Physics-Informed Neural Networks and Sequence Encoder: Application to heating and early cooling of thermo-stamping process

Mouad Elaarabi; Domenico Borzacchiello; Philippe Le Bot; Nathan Lauzeral; Sebastien Comas-Cardona

cs.CEMar 27, 2026

Physics-Informed Neural Networks and Sequence Encoder: Application to heating and early cooling of thermo-stamping process

Authors: Mouad Elaarabi, Domenico Borzacchiello, Philippe Le Bot, Nathan Lauzeral, Sebastien Comas-Cardona

Organizations: Nantes Université, Ecole Centrale Nantes, CNRS, GeM, UMR 6183, Nantes, 44321, France · Nantes Université, IRT Jules Verne, Bouguenais, 44340, France

Abstract

In a previous work (Elaarabi et al., 2025b), the Sequence Encoder for online dynamical system identification (Elaarabi et al., 2025a) and its combination with PINN (PINN-SE) were introduced and tested on both synthetic and real data case scenarios. The sequence encoder is able to effectively encode time series into feature vectors, which the PINN then uses to map to dynamical behavior, predicting system response under changes in parameters, ICs and BCs. Previously (Elaarabi et al., 2025b), the tests on real data were limited to simple 1D problems and only 1D time series inputs of the Sequence Encoder. In this work, the possibility of applying PINN-SE to a more realistic case is investigated: heating and early cooling of the thermo-stamping process, which is a critical stage in the forming process of continuous fiber reinforced composite materials with thermoplastic polymer. The possibility of extending the PINN-SE inputs to multimodal data, such as sequences of temporal 2D images and to scenarios involving variable geometries, is also explored. The results show that combining multiple encoders with the previously proposed method (Elaarabi et al., 2025b) is feasible, we also show that training the model on synthetic data generated based on experimental data can help the model to generalize well for real experimental data, unseen during the training phase.

Figures & tables

Figure 1: Overview of the setup used to validate PINN-SE. This setup includes the three main steps of the thermo-stamping process, focusing only on the thermal evolution. An IR camera is mounted in a fixed position above the mold using a support arm, while the mold is actuated using a hydraulic cylinder and controlled by a control unit. The steps can be summarized as follows: (a) The sheet is first heated to a certain temperature $T_{h}$ using an infrared (IR) oven. (b) The sheet is then taken out of the oven, where it is exposed to natural convection during the transfer phase . (c) Finally a cold mold is raised to establish thermal contact, emulating the cooling step of the stamping phase.

Material	Fiber	Polymer Matrix	Thickness (mm)	Fabric	Warp to Weft Ratio	Number of Plies
PA66GF	E-glass	PA66	$1.5\pm 0.008$	Twill	2/2	3
PPGF	E-glass	PP	$2.3\pm 0.05$	Twill	2/2	2

Table 1: Summary of sheets: fiber type, polymer matrix, thickness, fabric type, warp-to-weft ratio and number of plies.

Property	E-glass	PP	PA66	Units
Density ( $\rho$ )	2580	910	1140	kg/m 3
Specific Heat Capacity ( $C_{p}$ )	830	1610	1460	J/(kg $\cdot$ K)
Thermal Conductivity ( $\lambda$ )	1.0	0.22	0.2	W/(m $\cdot$ K)

Table 2: Material properties of E-glass Fiber [ 20 ] , PP [ 21 , 22 ] and PA66 [ 21 , 22 ] .

Panel of Figure 2 — Figure 2: Top view of the mold geometry and thermocouple positions.

Panel of Figure 3 — Figure 3: Top view of Temperature patterns after heating of the PA66GF and PPGF sheets of the samples used in these experiments .

Panel of Figure 4 — Figure 4: Top view of temperature patterns at the surface 15 seconds after contact with the mold of the PA66GF and PPGF sheets for all the samples used in this experiments .

Figure 5: Illustration of heat transfer between two surfaces with asperities. The thermal contact resistance defines the quality of the contact and consequently the efficiency of heat transfer.

Figure 6: Example of the top view normalized temperature difference , $\text{normalized}(T^{H}_{\text{init}}-T^{H}_{\text{end}})$ , for PPGF sheets used to approximate the contact heat transfer coefficient $h_{c}$ .

Figure 7: The Quality of Contact representation for all PA66GF sheets experiments.

Figure 8: The Quality of Contact representation for all PPGF sheets experiments.

Figure 9: Example of data generated by simulation versus a processed experimental data .

Figure 10: Overview of the CNN-VAE architecture.

Figure 11: Examples of the QC generated using the VAE model.

