input variables, and governing equation structures. Additionally, two recent studies that applied PINNs to welding processes were introduced to explore their potential future applications in this field. Fig. 1 categorizes the reviewed studies into additive manufacturing and welding based on the process where PINNs were applied. Further classification was performed in additive manufacturing based on research objectives: predicting temperature distribution, molten pool behavior, and mechanical properties. Conversely, the welding studies were primarily focused on predicting temperature fields and are discussed separately.

Fig. 1

Classification of PINN research papers on additive manufacturing and welding processes

2. Theoretical Background of PINN

2.1 Basic Structure of PINN

PINNs are artificial intelligence models incorporating governing equations into traditional neural networks to predict approximate solutions that adhere to these equations⁷⁾. The basic structure of a PINN is similar to that of a typical deep neural network (DNN) (Fig. 2), but the loss function setting is notably different. Traditional neural networks are trained with labeled data using a loss function to calculate the error between predicted and actual results. After that, this error is used in backpropagation to adjust the weights and biases of the network, improving the prediction accuracy. However, PINNs embed governing equations into the loss function, ensuring predictions align with training data and the governing equations. These equations include partial differential equations (PDEs) such as the heat transfer equation, conservation of energy law, Navier–Stokes equations, initial and boundary conditions, and other physical constraints or semi-empirical models. By incorporating these equations into the loss function, PINNs are trained to generate solutions that approximate the true behavior of the modeled physical system.

Fig. 2

Basic structure of PINN

2.2 Basic Principle of PINN

PINNs, like traditional artificial neural networks, use backpropagation to compute approximate solutions that satisfy the governing equations. PINNs integrate these governing equations into the loss function to facilitate effective learning. The fundamental principles of PINNs are as follows: Only data loss is calculated in a standard AI model without governing equations incorporated into the loss function, as shown in Eq. (1).

(1)

MSEdata:1N∑i=1N(uNN(xi;θ)−utrue(xi))2

where N represents the number of data points, u_NN(x_i; θ) denotes the predicted result of the neural network for the input variable x_i, with weights and biases parameterized by θ.u_true(x_i) represents the actual true value of the input variable x_i. On the contrary, PINNs incorporate governing equations into the error calculations using the loss function. For instance, when predicting an approximate solution for the differential equations (Eq. (2)) using the PINN, Eq. (3) is applied to the PINN loss function to compute the error between the differential equation and predicted values. The loss function value is calculated at preselected points, known as collocation points, rather than at all coordinates of the differential equation.

(2)

Differential equation:md2udx2+μdudx+ku=0

(3)

MSEf:1M∑j=1M([md2uNN(xj;θ)dx2+μduNN(xj;θ)dx+kuNN(xj;θ)−0])2

where u corresponds to the true solution; m,μ,k denote constants and M represents the number of collocation points. Increasing the number of collocation points enhances the training sample size of the PINN; however, it also increases computational cost. Therefore, users adjust the number of collocation points based on the predictive performance of the model. Consequently, Eq. (3) calculates the physical loss by substituting the approximate solution u_NN, predicted by the neural network, into the differential equation. The loss function of the PINN was configured to consider all differential equations, initial conditions, and boundary conditions that the approximate solution must physically satisfy, similar to a numerical model. The neural network parameters θ were adjusted to minimize the loss based on the computed physical loss. This process continued until the loss reached its minimum, training the neural network to approximate the solution.

3. Applications of PINNs in Additive Manufacturing Processes

3.1 Temperature Field Prediction in Additive Manufacturing Processes Using PINNs

In additive manufacturing processes, the temperature of the deposit significantly influences various aspects of the final product, including thermal distortion, residual stress, and microstructure formation⁸⁾. Therefore, accurately predicting the temperature field during these processes is essential. As summarized in Table 1, researchers have developed models to predict the temperature fields during additive manufacturing by incorporating governing equations, such as the conservation of energy and heat conduction equations, into PINNs. This section analyzes the input variables used in the PINNs, the structure of the neural networks, and the governing equations employed to predict the temperature field of the deposit.

