PiGRAND: Physics-informed Graph Neural Diffusion for Intelligent Additive Manufacturing

Benjamin Uhrich; Tim Häntschel; Erhard Rahm

cs.LGMar 16, 2026

PiGRAND: Physics-informed Graph Neural Diffusion for Intelligent Additive Manufacturing

Authors: Benjamin Uhrich, Tim Häntschel, Erhard Rahm

Organizations: Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig, Germany. · Leipzig University, Leipzig, Germany.

Abstract

A comprehensive understanding of heat transport is essential for optimizing various mechanical and engineering applications, including 3D printing. Recent advances in machine learning, combined with physics-based models, have enabled a powerful fusion of numerical methods and data-driven algorithms. This progress is driven by the availability of limited sensor data in various engineering and scientific domains, where the cost of data collection and the inaccessibility of certain measurements are high. To this end, we present PiGRAND, a Physics-informed graph neural diffusion framework. In order to reduce the computational complexity of graph learning, an efficient graph construction procedure was developed. Our approach is inspired by the explicit Euler and implicit Crank-Nicolson methods for modeling continuous heat transport, leveraging sub-learning models to secure the accurate diffusion across graph nodes. To enhance computational performance, our approach is combined with efficient transfer learning. We evaluate PiGRAND on thermal images from 3D printing, demonstrating significant improvements in prediction accuracy and computational performance compared to traditional graph neural diffusion (GRAND) and physics-informed neural networks (PINNs). These enhancements are attributed to the incorporation of physical principles derived from the theoretical study of partial differential equations (PDEs) into the learning model. The PiGRAND code is open-sourced on GitHub: https://github.com/bu32loxa/PiGRAND

Figures & tables

Figure 1: Graph Construction Algorithm - Step 1: Generating point cloud, Step 2: Pruning method, Step 3: Delaunay triangulation, Step 4: Alpha shape, Step 5: Simplicial 3-complex.

Figure 2: Flowchart - Graph construction and physics-informed graph neural diffusion

Figure 3: Geometry of Printed Objects - Photographs of high quality components, strongly deformed component P9 and CAD models

P	$h_{0}$ [mm]	$\alpha$ [°]	$l_{1}$ [mm]	$l_{0}$ [mm]	$h_{1}$ [mm]	Material
4	10	70	20	6.7	18.27	Stainless Steel
7	10	70	25	8.3	22.94	Stainless Steel
9	10	50	25	8.3	9.95	Stainless Steel
7M	10	70	25	8.3	22.94	Nickel-based Alloy

Table 1: Comparison of the measures of the 3D-printed objects P4, P7, P9 and P7M

Figure 4: Temporal-Spatial Graph - graph model development for layer 100, 250, 500 (rows). Left column: point cloud, middle column: Simplicial Complex, right column: Alpha Shape

Panel of Figure 5 — Figure 5: 4D-Heat Transport Prediction - Heat transport evolution for layer 100, 250, 350, 500. As time progresses, the component’s temperature increases, reaching a maximum at the upper portion and subsequently declining towards the base.

Panel of Figure 6 — Figure 6: Heat Transport Prediction and Error Plots – A comparison of the heat transport prediction for print layer 232 ( $t=7$ s) with a PINN, GRAND, PiGRAND and the real measurement data.

Figure 7: Relative Error - Comparison of

\epsilon_{r}

between GRAND, PiGRAND and PINN over the entire printing layer 232 for each timepoint.

	PINN	GRAND	PiGRAND
$\frac{1}{N_{T}}\sum_{k=0}^{N_{T}}\epsilon_{r}$	0.041	0.003	0.001

Table 2: Comparison of the mean value of

\epsilon_{r}

between PINN, GRAND and PiGRAND, where

N_{T}

is the number of time points during printing layer 232.

Panel of Figure 8 — Figure 8: Comparison of explicit Euler and implicit Crank-Nicolson inspired graph diffusion network - Left top: Comparison of $\epsilon_{r}$ for 500 print layers, Right top: Comparison of $\mathcal{L}_{\text{Energy}}$ for 500 print layers. Left bottom: Max temperatures for print layers 3, 4, 5.

