Accurate long-horizon prediction of spatiotemporal fields on complex geometries is a fundamental challenge in scientific machine learning, with applications such as additive manufacturing where temperature histories govern defect formation and mechanical properties. High-fidelity simulations are accurate but computationally costly, and despite recent advances, machine learning methods remain challenged by long-horizon temperature and gradient prediction. We propose a deep learning framework for predicting full temperature histories directly on meshes, conditioned on geometry and process parameters, while maintaining stability over thousands of time steps and generalizing across heterogeneous geometries. The framework adopts a temporal multiscale architecture composed of two coupled models operating at complementary time scales. Both models rely on a latent recurrent graph neural network to capture spatiotemporal dynamics on meshes, while a variational graph autoencoder provides a compact latent representation that reduces memory usage and improves training stability. Experiments on simulated powder bed fusion data demonstrate accurate and temporally stable long-horizon predictions across diverse geometries, outperforming existing baseline. Although evaluated in two dimensions, the framework is general and extensible to physics-driven systems with multiscale dynamics and to three-dimensional geometries.
Figures & tables
Figure 1: Example of printed geometry. Metallic powder is in gray, the final printed part is in blue and the part boundaries are highlighted in green.
Figure 2: Temperature field snapshots for a fixed printed geometry at multiple time steps. The green lines indicate the part boundaries, enclosing the solid metallic region. The temperature is simulated for both metallic part and the surrounding powder.
Figure 3: Overview of the LM-RGNN architecture and its main components, including the latent-RGNN and the VGAE.
Model
MAE (°C)
MAPE (%)
MME (°C)
mpIoU
MAE∇xy (°C/mm)
MAE∇t (°C/s)
Parameters
Interlayer
7.29 ± 0.17
4.48 ± 0.22
44.5 ± 0.5
2.10 ± 0.05
0.76 ± 0.09
85k
↪ w/o latent
17.80 ± 9.16
4.73 ± 1.57
95.0 ± 40.6
4.65 ± 1.54
0.40 ± 0.18
91k
Intralayer
6.48 ± 0.19
2.20 ± 0.010
108.7 ± 2.80
0.87 ± 0.0016
2.46 ± 0.030
1.61 ± 0.020
42k
↪ w/o latent
6.80 ± 0.34
2.43 ± 0.18
50.6 ± 2.0
0.91 ± 0.0081
2.27 ± 0.10
1.28 ± 0.064
41k
LM-RGNN
6.50 ± 0.10
2.58 ± 0.002
108.7 ± 0.28
0.85 ± 0.0017
2.44 ± 0.030
1.98 ± 0.0031
127k
↪ w/o latent & multiscale strategy
26.28 ± 1.45
7.25 ± 1.12
111.46 ± 5.1
0.69 ± 0.097
5.98 ± 1.30
7.13 ± 0.84
131k
Table 1: Comparison of performance metrics on the test set across the proposed models, their ablated variants (marked by the ↪ symbol) , and the baseline.
Model
Inference time (ms)
Mem. pressure
Interlayer
30.4
Low
↪ w/o latent
30.9
Medium
Intralayer
12.35
Low
↪ w/o latent
14.28
Medium
LM-RGNN
14.47
Low
↪ w/o latent & multiscale strategy
68.19
High
Table 2: Comparison of per-time-step inference time and training-time memory pressure across the proposed models, their ablated variants (marked by the ↪ symbol), and the baseline.
Figure 4: Temporal MAE (t-MAE) on the test set for the interlayer model, compared with its full-dimensional (non-latent) counterpart.
Figure 5: Temporal MAE (t-MAE) on the test set for the intralayer model, compared with its ablated variants.
Figure 6: Temporal MAE on the test set for the LM-RGNN model, compared with its ablated variant and the Decoupled-RGNN baseline.
Appendix figures & tables17 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 7: Geometry parametrization. w is the part base width, hl and hr are the heights of left and right line-breaks respectively, αl1 , αr1 , αl2 , αr2 are line angles.
Figure 8: Examples of generated geometry. Metallic powder is in gray, the final printed part is in blue and the part boundaries are highlighted in green.
Number of elements
Runtime
Number of time steps
mean
29987
1h 38m 37s
9316
std_dev
5756
0h 53m 13s
1341
min
18126
0h 20m 42s
6544
max
43670
3h 51m 2s
12523
Appendix
Table 3: Simulations statistics
Figure 9: Thermal domain at time t .
Figure 10: Thermo-physical properties used in the numerical model.
Figure 13: (a) VGAE encoder with depth d=3 . (b) VGAE decoder with depth d=3 .