Table 1

Summary of studies on temperature field prediction using PINN in additive manufacturing processes

AM process	PINN input variables	PINN output results	Network	Governing equation	Ref. No
DED	Spatial-temporal coordinates (x, y, z, t), Process parameters, Material properties	3D Temperature field	DNN	Heat conduction	⁹⁾
LPBF	3D temperature field, 3D heat generation field	3D Temperature field at the next time step	CNN	Heat conduction, convection	¹⁰⁾
LPBF	Spatial-temporal coordinates (x, y, z, t), Gaussian beam profile parameters	3D Temperature field	DNN	Heat conduction	¹¹⁾
DED	Spatial-temporal coordinates (x, y, z, t), Partially observed temperature data from infrared (IR) camera	3D Temperature field, Identification of unknown material and process parameters	DNN	Heat conduction	¹²⁾
DED	Spatial-temporal coordinates (x, y, z, t), Process parameters, Material properties	3D Temperature field	RNN	Heat conduction, Thermal radiation, Convection	¹³⁾
LPBF	Greyscale images from the camera, Pyrometer heatmaps	2D Temperature field	CNN + PINN	Conservation of energy	¹⁴⁾

Xie et al.⁹⁾ developed a PINN to predict the three-dimensional temperature field of a deposit during DED. The PINN used spatiotemporal coordinates (x,y,z,t), process parameters, and material properties as inputs to predict the 3D temperature field. The neural network within the PINN was designed with a DNN structure consisting of five hidden layers, each containing 50 neurons. The loss function incorporated pre-calculated temperature distributions from a numerical model for data loss computation, while the heat conduction equation was used for physical loss computation. During the training process, 6000 training data points generated from the numerical model were used, and 2000 test data points were used to validate the predictive performance of the PINN. As a result, the PINN successfully predicted the 3D temperature distribution during the DED process with an average relative error of 4.83% than the numerical model. Notably, the PINN outperformed a conventional AI model that did not include physical information despite using 20% fewer data points, demonstrating the lower data dependency and higher predictive accuracy of the PINN.

Chen et al.¹⁰⁾ developed a PINN to predict the three- dimensional temperature distribution in an LPBF process. The PINN used the current temperature field and heat input distribution at a specific time (e.g., 100 s) in a 3D data format (20×7×7) to predict the temperature field at the next time step (e.g., 101 s). The neural network within the PINN employed a convolutional neural network (CNN) structure to handle 3D data input. The loss function incorporated pre-calculated temperature fields from numerical simulations for data loss calculation, heat conduction, and convection equations for the physical loss calculation. The experimental results indicated that the PINN achieved a mean absolute percentage error (MAPE) of 1.24%, which was 21.2% point lower than the MAPE of traditional data-based artificial neural network (ANN) models. Furthermore, the PINN maintained a MAPE of 3.57% under new process conditions with changed deposition paths by using the heat input distribution as an input variable without requiring additional training. This result demonstrates the robustness and adaptability of PINN to varying process conditions.

Voigt et al.¹¹⁾ developed a PINN to model the 3D heat transfer of deposits during the LPBF process. The PINN used spatiotemporal coordinates and parameters related to the laser beam heat source as inputs to predict the 3D temperature field of the deposit. The neural network within the PINN was structured as DNN. Data loss was not calculated in the loss function; instead, the heat conduction equation, boundary conditions, and initial conditions were employed for physical loss calculation. This study decreased the computational cost of training by applying Latin hypercube sampling (LHS) to select the collocation points, reducing their number to 20% of the original level. This strategy improved the training speed of the PINN while maintaining prediction accuracy, illustrating the appropriate collocation point selection can enhance the computational efficiency of PINNs.

Liao et al.¹²⁾ developed a PINN model capable of predicting the 3D temperature field of a deposit during the DED process, including process parameters and material properties. This PINN used spatiotemporal coordinates and partially measured temperature images from a thermal imaging camera to predict the 3D temperature field, process parameters, and material properties. The artificial neural network was designed with a DNN structure, incorporating a loss function that used partially measured temperature images for data loss calculation and the heat conduction equation, boundary conditions, and initial conditions for physical loss calculation. The prediction results of the PINN, when compared with those of the numerical model, demonstrated high accuracy with root mean square error (RMSE) and mean absolute error (MAE) of 14.07 K and 11.45 K, respectively. Further analysis suggested the application of additional data loss using the temperature field of the numerical model as auxiliary data to enhance computational speed. Consequently, the prediction accuracy was maintained even though the number of training iterations was reduced by one-third, decreasing computational costs. Additionally, high prediction accuracy was achieved with only one-fifth of the training time when the model trained under specific process conditions was transferred to new conditions via transfer learning. This indicates that using auxiliary data for training with numerical analysis models or employing transfer learning for different process conditions can reduce computational costs.