Figure 12: An overview of the PINN-SE architecture used in this application. The PINN-SE can be divided into three parts. The Encoder is composed of four sub modules: the first encodes a sequence of images representing the temperature during the convection phase of the process (part (b) in Fig. 1 ) using a CNN and Deep Sets. The second model is a CNN that encodes the Quality of Contact, $\text{QC}\in\mathbb{R}^{H\times W\times C}$ . The third and fourth models are simple MLPs that encode mold coordinates, sheet thickness and initial contact temperature. Each model produces a feature vector $F_{i}\in\mathbb{R}^{B\times f_{i}}$ , $i\in\{1,2,3,4\}$ , which are concatenated to form $Z\in\mathbb{R}^{B\times(\sum_{i=1}^{4}f_{i})}$ . The second main part is the Space-time Encoder, which encodes spatio-temporal data into Query Points using Fourier feature layers [ 28 ] and an MLP. The final part is the PINN, which takes the concatenation of QPs and $Z$ as input. In this work, two MLP are used and their outputs are summed to give the final temperature prediction.

Figure 13: Model performance across different spatial locations ( $z=\frac{kH}{4}\hskip 9.24994ptk\in\{0,1,2,3,4\}$ and $t=\frac{k{\color[rgb]{0,0,0}t_{f}}}{10}\hskip 9.24994ptk\in\{0,2,5,7,10\}$ ), using MAE metrics and using only 2 thermal images as input to the encoder. The figure summarizes the mean, median and 90th percentile errors of the model, as defined in Eq. 14 , across all validation datasets.

Figure 14: Model performance across different spatial locations ( $z=\frac{kH}{4}\hskip 9.24994ptk\in\{0,1,2,3,4\}$ and $t=\frac{k{\color[rgb]{0,0,0}t_{f}}}{10}\hskip 9.24994ptk\in\{0,2,5,7,10\}$ ), using MAPE (in %) metric and using only 2 thermal images as input to the encoder. The table summarizes the mean and 90th percentile errors of the model, as defined in Eq. 13 , across all validation datasets.

Figure 15: Exemple of model results of top view surface temperature for validation dataset at $z=\frac{H}{4}$ and $t=\frac{k{\color[rgb]{0,0,0}t_{f}}}{10}\hskip 9.24994ptk\in\{0,2,5,7,10\}$ (columns), the first line represents the model prediction, the second presents the simulation results (ground truth) and the last one represents the Absolute Error between prediction and simulation.

Figure 16: Exemple of model results of top view surface temperature for validation dataset at $z=\frac{3H}{4}$ and $t=\frac{k{\color[rgb]{0,0,0}t_{f}}}{10}\hskip 9.24994ptk\in\{0,2,5,7,10\}$ (columns), the first line represents the model prediction, the second presents the simulation results (ground truth) and the last one represents the Absolute Error between prediction and simulation.

Figure 17: Model performance for PA66GF sheet experiments, across different instances $t=\frac{k{\color[rgb]{0,0,0}t_{f}}}{10}\hskip 9.24994ptk\in\{0,2,4,6,8,10\}$ and $z=H$ , using AE metric and using 4 thermal images as input to the encoder. The figure summarizes the mean, median and 90th percentile of AE of the model as given in Eq. 16 .

Figure 18: Model performance for PA66GF sheet experiments, across different instances $t=\frac{k{\color[rgb]{0,0,0}t_{f}}}{10}\hskip 9.24994ptk\in\{0,2,4,6,8,10\}$ and $z=H$ , using APE metric and using 4 thermal images as input to the encoder. The figure summarizes the mean, median and 90th percentile of APE of the model as given in Eq. 15 .

Figure 19: Model performance for PPGF sheet experiments, across different instances $t=\frac{k{\color[rgb]{0,0,0}t_{f}}}{10}\hskip 9.24994ptk\in\{0,2,4,6,8,10\}$ and $z=H$ , using AE metric and using 4 thermal images as input to the encoder. The figure summarizes the mean, median and 90th percentile of AE of the model as given in Eq. 16 .

Figure 20: Model performance for PPGF sheet experiments, across different instances $t=\frac{k{\color[rgb]{0,0,0}t_{f}}}{10}\hskip 9.24994ptk\in\{0,2,4,6,8,10\}$ and $z=H$ , using APE metric and using 4 thermal images as input to the encoder. The figure summarizes the mean, median and 90th percentile of APE of the model as given in Eq. 15 .

Figure 21: Example of model results of top view surface temperature for PPGF sheet sample at $z=H$ and $t=\frac{k{\color[rgb]{0,0,0}t_{f}}}{10}\hskip 9.24994ptk\in\{0,2,4,6,8,10\}$ (columns). The first line represents the thermal images from experiment (ground truth), the second represents the model prediction, the third represents the Absolute Percentage Error (0-1) and the last line represents $\textbf{APE}>0.12$ (binary values 0 or 1).