GraphPool hidden size
32
GraphPool hidden layers
2
GraphUnpool hidden size
32
GraphPool hidden layers
2
Input features size
3 (temperature, mask, material mask)
Output features size
1 (temperature)
Encoder hidden size
12, 24, 32
Appendix
Table 4: VGAE hyper-parameters.
Interlayer latent RGNN
Intralayer latent RGNN
Hidden feature size
48
32
Geometry encoder: GraphPool hidden size
16
16
Geometry encoder: GraphPool hidden layers
2
2
Geometry encoder hidden size
8, 16, 16
8, 16, 16
Geometry encoder input features size
2
2
Geometry encoder output features size
16
16
Appendix
Table 5: Architecture details of the interlayer and intralayer models. The geometry encoder is used to encode geometrical inputs (active nodes mask, material mask) to the latent mesh. MPGNN denotes Message Passing Graph Neural Network and is used in the GraphGRU architecture.
Model
Learning rate
Batch size
LR warmup
LR scheduler
VGAE
1e-3
64
Yes
Cosine annealing ( Tmax=130 , ηmin=1e−4 )
Interlayer
2e-3
4
Yes
Cosine annealing ( Tmax=140 , ηmin=1e−5 )
Intralayer
4e-3
64
Yes
Cosine annealing ( Tmax=180 , ηmin=1e−5 )
Decoupled-RGNN
1e-3
4
Yes
Cosine annealing ( Tmax=130 , ηmin=1e−3 )
Appendix
Table 6: Training hyper-parameters. The LR warmup is a linear LR scheduler with start factor = 0.1 and duration = 8.
TBPTT steps
8
Early stopping patience
25
SOAP optimizer betas
0.95, 0.95
SOAP optimizer weight decay
0.01
SOAP optimizer preconditioning frequency
8
Appendix
Table 7: Training hyper-parameters with a fixed value.
Figure 14: The blue curve is the temporal MAE (t-MAE) on the test set for the VGAE with latent dim. 4. The red curve is the function f(t)=t-MAE(0)/domain size(t) .
Latent dim.
MAE (°C)
MAPE (%)
MME (°C)
mpIoU
MAE∇xy (°C/mm)
MAE∇t (°C/s)
Parameters
4
7.05 ± 1.68
2.44 ± 0.51
118.6 ± 22.5
0.86 ± 0.02
3.01 ± 0.30
3.68 ± 0.84
19.0k
8
5.74 ± 0.93
2.08 ± 0.31
98.3 ± 12.7
0.88 ± 0.003
2.62 ± 0.28
2.86 ± 0.32
19.8k
16
5.08 ± 0.03
1.82 ± 0.10
94.5 ± 2.5
0.89 ± 0.003
2.35 ± 0.005
2.59 ± 0.05
21.5k
Appendix
Table 8: VGAE performance on the test set for varying latent space dimensionalities.
Latent dim.
Sim. data footprint
Mem. reduction factor
4
92 MB
0.06
8
185 MB
0.12
16
369 MB
0.23
No compress.
795 MB
Appendix
Table 9: VGAE compression performance on the test set for varying latent space dimensionalities. “Sim. data” refers to the combined storage of the simulation mesh description and the corresponding sequence of temperature fields.
Figure 15: Temporal MAE of the VGAE on the test set for different latent dimensionalities. The initial peak in error is attributable to the concentration of reconstruction errors in the vicinity of the laser beam combined with the small domain size at early time steps, as discussed in Appendix G.1 .
Figure 16: Comparison of ground-truth temperature fields with predictions from the LM-RGNN and the Decoupled-RGNN for one of the test geometries at t=700 s. The corresponding temperature MAE is reported for each model. For visualization, the MAE color scale is clipped at 100°C, as the Decoupled-RGNN can produce substantially higher maximum errors compared to the LM-RGNN. The green lines indicate the part boundaries, enclosing the solid metallic region.
Figure 17: Comparison of ground-truth temperature fields with predictions from the LM-RGNN and the Decoupled-RGNN for one of the test geometries at t=1800 s. The corresponding temperature MAE is reported for each model. For visualization, the MAE color scale is clipped at 100°C, as the Decoupled-RGNN can produce substantially higher maximum errors compared to the LM-RGNN. The green lines indicate the part boundaries, enclosing the solid metallic region.
Department of Computer Science and Engineering, Lehigh University, Bethlehem PA 18015, USA · Department of Civil and Environmental Engineering, Lehigh University, Bethlehem2026 PA 18015, USA