Physics-informed machine learning-based real-time long-horizon temperature fields prediction in metallic additive manufacturing

Abstract

Real-time long-horizon temperature prediction in wire arc additive manufacturing is critical for process control and quality assurance. However, finite element methods are computationally expensive, and the existing data-driven models suffer from error accumulation and poor adaptability. Here we propose a physics-informed geometric recurrent neural network that integrates geometric characteristics and physical constraints, captures spatiotemporal characteristics via convolutional long short-term memory cells, and enforces physical consistency through hard-encoding initial/boundary conditions and physics-informed loss function. The model can predict the temperature field for future 1.25 s based on current 1.25 s data, and has also been evaluated for more long-horizon predictions. Transfer learning was used to enhance the model’s efficiency in practical applications. Results demonstrate that the proposed model achieves 4.5−13.9% maximum prediction error in simulations and experimental data. Including geometric characteristics and physical information reduces maximum error by about 1%, while the integrated model lowers it by 4%. Furthermore, transfer learning reduces the training time by approximately 50% while achieving the same loss level.

Real-time long-horizon temperature prediction in metal additive manufacturing is critical for process control and quality assurance. Mingxuan Tian and colleagues propose a physics-informed machine learning model to predict temperature field for future 1.25 s.

Introduction

With the accelerated advancement of Industry 4.0 and intelligent manufacturing, the Additive Manufacturing (AM) field is entering a critical phase of technological innovation and application expansion. Against the backdrop of intelligent manufacturing, AM integrates cutting-edge technologies such as Digital Twin (DT) and Machine Learning (ML) to promote the automation, intelligence, and digital transformation of production processes^{1, 2–3}. Wire Arc Additive Manufacturing (WAAM), also known as Directed Energy Deposition-Arc (DED-Arc), produces near-net-shape parts by melting wire feedstock using an electric arc⁴. Characterized by high deposition efficiency and material utilization, as well as low equipment costs, this process can process high-strength materials. Therefore, it has been widely applied in fields such as aerospace, maritime, energy, and heavy industry, providing feasible solutions for the production of complex components⁵.

Although WAAM has certain advantages compared with other AM technologies, minimizing the impact of the temperature field during the WAAM process on the properties of the parts remains a challenge⁶. The spatiotemporal distribution of the temperature field during the WAAM process directly affects the quality and performance of the printed parts. For instance, high temperatures and rapid cooling lead to the formation of different microstructures during material solidification, such as coarse columnar grains or fine equiaxed grains, affecting the mechanical properties of the material⁷. The non-uniformity of the temperature field induces uneven thermal expansion, resulting in thermal deformation and the generation of residual stress, potentially inducing defects such as cracks⁸. Real-time long-horizon temperature field prediction, which involves predicting future temperature distributions over periods longer than the molten pool solidification process, is crucial for effective thermal control and quality assurance in WAAM. In this study, long-horizon is defined as 10 s for thin-wall structures because the heat source has traveled far enough in 10 s, as well as the period when stress evolution dramatically changes. Long-horizon prediction enables optimized interlayer cooling, path planning, and residual stress management while facilitating feed-forward process control and DT development to reduce defects. However, accurate long-horizon prediction remains challenging due to uncertainties such as temperature-dependent thermal conductivity, surface emissivity, and ambient conditions.

Various temperature prediction methods have been developed and a comparison has been summarized in Table 1. Numerical simulations (e.g., Finite Element Method (FEM)⁹, Finite Volume Method (FVM)¹⁰, and Computational Fluid Dynamics (CFD))¹¹ are the most commonly used approaches due to their superior reliability for unseen deposition scenarios. These methods discretize the continuous heat conduction problem into a finite number of elements, control volumes, or grid points, and then use physical equations to calculate the temperature distribution within each discrete region. FEM excels in accuracy, time-efficiency, error control, and reliability due to its ability to accurately simulate complex geometries and boundary conditions (BCs), and handle nonlinear problems. For example, three-dimensional (3D) thermodynamic model was developed to analyze scanning path effects on thermal gradients in laser Directed Energy Deposition (DED)¹², multi-layer WAAM of IN718 was simulated using FEM combined with a Johnson-Mehl-Avrami-Kolmogorov-based transformation kinetics model¹³, and Inconel 718 WAAM model was established to investigate position-dependent thermal histories¹⁴. CFD is specifically used to simulate the flow and heat transfer processes in the molten pool, enabling detailed simulation of complex fluid dynamics behavior. Compared to FEM, CFD generally offers higher accuracy but sacrifices computational efficiency as a trade-off due to its specialized algorithms, finer temporal and spatial resolution. A 3D CFD model was constructed for temperature field prediction in the Cold Metal Transfer-WAAM (CMT-WAAM) process¹¹. A 3D heat transfer and fluid flow model were developed to calculate the temperature field in the WAAM process with circular and triangular paths¹⁵. However, numerical simulations lack adaptiveness as they need to be resolved when deposition scenario changes (e.g., changing deposition parameters or path, or calibrate with temperature feedback), which is the major limitation for their applications in real-time predictions.

Table 1. Comparison of temperature prediction methods for metal AM

Methods	Numerical methods		Traditional ML		PIML45
	FEM9	CFD11	ANN22	LSTM25
Accuracy	++	+++	+	++	++
Time-efficiency	++	+	+++	+++	+++
Error accumulation	+++	+++	+	+	++
Adaptiveness	+	+	++	++	+++

+++: best; ++: good; +: moderate.

In recent years, ML approaches are used as a surrogate model for numerical approaches to simulate thermal behaviors between similar deposition scenarios^{16, 17, 18, 19, 20–21}. For instance, Artificial Neural Network (ANN) was employed to estimate temperature curves for points on the layer to be printed²², Multilayer Perceptron (MLP) has been used for online thermal field prediction²⁰, Extremely Randomized Trees models learn the relationship between designed characteristic from simulation data and the thermal field²³, Feedforward Neural Network (FFNN) have been used for replicate temperature field generated by FEM²⁴. To extract temporal characteristics, Long Short-Term Memory (LSTM) was leveraged to learn the spatiotemporal characteristics of the scanning path²⁵, Graph Neural Network (GNN) was employed to predict thermal histories through spatiotemporal dependency capture^{26, 27–28}, the combination of Finite Element Method (FEM) with Bidirectional Gated Recurrent Unit (Bi-GRU) was proposed to enhance prediction accuracy²⁹, Bayesian LSTM was developed for spatial temperature prediction in DED processes³⁰. Nevertheless, error accumulation is still a main hinderance for using traditional ML approaches in long-horizon prediction due to the lack of physical constraints in thermal histories.