Li et al.¹³⁾ developed a PINN to predict the 3D temperature field of deposits in a DED process. The inputs for the PINN included spatiotemporal coordinates, process parameters, and material properties. Among various neural network structures, the recurrent neural network (RNN) structure, which showed the highest prediction accuracy, was selected for the PINN. The loss function did not calculate data loss; instead, it used the heat conduction, convection, and radiation equations, along with boundary and initial conditions, to calculate physical loss. Material properties that vary with temperature, such as density, are implemented as functions of temperature and incorporated into the governing equations within the loss function. Comparing the PINN prediction results with those of the numerical model revealed an error of approximately 2%. Moreover, the training was completed in 5,000 iterations-one-third of the iterations required without transfer learning-by employing transfer learning under process conditions with changed laser power. This study demonstrated that the PINN could reflect temperature-dependent material properties in governing equations, similar to numerical models.

Bauer et al.¹⁴⁾ developed a multimode AI system integrating PINNs and autoencoders to monitor process quality during LPBF. The PINN predicted the temperature field at subsequent time steps based on the current surface temperature distribution data, measured using an infrared camera. The neural network within the PINN was structured as a DNN. The current surface temperature distribution data were used for the data loss calculations in the loss function, while the conservation of energy law, initial conditions, and boundary conditions were used for physical loss calculation. An autoencoder generated labels representing the state of the grayscale images collected during the additive manufacturing process, classifying them as of the process as success, warning, or failure. These labels were used to train an independent CNN to diagnose the state of the LPBF process (normal, warning, or failure) based on the temperature field data predicted by the PINN. This study illustrates the potential of monitoring the process state by predicting the in situ temperature field of a deposit using a PINN.

3.2 Melt Pool Behavior Prediction in Additive Manufacturing Processes Using PINNs

Thermofluid analysis to predict the shape and behavior of the melt pool in additive manufacturing processes necessitates using the Navier-Stokes and heat transfer equations, leading to significant computational costs¹⁵⁾. This section reviews studies employing PINNs to predict the shape and behavior of the melt pool in additive manufacturing processes, which are summarized in Table 2. It discusses the advantages of PINNs by examining the characteristics and predictive performance of the PINN models used in each study.

Table 2

Summary of studies on melt pool behavior prediction using PINN in additive manufacturing processes

AM process	PINN Input variables	PINN output results	Network	Governing equation	Ref. No
LPBF	Spatial-temporal coordinates (x, y, z, t) Process parameters, Material properties	Melt pool dimensions, Temperature profiles	DNN	Conservation of energy	¹⁵⁾
LPBF	Spatial-temporal coordinates (x, y, z, t) Process parameters, Material properties	Predicted melt pool fluid dynamics and temperature field	DNN	Conservations of mass, energy, and momentum	¹⁶⁾
LPBF	Spatial-temporal coordinates (x, y, t)	Predicted melt pool fluid dynamics and temperature field	DNN	Navier-Stokes equations, Conservations of mass and energy	¹⁷⁾

Hossieni et al.¹⁵⁾ developed a PINN to predict the dimensions of the melt pool and the temperature field in the LPBF process, considering variations in process conditions and welding materials. This PINN used spatiotemporal coordinates, process parameters, and material properties as inputs to forecast the width and length of the melt pool and the temperature field of the deposit. The neural network was structured as a DNN. The conservation of thermal energy equation was employed in the loss function for physical loss calculation without computing data loss. Collocation points for physical loss calculation were densely distributed within a 300 μm radius of the laser heat source, where the temperature gradient was steepest, to enhance the efficiency of PINN training. This method successfully predicted the temperature profile of the build and shape of the melt pool. Validation against a numerical model under identical process conditions showed an MAE of less than 5%.

Zhu et al.¹⁶⁾ developed a PINN to predict the melt pool behavior during LPBF. This PINN used spatiotemporal coordinates, process parameters, and material properties as inputs to predict the temperature, pressure, and velocity distributions within the melt pool. The neural network was a DNN with five hidden layers comprising 250 nodes. The loss function incorporated data loss calculated using pre-computed numerical model data, while the conservation laws of mass, energy, and momentum, along with boundary and initial conditions, were used for physical loss calculation. The PINN was tested under three different process conditions by varying laser power and deposition speed: A (150 W, 0.4 m/s), B (195 W, 0.8 m/s), and C (195 W, 1.2 m/s). It predicted the 3D shape and cooling rates of the melt pool for each condition. Comparing PINN predictions with a numerical model and performing individual analyses for each process condition showed that the PINN demonstrated superior accuracy in predicting the melt pool shape across all cases. The prediction error of the PINN for cooling rates was within 5% for Cases B and C, significantly lower than the numerical model’s error of approximately 20%. In Case A, both models exhibited similar errors (within 10%), demonstrating that, unlike numerical models that necessitate separate analyses for different process conditions, the PINN can predict the temperature field and melt pool behavior using process parameters as input variables without requiring individual training for each condition.