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Stage	Layer	Activation
Encoder
Input	Input $(161,161)$	–
Conv2D + MaxPool	32 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Conv2D + MaxPool	64 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Conv2D + MaxPool	128 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Flatten	–	–

Appendix

Table 3: CNN-VAE Architecture for ICs Configuration

Stage	Layer	Activation
Encoder
Input	Input $(91,59)$	–
Conv2D + MaxPool	32 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Conv2D + MaxPool	64 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Conv2D + MaxPool	128 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Flatten	–	–

Appendix

Table 4: CNN-VAE Architecture for QC Configuration

Stage	Layer	Activation
Input	Input $(161,161)$	–
Conv2D + MaxPool	32 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Conv2D + MaxPool	64 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Conv2D + MaxPool	128 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Flatten	–	–
Dense $\times 4$	32 nodes each	swish

Appendix

Table 5: CNN Encoder for ICs Configuration (PINN-SE).

Stage	Layer	Activation
Input	Input (sequence size, 9 features)	–
Dense	64 nodes with L1 regularization layer	swish
Dense	64 nodes with L1 regularization layer	swish
Dense	64 nodes with L1 regularization layer	swish
Dense (output)	32 nodes with L1 regularization layer	–
Aggregation	Max pooling across time dimension	–

Appendix

Table 6: Deep Set Architecture.

Stage	Layer	Activation
Input	Input $(91,59)$	–
Conv2D + MaxPool	32 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Conv2D + MaxPool	64 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Conv2D + MaxPool	128 filters, 3 $\times$ 3 same padding, pool (2,2)	swish
Flatten	–	–
Dense $\times 4$	64 nodes each	swish

Appendix

Table 7: CNN Encoder for QC Configuration (PINN-SE).

Stage	Layer	Activation
Input	Input (2 features)	–
Dense	64 nodes with L1 regularization layer	swish
Dense	64 nodes with L1 regularization layer	swish
Dense (output)	32 nodes with L1 regularization layer	–

Appendix

Table 8: Thickness and initial mold temperature MLP Architecture.

Stage	Layer	Activation
Input	Input (16 features)	–
Dense	64 nodes with L1 regularization layer	swish
Dense	64 nodes with L1 regularization layer	swish
Dense (output)	32 nodes with L1 regularization layer	–

Appendix

Table 9: Polygon coordinates Model Architecture.

Stage	Layer	Activation
Input	Input (104 features)	–
Dense	128 nodes with L1 regularization layer	swish
Dense	128 nodes with L1 regularization layer	swish
Dense (output)	64 nodes with L1 regularization layer	linear

Appendix

Table 10: Post feature encoding MLP Architecture.

Stage	Layer	Activation
Fourier Features	Random Fourier Features, 30 nodes (scale = 0.1)	–
Concatenate	–	–
Input	Input (32 features)	–
Dense	100 nodes with L1 regularization layer	swish
Dense	100 nodes with L1 regularization layer	swish
Dense (output)	32 nodes with L1 regularization layer	–

Appendix

Table 11: Space-time Encoder Architecture with Fourier Features layer.

Stage	Layer	Activation
Input	Input (86 features)	–
Dense	100 nodes with L1 regularization	swish
Dense	100 nodes with L1 regularization	swish
Dense	100 nodes with L1 regularization	swish
Dense	100 nodes with L1 regularization	swish
Dense	100 nodes with L1 regularization	swish

Appendix

Table 12: The PINN Architecture.

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Domain-informed Adaptive Sampling for Generalizable PINNs in Metal Additive Manufacturing via Conditional Flow Matching

Hyeonsu Lee, Jihoon Jeong

Wm Michael Barnes ’64 Department of Industrial and Systems Engineering Texas A&M University College Station, TX 77843, USA

Adaptive Sampling Physics-Informed ML PDE Surrogate Modeling Additive Manufacturing

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Accurate thermal modeling is essential in metal additive manufacturing (AM) for understanding the process-structure-property chain. Physics-informed neural networks (PINNs) offer effective surrogate thermal modeling by minimizing physics-based residual losses at collocation points. However, prior works typically rely on manually-crafted, static collocation sampling strategies, which are neither principled nor scalable across process conditions, hindering their generalization capability. In this work, we provide theoretical analysis through empirical risk minimization, showing that process condition-aware adaptive sampling is strictly more favorable than conventional static sampling for generalization. Building on this insight, we propose an adaptive sampling strategy within a two-stage framework: (1) a conditional Flow Matching model that learns approximate high-residual distributions across different process conditions, and (2) a mixed sampling strategy combining this distribution with a domain-informed base distribution to generate adaptive collocation points for refining the PINN predictor. Experiments on metal AM numerical benchmarks demonstrate that our method consistently outperforms state-of-the-art PINN baselines, achieving an average 62.1% reduction in relative

L_2

error under an identical collocation budget, by capturing process-dependent heat dissipation regions often overlooked in the literature. To the authors' knowledge, this is the first adaptive sampling strategy for PINNs in metal AM, contributing to the enhanced generalization and broader applicability.

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