To address these limitations, Physics-informed Machine Learning (PIML) has become a promising approach, which integrates physical laws, domain knowledge, or mathematical models with ML to enhance model accuracy and adaptiveness^{31, 32, 33, 34, 35–36}. By embedding physical constraints (e.g., conservation laws and Partial Differential Equation (PDE) of heat conduction) directly into the model architecture or loss function to regulate training parameters and provide constraints for model outputs. For instance, Physics-informed Neural Network (PINN) was applied to predict temperature and molten pool fluid dynamics in Laser Powder Bed Fusion (L-PBF) processes³⁷, a novel PINN framework was developed for 3D temperature field prediction in both single or multi-layer DED processes³⁸, a hybrid physics-data driven thermal model based on PINN was established for metal additive manufacturing, enabling full-field temperature history prediction and process parameter identification from partial temperature observations³⁹, a label-free PINN framework was proposed for DED temperature prediction using physical losses⁴⁰. Thus, PIML is capable of exhibiting lower error accumulation and greater adaptiveness while maintaining benefits of the high computing efficiency of traditional ML approaches. This paper aims to improve the current real-time temperature field prediction methods in metallic AM from the following aspects:

1. Process-informed: The WAAM process encompasses rich process information that the model needs to effectively identify and learn.

2. Long-horizon prediction: The model must have long-horizon prediction capabilities with minimal error accumulation over time.

3. Adaptiveness: The model should generalize well to new geometric characteristics and process parameters.

4. Interpretability: PDE and Initial/Boundary Conditions (I/BCs) must be enforced as physical constraints, via soft or hard constraint methods.

This study aims to fill the gap of real-time long-horizon temperature fields prediction in metallic AM through a PIML method. The model can not only effectively learn geometric characteristics and physical information, but also accurately recognize the spatiotemporal correlation information of the data, while ensuring that the long-horizon prediction results strictly follow the physical laws.

Methods

System overview

Figure 1a illustrates the overall workflow of the real-time long-horizon temperature field prediction system for WAAM based on PIML. It primarily consists of five stages: data generation, data collection, data processing, basic training, and Transfer Learning (TL). The system first provides the material parameters, geometric characteristics, and process parameters required for data generation, followed by generating process data from FEM and practical experiments. Simulation data is exported via command streams, while experimental data is collected using a thermal camera. The data processing stage refines the collected data from both modes into temperature field, geometric characteristics, and physical information using post-processing techniques. Basic training stage involves training the model with simulation data and saving the model parameters. TL stage uses the saved model parameters to train with experimental data, thereby enhancing learning efficiency and performance.

[See PDF for image]

Fig. 1

Overview of Physics-informed machine learning-based real-time long-horizon temperature fields prediction systems in metallic Additive Manufacturing (AM).

a Flowchart of Wire Arc Additive Manufacturing (WAAM) real-time temperature fields prediction system. b Top view and main view of the model with dimensions (mm), and temperature field at a certain time. c Workflow for FEM simulation data collection and processing. d Proposed Physics-informed Geometric Recurrent Neural Network (PIGeoRNN) model architecture.

WAAM FEM modeling

To establish a FEM simulation dataset for the temperature field, a detailed command stream for the temperature field prediction of thin-wall structures was executed. The selection of thin-wall structures for this investigation was primarily driven by their extensive demand for lightweight applications in aerospace engineering, coupled with the fundamental role that temperature field prediction modeling plays in establishing theoretical foundations for AM process optimization and quality control of intricate thin-wall components. Thin-wall structures utilized ER70S-6 low-carbon steel wire, with the base plate made of Q235b low-carbon steel. Heat source followed Goldak’s double-ellipsoidal model⁴¹. The shape parameters of the heat source were calibrated using the wire feed speed (WFS) and travel speed (TS).

Figure 2 presents the established FEM model, with meshing detailed in Fig. 1b. The model consists of a 300 × 300 × 10 mm³ base plate and a 100 mm thin-wall structures, with deposition height and width obtained from experimental data. The model was discretized elements with specified dimensions, and the deposition process was simulated in sequential steps. The base plate was meshed using tetrahedral free meshing to balance meshing efficiency and computational accuracy. For the thin-wall structures, a uniform hexahedral mesh with dimensions of 2 × 2 × 2.5 mm was selected. This mesh size has been experimentally verified to balance computational accuracy and cost. During modeling, the moving characteristics of the dual-ellipsoid heat source were also taken into account to more accurately reflect the actual physical process⁴¹. Simulation results are clearly depicting the thermal gradients and molten pool within the temperature field. Each layer is deposited in the opposite direction to the previous one, with odd layers being deposited from right to left and even layers from left to right. This deposition sequence is adopted because unidirectional deposition can lead to imbalances in heat accumulation (i.e., the temperature at the arc ignition point is higher than on the opposite side) and compromised dimensional accuracy of the part. To ensure that the interlayer temperature remains around 250 °C, a 60 s interlayer cooling time is set after each layer is printed. The FEM simulation takes ~1 h to complete.

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Fig. 2

Hard-encoding of Boundary Conditions (BCs).

Data collection and processing

After simulating the temperature field of the thin-wall structures, the data were collected and post-processed for training and testing the proposed model. The structure was discretized into $n_x\times n_y\times n_z$ elements, with $\left(n_x+1\right)\times \left(n_y+1\right)\times \left(n_z+1\right)$ nodes. Data were collected at each node with a time resolution of 0.25 s. To standardize dimensions, a 2D grid matching the structure’s geometry was designed, with grid nodes aligned to FEM nodes. Nodes without FEM counterparts were treated as non-deposited. The size of the 2D grid was $Ly\times Lz\left(Ly=Lz=n_{\max }+1\right)$, where n_max was the maximum number of elements along the n_y or n_z axes. To fully collect data from the WAAM process, the 2D matrix must capture maximal process information. This study uses Y-Z planes for data extraction. By dividing the X-axis into n_x elements, data from $n_x+1$ nodes can be collected. Therefore, for each time, $3\times \left(n_x+1\right)$ 2D matrices can be obtained for each type of matrix. The FEM simulation data processing workflow is shown in Fig. 1c, where temperature field matrices ($T_t$), geometric characteristics matrices ($G_t$), and PI matrices ($S_t$) can be collected. During the temperature field simulation, the activation of elements and temperature changes represent the state and temperature variations of the 2D grid nodes. Thus, the deposition phase and its corresponding temperature field can be represented by the states of all 2D grid nodes. Additional data processing is required before the collected data can be input into the model. The final form of the input dataset for this study is as follows:

1

$X_t=\left[T_t^{*},G_t^{*},S_t^{*}\right]$

This study designs a geometric characteristics matrix $G_t$ to describe the geometric characteristics of the thin-wall structures at time $t$. Each node in the $G_t$ represents the state of material deposition at that location, where “1” indicates the deposited state of the manufactured parts, and “0” indicates the undeposited state. The geometric characteristics matrix sequence can be used to represent the geometric evolution and path information of the workpiece during the AM process. To enforce force the model to learn the geometric characteristics and deposition rates of future time series, the geometric characteristics matrix sequence of the current input sequence at the next moment is normalized by mean processing.