Sharma et al.¹⁷⁾ developed a PINN to predict the melt pool behavior in the LPBF process, incorporating the effects of protective gas flow on fluid velocity above the melt pool. This PINN used spatiotemporal coordinates as inputs to predict the temperature, velocity, and pressure of the melt pool. The neural network employed a DNN structure with ten hidden layers, each containing 50 nodes. The loss function calculated the data loss based on results from a numerical model, while physical loss was derived from the Navier-Stokes equation, energy conservation equation, mass conservation equation, boundary conditions, and initial conditions. The study assessed the prediction accuracy of the PINN concerning the number of collocation points to identify the minimum required for thermo-fluid analysis. The findings recommended using at least 7 time steps and 850 collocation points per time step to perform accurate thermo-fluid analysis with a PINN.

3.3 Mechanical Property Prediction in Additive Manufacturing Processes Using PINNs

The mechanical properties in AM processes critically affect structural stability and durability of the product. Consequently, research has focused on accurately predicting the mechanical properties of deposits using PINNs. This section reviews studies employing PINNs to predict mechanical properties in additive manufacturing processes, analyzing the structure of the PINN models and loss functions used in each study, which are summarized in Table 3.

Table 3

Summary of studies on mechanical property prediction using PINN in additive manufacturing processes

AM process	PINN Input variables	PINN output results	Network	Governing equation	Ref. No
LPBF	Defect characteristics from CT scans, Fatigue testing conditions, Other defect features	Fatigue life	DNN	Linear Elastic Fracture Mechanics-based governing equations	¹⁸⁾
LPBF	Process parameters, Heat treatment properties	Fatigue strength (S-N curve) prediction	DNN	Murakami equation	¹⁹⁾
LPBF	Process parameters, Heat treatment properties, Surface treatments	Fatigue strength (S-N curve) prediction	DNN	Murakami equation	²⁰⁾
LPBF	Process parameters, Heat treatment Parameters	Fatigue life	DNN	Murakami equation	²¹⁾
LPBF	Strain amplitude, Strain rate, Temperature, Stacking fault energy	Low-cycle Fatigue life	DNN	Graphical feature of S-N curve	²²⁾

Salvati et al.¹⁸⁾ developed a PINN to minimize the data required and enhance the accuracy of predicting the fatigue life of deposits produced by LPBF. This PINN employed defect characteristic analysis from CT scans, fatigue life test conditions, and other defect-related features as inputs to predict fatigue life. The neural network featured a DNN structure. The loss function incorporated governing equations related to defect geometry, such as the stress intensity factor from linear elastic fracture mechanics (LEFM), for physical loss calculation. Performance evaluations revealed a maximum relative error of less than 2%, signifying high predictive accuracy. Additionally, the PINN model showed superior RMSE and R² values compared to traditional AI models, underscoring its reliability.

Ciampaglia et al.¹⁹⁾ developed a PINN to forecast the fatigue strength of deposits produced by the LPBF process. This PINN used process parameters (layer thickness, beam diameter, hatch distance, energy density, and build orientation) along with heat treatment characteristics (heating temperature and duration) to predict the fatigue strength of the deposits. The neural network employed a DNN structure. The loss function calculated physical loss using Murakami’s equation, which predicts the fatigue limit of metals based on small cracks and defects. Experimental results demonstrated that the PINN model, predicting the S-N curve based on heat treatment conditions and process parameters, exhibited high predictive accuracy regarding the mean squared error (MSE) and RMSE. The model displayed average and maximum errors of 4% and 17%, respectively, indicating excellent performance. Additionally, the PINN model achieved a 20% lower MSE than traditional AI models that rely solely on data.

In a subsequent study, Ciampaglia et al.²⁰⁾ employed a previously developed PINN to predict the fatigue strength of deposits made from various materials using the LPBF process. The input data included process parameters (laser power, build speed, hatch distance, layer thickness, and build orientation), heat treatment characteristics (heat treatment temperature and duration, hot isostatic pressing conditions), and surface treatments (sandblasting, shot peening, laser peening, surface mechanical attrition, electric discharge machining, and surface polishing). The findings confirmed the robust performance and versatility of the PINN model, maintaining excellent predictive accuracy under new process conditions and materials.