	$\frac{1}{N_{L}}\sum_{k=0}^{N_{L}}\epsilon_{r}$	$\frac{1}{N_{L}}\sum_{k=0}^{N_{L}}\hat{\mathcal{L}}_{\text{Energy}}$
Euler	0.010	24.473
Crank-Nicolson	0.006	23.449

Table 3: Comparison of the mean of

\epsilon_{r}

and

\mathcal{L}_{Energy}

between Euler and Crank-Nicolson inspired Graph Neural Diffusion, where

N_{L}

is the number of print layers.

regularization	weight	$\mathcal{L}_{\text{data}}$	$\mathcal{L}_{\varphi}$	$\mathcal{L}_{\psi}$	$\mathcal{L}_{\text{energy}}$
	high	1.085	1.134	0.375	0.000
all	normal	1.128	1.211	0.175	0.076
	low	1.111	1.206	0.231	0.588
	high	1.107	1.205	0.238	0.367
$\mathcal{L}_{\varphi},\,\mathcal{L}_{\psi},\,\mathcal{L}_{\text{heat}}$	normal	1.109	1.205	0.238	0.335
	low	1.108	1.205	0.269	0.667

Table 4: Model fit for the foundation model using different sets of weighted regularization functions for training. ’High weights’ uses a factor of

10

for the regularization terms in the total loss function, ’low weights’ uses a factor of

0.1

regularization	weight	$\mathcal{L}_{\text{data}}$	$\mathcal{L}_{\varphi}$	$\mathcal{L}_{\psi}$	$\mathcal{L}_{\text{energy}}$
	high	1.067	1.021	0.133	0.000
all	normal	1.065	0.979	0.127	0.008
	low	1.063	0.981	0.114	0.291
	high	1.063	0.979	0.112	0.029
$\mathcal{L}_{\varphi},\,\mathcal{L}_{\psi},\,\mathcal{L}_{\text{heat}}$	normal	1.063	0.979	0.108	0.024
	low	1.063	0.981	0.104	0.283

Table 5: Model fit for P4 using different sets of weighted regularization functions for training. ’High weights’ uses a factor of

10

for the regularization terms in the total loss function, ’low weights’ uses a factor of

0.1

Panel of Figure 9 — Figure 9: Inference and transfer learning - Left: Inference for pyramid P4, print layer 380, Right: Inference for pyramid P9, print layer 200, Bottom: Transfer learned heat transport for pyramid P7M, print layer 232.

Figure 10: A comparison of the P7 foundation model inspired by Crank-Nicolson, the P4 and P9 inference and the P7M transfer-learned model. - Left: Comparison of

\epsilon_{r}

for 500 print layers, Right: Comparison of

\mathcal{L}_{\text{Energy}}

for 500 print layers.

Panel of Figure 11 — Figure 11: Performance of PiGRAND - Left: The total solving time of the PiGRAND foundation model is compared with the inference times of the other printed components P4, P9 and the transfer-learned temperature distribution of P7M, Right: The average computational effort associated with the prediction of the temperature distribution of a single printed layer, contrasting the performance of a transfer-learned PINN with our proposed PiGRAND, Bottom: The total solving time of PiGRAND (Crank-Nicolson) compared to a FVM solver and a vanilla PINN for one printing benchmark layer

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Developing reliable predictive models for multi-physics systems often forces a compromise between computationally demanding physical simulators and data-hungry, opaque machine learning (ML) algorithms. While incorporating dimensional analysis into directed computation graphs offers a structurally sound alternative to explicitly solving partial differential equations (PDEs), such formulations typically yield underdetermined systems that act as imperfect simulators. We introduce pi-Net, a physics-informed latent variable model that resolves these imperfect simulators. The hybrid pi-Net architecture consists of an encoder that maps inputs into physically normalized latent space, which are then decoded into final predictions by a differentiable physics head. Evaluated on a real-world, data-scarce Gas Metal Arc Welding (GMAW) case study, the framework achieves an R2 of 0.864 with only 60 experimental samples. The results demonstrate that pi-Net significantly outperforms purely data-driven baselines in data-scarce regimes while providing causal, process-level explainability.

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