2

$G_t^{*}=\frac{G_1+G_2+\cdots +G_n+G_{n+1}}{n+1}$Where $G_1$ represents denotes the mean-processed matrix of future geometric characteristics at the initial time step, $G_{n+1}$ represents the mean-processed matrix of future geometric characteristics at the last input time step, and $G_t^{*}$ is employed to signify the mean-processed matrix of future geometric characteristics for the input sequence, thereby compelling the model to learn information regarding future geometric characteristics and deposition rates. As a result, the original sequence of geometric matrices from $G_1$ to $G_{n+1}$ are transformed into $n+1$ matrices of $G_t^{*}$.

This study designs a 2D matrix $T_t$ to represent the temperature field of the thin-wall structures at time $t$. The temperature field matrix $T_t$ records nodal temperatures at each timestep, with non-deposited regions assigned ambient values. Additionally, a PI matrix $S_t$ is designed to encode the physical information of the thin-wall structures at time $t$. Arc power, heat source shape parameters, and heat source position are considered important physical parameters affecting WAAM temperature field modeling. Instead of direct input, these parameters are embedded as auxiliary characteristics by constructing $S_t$, which integrates heat input physics into the model’s input layer. This intermediate characteristic integrates the effects of process parameters to enhance the model’s performance and interpretability. Prior to model input, both $T_t$ and $S_t$ are normalized to [0, 1].

3

$T_t^{*}=\frac{T_t-T_{\min }}{T_{\max }-T_{\min }}$

4

$S_t^{*}=\frac{S_t-S_{\min }}{S_{\max }-S_{\min }}$Where $T_{\max }$ is set at 1850 °C, which is based on the highest temperature achievable during experiments conducted using the Cold Metal Transfer (CMT) process with the materials employed in this study; $T_{\min }$ is set at the ambient temperature; $T_t^{*}$ is the normalized temperature field matrix. This normalization strategy, using carefully determined $T_{\max }$ and $T_{\min }$, ensures consistent denormalization across datasets, facilitating accurate reverse normalization of prediction results. $S_{\max }$ and $S_{\min }$ represent the maximum and minimum heat fluxes across all time in the PI matrix, respectively, and $S_t^{*}$ is the normalized PI matrix.

Physics-informed machine learning model

Model structure

As a variant of RNN, Convolutional Long Short-Term Memory (ConvLSTM) cells employ a hybrid architecture that combines the spatial characteristics extraction capabilities of Convolutional Neural Network (CNN) with the memory and sequential modeling capabilities of LSTM, enabling the processing of both temporal and spatial data⁴². The task in this study is to take the temperature field, geometric characteristics, and PI as inputs, with the goal of predicting the long-horizon temperature field in the future. The structure of the constructed PIGeoRNN model is shown in Fig. 1d. It uses two ConvLSTM cells to extract spatiotemporal characteristics. The encoder-decoder architecture in our model, featuring CNN, extracts key characteristics from input data and reconstructs the output to original dimensions. The CNN also reduces noise, enhancing the model’s accuracy with noisy inputs. $X_0$ to $X_n$ use real input data, while subsequent times use the model’s predictions as the input for the next time. The ConvLSTM cells enable simultaneous standard temporal memory and spatiotemporal memory flow.

PINN often enforce BCs through soft constraints where additional loss terms are defined at collocation points on the boundary. However, this method has two key limitations: (1) it cannot guarantee strict satisfaction of BCs, and (2) the choice of loss weights lacks theoretical guidance, impacting learning efficiency. To address these issues, this study adopts a hard-encoding strategy, where I/BCs are directly embedded into the model through a matrix-based discretization of the temperature field. By imposing I/BCs as hard constraints, the method ensures exact compliance and has been proven to improve the accuracy of the solution at the boundary⁴³. Furthermore, this approach promotes a well-posed optimization problem during training, which enhances both convergence and overall precision. Figure 2 illustrates the approach to hard-encoding of BCs. To facilitate the calculation of gradients using the Finite Difference Method (FDM), Ghost nodes are introduced for boundary filling. The upper boundary applies Dirichlet BCs, where the known temperatures on the boundary are used to fill the Ghost nodes.

5

$T_{\mathrm{ghost}}\left(x_i,y_j\right)=T_{\mathrm{boundary}}\left(x_i,y_j-d\right)$Where $T_{\mathrm{ghost}}$ and $T_{\mathrm{boundary}}$ represent the ghost and boundary nodes temperature respectively, $d$ is the size of the grid space distance. The lower boundary employs Robin BCs, and the values of the Ghost nodes can be derived from the temperature values of the lower boundary nodes using the FDM.

6

$T_{\mathrm{ghost}}\left(x_i,y_j\right)=T_{\mathrm{boundary}}\left(x_i,y_j+d\right)-\frac{d{h}_c}{k}\left(T_{boundary}\left(x_i,y_j+d\right)-T_0\right)$Where $k$ is the thermal conductivity, $h_c$ is the thermal conductivity of substrate material, $T_0$ is ambient temperature. Considering the effects of convective and radiative heat transfer, the left and right boundaries employ convective-radiative BCs, and the values of the Ghost nodes are derived from the temperature values of the left and right boundary nodes.

7

$T_{ghost}\left(x_i,y_j\right)=T_{boundary}\left(x_i,y_j\pm d\right)\pm \frac{d}{k}\left(\right)\sigma \epsilon \left(T_{boundary}{\left(x_i,y_j\pm d\right)}^4-T_0^4\right)+h_a\left(T_{boundary}\left(x_i,y_j\pm d\right)-T_0\right)\left)\right)$Where $\sigma$ is thermal emissivity, $\epsilon$ is Stefan-Boltzmann constant, and $h_a$ is heat convection coefficient of substrate and air interaction. Although the relationship between the Ghost nodes and the internal nodes is fixed, their specific values change as the computation progresses because they depend on the values of the internal nodes. Purpose of boundary filling operations is to ensure the minimization of the residual of the PDE across the entire solution domain without compromising any boundary nodes.

The hard-encoding strategy ensures strict adherence to the BCs, which is crucial for accurate modeling⁴³. This method guarantees that the model’s predictions align precisely with the physical constraints, leading to more reliable and accurate results. Meanwhile, the PI loss function is instrumental in integrating the underlying physical laws into the learning process, guiding the model towards solutions that not only satisfy the geometric constraints but also adhere to the physical principles governing the process³¹. This integration is vital for enhancing the model’s ability to generalize and perform accurately across different scenarios within the WAAM process. Both components are essential: the hard-encoded BCs provide a robust foundation for accurate boundary handling, while the PI loss function ensures that the model’s performance is physically meaningful and consistent with the laws of physics.

In this study, physical information is not directly input into the model as raw features, but is embedded through an intermediate feature related to heat input. This intermediate feature integrates the effects of process parameters and is combined with other feature matrices. The higher-order parameter selected here is the arc heat source term, which integrates arc heat source information into the historical temperature field to improve the model preformance Additionally, to enable the model to learn geometric characteristics of the structure, a characteristic matrix of the same size as the temperature field matrix is constructed, with its values defined by the deposition state of the material.