Centola et al.²¹⁾ developed multiple machine-learning algorithms to predict the fatigue life of Ti6Al4V components fabricated via an LPBF. Models were designed to predict the fatigue life based on process parameters (layer thickness, laser power, scan speed, hatch distance, and build orientation) and post-processing conditions (heat-treatment temperature and duration, hot- isostatic pressing conditions, and surface treatments). Three machine-learning algorithms were used: a feed- forward neural network (FFNN), a PINN based on the Murakami equation, and a bilinear feed-forward neural network (BLFFNN) to estimate the coefficients of the S-N curve. The PINN model integrated the physical constraints of the fatigue life problem into the loss function, including the first derivative of the S-N curve representing the fatigue life trends. Experimental results indicated that the PINN model achieved a 20% lower MSE than the FFNN model and maintained excellent predictive performance under new process conditions. Additionally, the PINN accurately predicted the shape of the S-N curve, closely matching the actual form.

Jiang et al.²²⁾ employed a PINN to predict the low-cycle fatigue life of 316 stainless steel processed by LPBF. The PINN used input data such as strain amplitude (εa), strain rate (ε̇), temperature (T), and stacking fault energy (SFE) to predict the fatigue life. Governing equations incorporated into the loss function included the the trends of the S-N curves, such as the first derivative of fatigue life for strain amplitude being negative, the second derivative being positive, the first derivative concerning the strain rate being positive, and the first derivative concerning the temperature being negative. The performance evaluation of the PINN model demonstrated high predictive accuracy in terms of MSE and R² indicators, outperforming traditional neural network models. This study illustrates that the model can be effectively applied to mechanical property prediction by directly defining and applying physical constraints that were previously unformulated to the PINN.

4. Applicability of PINNs in Welding Processes

Preceding sections discussed cases where PINNs were developed to predict the temperature field, melt-pool behavior, and mechanical properties in additive manufacturing processes. These studies underscore the potential of PINNs to supplant numerical models in elucidating complex physical phenomena inherent in additive manufacturing processes. Nonetheless, the application of PINNs to welding processes remains sparse. Kim et al.⁶⁾ applied PINNs to study the effect of tool rotation speed on heat transfer in friction stir welding (FSW), successfully predicting the temperature field and heat diffusion by incorporating a heat conduction equation into the loss function. Zhu et al.²³⁾ also used a PINN to forecast melt pool size and temperature distribution during laser welding. These instances suggest the feasibility of employing PINNs to analyze heat transfer in specific welding processes.

The significant physical similarities between welding and additive manufacturing processes suggest potential effectiveness despite limited research on applying PINNs to welding. Both processes use high-temperature heat sources to melt and solidify metals, leading to analogous physical phenomena such as heat transfer and melt pool dynamics. Additionally, both require the prediction and control of melt-pool formation, residual stresses during cooling and solidification, and mechanical property changes while preventing defects²⁴^,²⁵⁾. Hence, we anticipated that PINNs, successfully applied in additive manufacturing, can similarly predict temperature fields, mechanical properties, and melt-pool behavior in welding processes, thereby enhancing the reliability and efficiency of welding.

5. Summary and Outlook

This review examines the application of PINNs to predict temperature distribution, melt pool behavior, and mechanical properties in additive manufacturing and welding processes. Summarizing recent research, the following trends are evident:

1) Temperature field prediction: Accurate temperature field predictions were achieved across various material and process parameters without requiring individual training through transfer learning or incorporating temperature-dependent material properties into PINNs. This approach suggests that PINNs can reduce the computational cost associated with predicting tempe- rature fields under diverse process conditions compared to traditional numerical models.
2) Melt-pool prediction: Predicting melt-pool behavior necessitates thermo-fluid analysis, increasing the complexity of the physical phenomena. Various techniques have been proposed to enhance accuracy and speed by integrating them with PINNs.
3) Mechanical property prediction: Unlike heat conduc- tion or Navier-Stokes equations, theoretical governing equations for mechanical properties are often poorly established. In such scenarios, PINNs effectively predict mechanical properties by applying newly established physical constraints, such as semi-empirical models or graph characteristics derived from experimental data.

Compared to numerical models, PINNs provide substantial advantages by enabling predictions within milliseconds under trained process conditions. This rapid prediction capability indicates that PINNs could be instrumental in advancing real-time quality control technologies, markedly improving the reliability and efficiency of additive manufacturing and welding processes through real-time process monitoring and thermomechanical interaction modeling.