Loss function design

The loss function of a PIML model consists of two core elements: data-based loss and physics-based loss. The total loss of the model is defined as:

8

$L_{total}=w_p{L}_{pde}+w_d{L}_{data}$Where $w_p$ and $w_d$ are the weights of $L_{pde}$ and $L_{data}$, and the default values are set to 0.3 and 0.7. These values are determined based on the relative magnitudes of the respective loss terms to balance their contributions during training.

Due to the hard-encoding of I/BCs, the physical loss only needs to control the PDE loss term. To compute the temperature gradients in time and space, the computational domain is discretized into m×n grid with a spatial distance of $d$. Following the FDM, differential approximations are employed to discretize the derivative terms of the heat conduction governing equations. The PDE loss can be expressed as a difference equation, whose discretized form is:

9

$R_t\left(x_i,y_j\right)=\rho C_p\frac{\partial T}{\partial t}-k{\nabla }^{2}T-\stackrel{°}{Q}$

10

$\frac{\partial T}{\partial t}=\frac{\left({\widehat{T}}_{t+1}\left(x_i,y_j\right)-{\widehat{T}}_{t-1}\left(x_i,y_j\right)\right)}{2\mathrm{\Delta }t}$

11

${\nabla }^{2}T=\frac{1}{d^2}\left({\widehat{T}}_t\left(x_i+d,y_j\right)+{\widehat{T}}_t\left(x_i-d,y_j\right)+{\widehat{T}}_t\left(x_i,y_j+d\right)+{\widehat{T}}_t\left(x_i,y_j-d\right)-4{\widehat{T}}_t\left(x_i,y_j\right)\right)$Where $\widehat{T}\left(x_i,y_j\right)$ represents the predicted temperature field at position $\left(x_i,y_j\right)$ at time $t$, and ${\widehat{T}}_{t+1}\left(x_i,y_j\right)$ and ${\widehat{T}}_{t-1}\left(x_i,y_j\right)$ represent the temperature at the same node at t + 1 and t - 1, respectively, $\rho$ is the density of the part material, $C_p$ is the specific heat, $\stackrel{°}{Q}$ is the arc heat source term. Since the model’s output is a 2D matrix, the PDE residuals must be computed at each grid point. Ideally, the model’s predicted temperature should be such that $R_t\left(x_i,y_j\right)$ is close to zero. Physical consistency is enforced by minimizing the PDE loss $L_{pde}$, defined as the mean squared PDE residuals over the spatio-temporal domain:

12

$L_{pde}=\frac{1}{nmT}\sum _{i=1}^n\sum _{j=1}^m\sum _{t=1}^T{R_t\left(x_i,y_j\right)}^2$Where $n$ and $m$ represent the number grid nodes along the height and width direction, respectively, while T is the total prediction time duration. The data loss term $L_{data}$ is defined as:

13

$L_{data}=\frac{1}{nmT}\sum _{i=1}^n\sum _{j=1}^m\sum _{t=1}^T{\left({\widehat{T}}_t\left(x_i,y_j\right)-T_t\left(x_i,y_j\right)\right)}^2$Where $T_t\left(x_i,y_j\right)$ is the true temperature.

Training and testing methods

Initially, the real-time long-horizon temperature field prediction capabilities of four models were evaluated using FEM simulation data: the Baseline model using ConvLSTM, the GeoRNN model incorporating geometric characteristics, the PIRNN model integrating physical information, and the PIGeoRNN model combined geometric characteristics and physical information. Subsequently, the proposed models were trained and tested on experimental data. Building on models trained with FEM simulation, further TL training was conducted using experimental data⁴⁴. The experimental data provide more realistic and specific scenarios, helping the model adapt to the complexities of actual manufacturing processes and enhancing its predictive accuracy and adaptability. The dataset was split into training and testing sets in a 7:3 ratio. The learning rate was set to 0.01, the batch size to 8, and training was set to halt after 3000 epochs. The test set was evaluated every 500 epochs, recording the mean squared error (MSE), saving the trained model parameters, and the test results. For all models, the patch size was set to 3 × 3 so that each 51 × 51 characteristics matrix is reshaped into a 17 × 17 × 9 tensor. The same hyperparameters were considered for the all models proposed in this study, with the networks trained using the stochastic gradient descent Adam optimizer.

To assess the performance of the proposed model based on the prediction results, the use of MSE as an evaluation metric for the model can effectively assess the prediction performance of the model at each point in time. This provides comprehensive insights into precision, error magnitude, and relative accuracy, thus enhancing the overall assessment of the model. When evaluating the long-horizon temperature field prediction capabilities of different models, the Root Mean Square Error (RMSE) is used to quantify the differences between predicted and true temperatures. The Kullback-Leibler (KL) divergence is used to evaluate the difference between the predicted and true probability distributions. True and predicted temperatures are normalized to probability distributions, resulting in the predicted probability distribution $Q_t$ and the true probability distribution $P_t$.

14

$Q_t\left(x_i,y_j\right)=\frac{{\widehat{T}}_t\left(x_i,y_j\right)}{{\sum }_{i^{\prime }=1}^n{\sum }_{j^{\prime }=1}^m{\widehat{T}}_t\left(x_{i^{\prime }},y_{j^{\prime }}\right)}$

15

$P_t\left(x_i,y_j\right)=\frac{T_t\left(x_i,y_j\right)}{{\sum }_{i^{\prime }=1}^n{\sum }_{j^{\prime }=1}^m{T}_t\left(x_{i^{\prime }},y_{j^{\prime }}\right)}$

Therefore, the spatial discrepancy between the predicted and true temperature field is quantified usings the KL divergence between $Q_t$ and $P_t$. In this study, a smaller KL divergence value suggests better agreement between the true and predicted temperature field distributions.

16

${\mathrm{D}}_{\mathrm{KL}}\left(P_t\mathrm{\mid \mid }{Q}_t\right)=\sum _{t=1}^T{P}_{t}log\frac{P_t}{Q_t}$

System setup and data collection

Experiment system

The experiment system comprised an ABB IRB 2600 robot, table, Fronius CMT welder, CMT wire feeder, robot controller, computer, FLIR A655sc thermal camera, and protective gas tank, as shown in Fig. 3a. The welding torch is held by an ABB IRB 2600 industrial robot and moves along a pre-programmed trajectory. A Fronius CMT Advanced 4000 welder was used as the welding power source. A thermal camera was used for temperature field in-situ monitoring and data collection. A computer was used as the main controller to coordinate the industrial robot, welder and thermal camera to perform the deposition tasks. The welding technique was CMT. The computer hardware specifications used in this study are shown in Table 2. The FEM solver was executed on the CPU, while the ML models were trained on the GPU.

[See PDF for image]

Fig. 3

Illustration of the experimental system and deposition cases.

a Experiment setup for WAAM system. b Tested thin-wall structure.

Table 2. Computer hardware specifications

Component	Specification
Operating system	Microsoft windows 10
Computer processing unit (CPU)	13th Gen Intel® Core™ i7-13620H
CPU speed	2.40 GHz
On-board Random Access Memory (RAM)	2×8 GB DDR5
Graphics Processing Unit (GPU)	NIVIDIA GeForce RTX4050
GPU Dedicated RAM	6.0 GB

Practical experiments

This study employed WAAM to deposit thin-wall structures. Deposition used ER70S-6 welding wire with a diameter of 1.2 mm. The substrate was selected as 300 × 300 × 10 mm³ mild steel plate made of Q235b. The main chemical compositions of the wire and substrate are shown in Table 3.

Table 3. Main chemical composition of ER70S-6 and Q235b

Elements	C	Mn	Si	Cu	Ni	Cr
ER70S-6	0.15	1.85	1.15	0.50	0.025	0.50
Q235b	0.2	1.4	0.35	0.3	0.3	0.3

The experiments were designed to quantitatively assess the prediction accuracy and thermal adaptation capability of the temperature field prediction models. A total of 10 thin-wall structures were designed. Each simulated structure was 100 mm long, while each experiment structure was 200 mm in length (with a mid-section of 100 mm extracted as the dataset) because thin-wall structures tend to collapse on both sides during the experiment, and both consisted of 20 deposited layers. This difference has a negligible impact on the results of this study. These structures were created using different WFS and TS groups. Five of these structures were simulated, and four were deposited, all of which were used for training. Additionally, one thin-wall structure was deposited for testing. The thin-wall structures were deposited using a bidirectional path. Due to thermal accumulation, the average temperature of each layer increased. To eliminate the impact of thermal accumulation on part formation, the dwell time for each layer, including the time for the welding gun to move on each layer, was controlled at 60 s. Detailed process parameters for the actual experiment are shown in Table 4. The arc current in the experiment was 109 A, the arc voltage was 10.9 V, and a mixed gas, mainly composed of 50% argon and 50% carbon dioxide, with a flow rate of 17 L/min was used. The experiment used a FLIR A655sc thermal camera for in-situ monitoring and data collection of the temperature field during the WAAM process.

Table 4. Process parameters of the experiments

	WFS(m/min)	TS(m/min)
Thin-wall training 1	4	0.48
Thin-wall training 2	5	0.48
Thin-wall training 3	6	0.48
Thin-wall training 4	7	0.48
Thin-wall training 5	8	0.48
Thin-wall training 6	4.9	0.5
Thin-wall training 7	4.9	0.45
Thin-wall training 8	5.5	0.45
Thin-wall training 9	6	0.4
Thin-wall test 1	6	0.45

The model was trained using the first 15 layers of data from five FEM simulations, with the last 5 layers reserved for testing to ensure the model could generalize well to unseen data. This approach helps to evaluate the robustness and predictive accuracy of the model in real-world scenarios. The same strategy was applied to five experimental datasets, and the prediction results were evaluated using MSE to assess model performance. The emissivity of the thermal camera was calibrated using thermocouples placed on the base plate, and a consistent emissivity was applied across the entire imaging area. The geometric characteristics and physical information of the experimental data were designed to match those of the simulation data, which were defined by real-time captured geometric characteristics and welding gun positions. The experimental data were acquired using a thermal camera and then cropped to extract the Region of Interest (ROI), which was subsequently processed to match the size of the FEM simulation data. Meanwhile, the encoder-decoder architecture containing the CNN in the model is utilized to reduce noise and extract features, which helps maintain the accuracy of the model even in the presence of real-world imaging noise. Building on the training with FEM simulation data, the model was further trained with actual experimental data using TL, and the performance of the model before and after TL training was compared.

Results and discussion

Temperature fields prediction based on FEM data

Comparison of prediction results

For the given deposition geometry, the true temperature field from FEM simulation data was compared with the model’s predicted results. As shown in Fig. 4a, the true temperature fields are compared with the predictions of the Baseline, GeoRNN, PIRNN, and PIGeoRNN models at two future time points (0.25 s and 1.25 s). The true temperature fields and the predicted results from the four models all exhibit a relatively consistent temperature fields distribution. However, due to the lack of sufficient process-informed input, the Baseline model struggles to accurately predict the molten pool region. In contrast, the other three models can clearly delineate the molten pool region and the deposition boundaries. Results indicate that geometric and physical information characteristics can serve as effective process information for temperature fields prediction in WAAM. Figure 4b provided a visual representation of the errors in the predicted temperature fields for each model at these two time points, which highlighted the maximum temperatures errors for each model. The Baseline, GeoRNN, PIRNN, and PIGeoRNN models predict future temperature fields with maximum errors of 8.8% to 9.7%, 7.9% to 9.5%, 7.6% to 9.3%, and 4.6% to 8.9%, respectively. The Baseline model, relying solely on temperature field data and lacking understanding of geometric characteristics and physical information, shows a clear deficiency in capturing the dynamic changes of the temperature field, especially in the molten pool region with large temperature gradients. This lack of prior knowledge hinders the Baseline model from accurately predicting the complex variations in the temperature field, leading to relatively larger errors in these critical areas. The GeoRNN model enhances the model’s ability to learn real-time geometric changes by introducing geometric characteristics matrix, dynamically adjusting the weights of temperature data in the deposition area, and correcting the geometric bias in the model’s temperature predictions. The PIRNN model combines PIML with RNN, introducing PI inputs and a PI loss function to strictly adhere to physical laws. The GeoRNN and PIRNN models show similar performance in their predictions, with the maximum error reduced by approximately 1% compared to the Baseline model. The PIGeoRNN model combines the advantages of the GeoRNN and PIRNN models, reducing the maximum error by about 4% and providing the most accurate predictions.

[See PDF for image]

Fig. 4

Performance of temperature fields prediction based on FEM data.

a Comparison of the future 0.25 s and 1.25 s temperature fields predicted by Baseline, Geometric Recurrent Neural Network (GeoRNN), Physics-informed Recurrent Neural Network (PIRNN), and PIGeoRNN models with the true temperatures field at two time points. b Comparison of predicted temperature fields errors. c The temperature predictions for the middle node of the 10th layer over the future 1.25 s using the PIGeoRNN model, with a temporal resolution of 0.25 s, resulting in the temperature evolution at five time points. d The temperature errors between these predicted values and the true values. e Training loss evolution for Baseline, GeoRNN, PIRNN and PIGeoRNN models. f Time-series MSE of predictions using FEM data.

In terms of time efficiency, the simulation time of the FEM model is about 1 hour, while the training times of the Baseline, GeoRNN, PIRNN and PIGeoRNN models are 5 min, 8 min, 9 min and 18 min, respectively. This indicates that an increase of in model complexity requires more parameters to be optimized, which in turn necessitates more training time. The prediction time of the proposed model is about 12 ms, which can meet the real-time prediction demand.

To evaluate the accuracy of the PIGeoRNN model’s prediction for individual nodes at each time, the temperature history of the middle node on the tenth layer was extracted. Figure 4c illustrates the temperature history and the predicted temperature evolution at the middle node of the tenth layer. The predictions are made for a future duration of 1.25 s with a time resolution of 0.25 s. This means that the model predicts the temperature at five distinct time points: 0.25 s, 0.5 s, 0.75 s, 1 s, and 1.25 s from the current time. The model’s predicted values are highly consistent with the true values in terms of overall trend and can effectively capture the changes in temperature increases and decreases. In regions where the temperature fluctuates gently, the predicted values are extremely close to the true values. However, during periods of rapid temperature change, there is a noticeable deviation between the two. Nevertheless, the model is still able to generally follow the direction of the true value changes. Figure 4d shows the errors between the predicted and true temperatures at these five time points. The errors across all time points are controlled within the range of −100 °C to 100 °C, with the majority of the time falling within −25 °C to 25 °C. This indicates that the model’s prediction errors are within an acceptable range and that the overall reliability of the model is high. However, the error at time points 5 shows fluctuation and deviation. This may be due to the increased prediction time, which requires the model to process more historical information and complex spatiotemporal dependencies, leading to the accumulation and amplification of errors, thereby affecting the prediction accuracy.

Ablation experiment

The training loss history of the four models on FEM simulation data and the MSE of all prediction results at each time are compared in Fig. 9. Overall, the loss decreases as epochs increase, indicating that the models can learn the dataset well. Figure 4e shows that all training losses fluctuate greatly in the first 1000 epochs because the models have not yet learned the complex patterns in the data, especially for high-dimensional data features. After 1000 epochs, the loss fluctuates around 5e-5. The PIGeoRNN model shows a more gradual decrease in training loss, which is related to the complexity of the model. The PDE loss history during the training of the PIGeoRNN model stabilizes within a narrow range after about 500 epochs, indicating that the model can learn physical information within a short number of epochs. Figure 4f shows that the MSE of all prediction results at each time exhibits a trend of increasing error with time, indicating that errors are accumulated over time. Long-horizon prediction may lead to further accumulation of errors, but experiments have shown that predicting the temperature field for the next 1.25 s results in lower errors. The Baseline model performs poorly in temperature field prediction at each time point, mainly because the model only learns the spatiotemporal characteristics of the temperature field data. The GeoRNN model performs better than the Baseline in temperature field prediction at each time point, as the model can learn real-time geometric characteristics to enhance robustness against environmental noise in the data. Due to physical constraints, PIRNN models are more accurate than GeoRNN models in long-horizon predictions. The PIGeoRNN model is proven to have the best performance in prediction, with a more gradual increase in error at each time point. This is related to the model’s integration of geometric characteristics inputs, PI inputs, and PI loss functions, which allows the model to learn more process knowledge and be subject to physical constraints.

Long-horizon predictive evaluation

To further investigate the performance of the models in long-horizon prediction, this study evaluated the predictions of four models for future time of 1.25 s, 3.75 s, 6.25 s, 8.75 s, and 11.25 s (with a time resolution of 0.25 s). Figure 5a presents the long-horizon prediction accuracy assessment for the four models, using the RMSE with error bars as the evaluation metric. The error bars indicate the range of sub-time RMSE values when predicting different future time points. Results show that the baseline model has the highest RMSE values at all time points, indicating the lowest prediction accuracy. The GeoRNN and PIRNN models have similar performance in predictions up to 6.25 s, but beyond 6.25 s, the RMSE of the PIRNN model is slightly lower than that of the GeoRNN model, demonstrating better prediction accuracy. For the PIGeoRNN model, the RMSE increases with time and the error bars have a larger range in predictions up to 6.25 s, indicating an accumulation of errors. However, in predictions beyond 6.25 s, the RMSE of the PIGeoRNN model stabilizes and is slightly lower than the value at 6.25 s, with reduction in the range of the error bars, indicating that the error accumulation stabilizes over time. Overall, the PIGeoRNN model achieves the best prediction accuracy in long-horizon prediction. Additionally, the KL divergence was used to evaluate the temperature field distributions of the four models in long-horizon predictions, as shown in Fig. 5b. It can be observed that the Baseline and PIRNN models showed higher KL divergence in long-horizon predictions, attributed to the absence of long-horizon geometric characteristics inputs. Consequently, the GeoRNN model, which incorporates geometric characteristics, showed a relatively smaller KL divergence. After integrating geometric characteristics and physical information, the PIGeoRNN model effectively captured both geometric characteristics and molten pool dynamics in long-horizon predictions, achieving the lowest KL divergence.

[See PDF for image]

Fig. 5

Long-horizon predictive evaluation based on FEM data via RMSE and Kullback-Leibler (KL) divergence.

a RMSE with error bars and (b) KL divergence for evaluating the long-horizon predictions of Baseline, GeoRNN, PIRNN, and PIGeoRNN models on FEM simulation data. Error bars represent the RMSE calculated at each time steps with a time resolution of 0.25 s.

Temperature fields prediction based on experimental data

Comparison of prediction results

To evaluate the adaptability of the real-time long-horizon temperature field prediction model, a thin-wall structure was deposited using different process parameters, as shown in Fig. 3b. A 100 mm section from the middle part was extracted and used as the test dataset. During the process of depositing thin-wall structures, temperature gradients occur due to rapid heat dissipation. Spatter is minimized through optimized parameters. The initial layers maintain stable heights, but subsequent layers show decreasing height as the structure grows, likely due to cumulative thermal effects. Figure 6a, b presents a visual comparison of the prediction results and errors for the experimental thin-wall structures data across the four models. The true temperature fields reveal differences in temperature gradients in the molten pool area between experimental and simulation data, which is related to the emissivity of the thermal camera. The molten pool appears liquid during deposition, whereas the setting of 0.8 corresponds to the solid-state emissivity of the material. Consequently, the issue with the thermal camera’s emissivity setting leads to larger errors in the molten pool area of the temperature field and may also result in the physical information not being well learned. The Baseline, GeoRNN, PIRNN, and PIGeoRNN models predicted the temperature field for the next 1.25 s with maximum errors of 14.2% to 16.5%, 13.3% to 15.2%, 13% to 15.2%, and 13% to 14.2%, respectively, with the PIGeoRNN still demonstrating superior performance. During the WAAM deposition process, defects such as spatter and lack of fusion can affect the distribution of the temperature field, leading to uneven temperature distribution, which in turn impacts the model’s learning of geometric characteristics and physical information. Due to differences between the layer height of each deposition in the thin-wall structures and the ideal case, the geometric characteristics learned by the model may deviate from reality.

[See PDF for image]

Fig. 6

Performance of temperature fields prediction based on experimental data.

a Comparison of true and predicted temperature fields for Baseline, GeoRNN, PIRNN, and PIGeoRNN models at future times 0.25 s and 1.25 s. b Comparison of predicted temperature fields errors. c Training loss evolution for Baseline, GeoRNN, PIRNN and PIGeoRNN models; (d) Time-series MSE of predictions using experimental data.

Ablation experiment

The training loss history comparison of the four models on the experimental data and the MSE of all prediction results at each time. Overall, the loss decreases with the increase in epochs, indicating that the models can learn the experimental data well. It can be observed from Fig. 6c that all losses also fluctuate over the first 1000 epochs. The PIGeoRNN model shows a more gradual decrease in training loss and still requires more time for training. Figure 6d shows that the MSE of all prediction results at each time exhibits a trend of increasing error with time, indicating that errors are accumulated over time. The PIGeoRNN model does not show improvement in prediction accuracy compared to the GeoRNN and PIRNN models.

Long-horizon predictive evaluation

The long-horizon prediction performance of the four models was further evaluated using experimental data. As shown in Fig. 7a, the Baseline model exhibited the highest RMSE in long-horizon predictions at different times, indicating the poorest prediction accuracy. The GeoRNN and PIRNN models showed similar performance, both outperforming the Baseline model. The PIGeoRNN model achieved the lowest RMSE, with its error accumulation stabilizing over time, demonstrating the highest prediction accuracy and stability. Furthermore, Fig. 7b shows the evaluation of the prediction results using KL divergence. It can be observed that, after incorporating geometric characteristics, the GeoRNN and PIGeoRNN models have relatively close KL divergence in long-horizon predictions, indicating better spatial distribution.

[See PDF for image]

Fig. 7

Long-horizon predictive evaluation based on experimental data via RMSE and KL divergence.

a RMSE with error bars and (b) KL divergence for evaluating the long-horizon predictions of Baseline, GeoRNN, PIRNN, and PIGeoRNN models on experimental data. Error bars represent the RMSE calculated at each time steps with a time resolution of 0.25 s.

Transfer learning training

By comparing the performance of the four models on FEM simulation data and experimental data, it can be demonstrated that the proposed PIGeoRNN model exhibits the best performance. To leverage existing knowledge to accelerate learning from experimental data, help model in learn the fundamental characteristics and physical laws of the temperature field, thereby enhancing learning efficiency and performance. Building on the model trained with FEM simulation data, further TL training is conducted on experimental data using actual data. Figure 8a, b compares the true temperature fields and the predicted temperature fields with and without TL, along with the visualization of errors. Results show that the model’s predicted temperature distribution aligns well with the true field. The temperature field distributions in the molten pool area and near the deposition boundaries are clearly displayed. The No TL model’s predictions show maximum errors of 13% to 14.2%, while the TL model reduces errors to 11.2% to 14%. The loss history of models with and without TL training is shown in Fig. 8c, d, proving that initializing with model parameters trained on simulation data leads to much faster loss reduction. The total loss for models with TL training can be reduced to approximately 1e-5 in just 500 epochs, whereas models No TL require 1000 epochs to achieve the same level of MSE, reducing training time by 50%.

[See PDF for image]

Fig. 8

Performance of temperature field prediction results for Training Learning (TL) and No TL.

a Comparison of true and predicted temperature fields for TL and No TL models at future times 0.25 s and 1.25 s. b Comparison of predicted temperature fields errors. c Training loss evolution for TL and No TL models. d Time-series MSE of predictions using experimental data.

Comparison of FEM and PIML-based model

Figure 9a shows that the PIGeoRNN model achieves the highest accuracy by comparing its long-horizon predictions with those of FEM and experimental results. Since thermal conductivity varies with temperature, PIGeoRNN leverages temperature history for real-time calibration and incorporates physical constraints to reduce training costs and enhance prediction stability. Although FEM dynamically adjusts thermal conductivity through temperature-dependent coefficients, it struggles to capture nonlinear characteristics and may fail to converge. For instance, thermophysical parameters such as density and specific heat capacity, and these parameters are affected by the actual quality of the wire, the slight errors are further amplified by the numerical calculation and affect the simulation results.

[See PDF for image]

Fig. 9

Performance of temperature field prediction results for FEM and PIML-based model.

a Comparison of experimental and predicted temperature fields for FEM, and PIGeoRNN model at future times 0.25 s and 1.25 s. b Comparison of experimental results with thermal histories predicted by FEM and PIGeoRNN model.

In addition, there is a difference between the temperature gradient in the molten pool area of the FEM simulation results and the experimental measurement, which is mainly attributed to the thermal conductivity, the surface emissivity of the formed part during deposition, and the convective heat transfer loss. In FEM, the thermal conductivity differs from the actual value because neither the liquid-metal droplet region nor the insulating effect of the oxide layer is taken into account, resulting in faster heat diffusion along the weld path. Deviation from the actual setting of convective heat transfer coefficients or surface emissivity (due to ignored temperature dependence or surface oxide layer variation) can lead to underestimation or overestimation of heat loss. Because the central region of the molten pool is liquid metal, the surface emissivity is quite different from the solid metal emissivity set by the thermal camera, which leads to the deviation of the temperature measurement value.

However, results show that PIGeoRNN model can compensate these shortcomings: although the PIGeoRNN model exhibits same errors in the liquid metal region in the center of the molten pool, the temperature fluctuations have limited influence on the global temperature distribution and overall evaluation accuracy.

Figure 9b compares the thermal history predicted by FEM and PIGeoRNN with experimental data from an intermediate node of the first layer. The PIGeoRNN prediction exhibits good agreement with the experimental data, except for a slight discrepancy in the peak temperature. In contrast, the FEM result deviates from the measurement, primarily because uncertainties in the convective heat transfer coefficients distort the heat conduction rate, the surface emissivity error alters radiative heat loss, fluctuations in the ambient temperature modify the BCs, and instabilities in the applied voltage and current introduce transient deviations in the thermal input from the prescribed values.

Discussion

This study aims to propose PIML-based real-time long-horizon temperature field prediction model for the WAAM process using interpretable, adaptive, and robust data-driven approaches. Compared with other models that integrate information at different levels, this model exhibits greater interpretability and robustness. The variation in performance across different models highlights the importance of integrating geometric characteristics and physical information, which impacts the model’s performance. The temperature fields prediction results based on FEM and experimental data indicate that the information described by inputting temperature field data alone is limited; the models primarily rely on learning patterns from historical data. However, in the WAAM process, real-time geometric characteristics affect the path and efficiency of heat conduction. Thicker deposited layers require more time to cool because heat must pass through thicker material layers before it can dissipate into the environment. Neglecting real-time geometric changes in the WAAM process results in the Baseline model poor prediction accuracy. Therefore, the GeoRNN model is constructed to utilize a real-time geometric characteristics matrix as an additional input to extract geometric characteristics information from the AM process. It dynamically adjusts weights for deposition-area temperature data, enabling bias correction during prediction. At the cost of a slightly reduced in temporal efficiency, it improves the accuracy and adaptability of predictions over the Baseline model.

During the WAAM process, changes in the temperature field are influenced by various physical mechanisms, such as heat conduction, convection, and radiation. If the model fails to adequately understand these physical mechanisms, it may lead to discrepancies between the predicted results and the actual physical processes. The PIGeoRNN model proposed in this study integrates PI inputs based on the combination of GeoRNN and PIML, enabling the model to learn molten pool (heat source) characteristics and adjust input weights, thus accelerating gradient descent. This model is constrained by I/BCs and a PI loss function, compelling the model to minimize the physical discrepancies between predictions and true values. This endows the model with interpretability and robustness, and it has been proven to possess the best performance. Notably, when predicting with FEM simulated data, there’s no error increments when predicting horizon exceed 6.25 s. That can be explained as the predicting horizon have covered the most welding zone where heat flux is applied. Hence, further increasing predicting horizon will have limited impact to predicting accuracy, which mitigates the error accumulation of traditional ML approaches. While in practical, the noise of thermal camera will introduce extra noise and lead to minor error accumulation. However, that’s currently unavoidable due to the hardware challenge of direct capturing of molten pool temperature.

This study employs a model pre-trained on FEM simulation data and applies TL to experimental data, which can effectively improve computational efficiency and accelerate model convergence. Parameters pre-trained on simulation data offer an optimal initialization for training on experimental data, thereby reducing the number of iterations and time required for training on the experiment data. Since the simulation data typically cover a wide range of process conditions and geometric characteristics, the model has already learned the fundamental patterns under these conditions without needing to learn all characteristics and patterns from scratch. Moreover, models trained on simulation data can capture complex physical phenomena in the WAAM process, such as the effects of heat accumulation and the trends in thermal cycling curves. This information can be directly utilized in the TL process, further improving the efficiency of model training.

In long-horizon temperature field prediction, the PIGeoRNN model demonstrates better accuracy and stability compared to FEM. This advantage stems from its ability to dynamically calibrate temperature-dependent thermal conductivity using thermal histories while reducing training costs and enhancing generalization through physical constraints. Although FEM accounts for temperature-dependent thermophysical parameters, it struggles to capture nonlinear behaviors effectively. Numerical errors can amplify when these parameters are influenced by material fluctuations, distorting thermal gradient distributions in the molten pool region. Additionally, FEM’s simplified treatment of surface emissivity and convective heat transfer coefficients fails to fully reflect dynamic surface state changes and liquid metal radiation characteristics, leading to deviations in heat loss evaluation.

The PIGeoRNN model proposed in this study holds potential application value in feed-forward control and DT for the WAAM process. In feedforward control, the temperature fields result predicted by the model can be directly used to guide the adjustment of process parameters (such as WFS and heat source power). This allows for preemptive optimization of deposition paths or cooling strategies, thereby actively avoiding the generation of thermal deformation and residual stress during the deposition process, and enhancing the quality and performance of the formed parts. In the construction of DT, the model can serve as an integration interface between the physical engine and real-time data, enabling dynamic prediction of the process by establishing multi-physics field mappings of geometry-temperature-stress relationships. However, these potential applications require further experiment validation to assess their performance in actual operating conditions.

Conclusion

The PIGeoRNN model proposed in this paper demonstrates the feasibility and effectiveness of real-time long-horizon temperature fields prediction in the WAAM process, with considerable accuracy. By integrating the advantages of PIML and RNN, this model enhances prediction time-efficiency, interpretability, and robustness. Based on RNN architecture, the model achieves long-horizon temperature field prediction by fusing multimodal spatiotemporal characteristics. The model includes an encoder-decoder module, dual-layer ConvLSTM cells, and a physical constraint module, integrating geometric characteristics and physical information to predict the temperature field for future 1.25 s based on current 1.25 s data, with a prediction time of only 12 ms. Trained and tested using FEM simulation data, the model’s performance was compared against three ablated models. Compared with FEM models, PIGeoRNN improves predictive accuracy and interpretability through integrating PI and geometric characteristics inputs while maintaining low computational costs. In addition, the experimental evaluation assessed the model’s predictive performance on experimental data by additionally fabricating a thin-wall structure for testing. Experimental validation included fabrication of a thin-wall test structure. To enhance learning efficiency, TL was applied to experimental data training. Results confirm the model’s effectiveness in real-world applications, meeting real-time long-horizon prediction requirements. The main contributions of this study are as follows:

1. The PIGeoRNN model, which integrates geometric characteristics and PI inputs, a PI loss function, and ConvLSTM cells, is used for real-time long-horizon temperature field prediction in the WAAM process and holds great promise for applications requiring feed-forward control and DT.

2. The ConvLSTM cell can simultaneously extract spatial temperature distributions and temporal change trends, which allows for accurate prediction and understanding of the dynamic behavior of the temperature field, thus enabling long-horizon temperature field prediction.

3. The model employs a strategy that can solve spatiotemporal PDE, imposing I/BCs as hard constraints on the model, which enhances the model’s predictive accuracy and robustness.

4. Compared with Baseline, GeoRNN and PIRNN model, the PIGeoRNN model demonstrates the best performance in both simulation and experiment data temperature field prediction achieving a maximum error of only 4.5% at the cost of slightly reduced temporal efficiency.

5. Models trained using simulation data are trained in TL on experiment data, which improves the utility and efficiency of predictions and reduces the training time by 50%.

Future work can further extend the model to consider more geometric characteristics and process parameters, and optimize the AM process based on accurate and fast temperature field prediction results. Additionally, the construction of long-horizon predictive models for stress fields and deformations can be considered to provide a feasible solution for depositing high-quality parts. Although the prediction model has been validated only for WAAM, the model can be extended to other DED processes, such as laser DED (wire or powder feeding), Electron Beam AM, and other metal or non-metal DED processes where thermal cameras can be effectively integrated, by modifying the geometric characteristics and BCs of the model, adjusting the relevant physical parameters in the PI loss function, and retraining with data specific to the new process.

Acknowledgements

This work is financially supported in part by the National Natural Science Foundation of China (NSFC) under Grant 52375344.

Author contributions

M.T., H.M., and D.D. conceptualized, proved theory, designed the experiments. M.T., T.L., and M. L. conducted experiments, data collection and processing. M.T. implemented the code. The structure of the manuscript was designed by H.M. and D.D., and the paper was mainly written by M.T. and H.M., with substantial feedback provided by H.M., D.D., and J.Z. M.T. and H.M. were responsible for the Machine Learning side, while D.N., A.G. Supervision and coordination of the development of the method was provided by H.M. D.D., and J.Z., and the design and writing of the manuscript was supervised by H.M. and D.D. H.M. and D.D. are the corresponding author.

Peer review

Peer review information

Communications Engineering thanks Muhammad Arif Mahmood and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editors: [Philip Coatsworth and Ros Daw]. [A peer review file is available].

Data availability

The data supporting the model training of this research can be obtained from the corresponding author upon request.

Code availability

The codes supporting the model of this research can be obtained from the corresponding author upon request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s44172-025-00501-7.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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