Compositional Embedding Architecture for Physical Field Prediction in Componentized Aerospace Systems

Qineng Wang; Xinrui Zhou; Shuwen Yue; Kangli Bao; Hairun Xie; Yonghe Zhang

cs.CESep 23, 2026

Compositional Embedding Architecture for Physical Field Prediction in Componentized Aerospace Systems

Authors: Qineng Wang, Xinrui Zhou, Shuwen Yue, Kangli Bao, Hairun Xie, Yonghe Zhang

Organizations: Key Laboratory for Satellite Digitalization Technology, Innovation Academy for Microsatellites, Chinese Academy of Sciences, Shanghai, China · School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China

Abstract

Spacecraft thermal design requires repeated evaluation of how variations in the number and spatial arrangement of heat-generating components and in thermal boundary conditions affect the temperature field. High-fidelity numerical simulations are computationally expensive and therefore difficult to use for large-scale design screening. Although surrogate models can accelerate temperature-field prediction, existing approaches generally encode each complete configuration as a whole and do not explicitly exploit the reusability of local physical constituents across configurations, which limits their accuracy for component counts and combinations not covered during training. To address this issue, we propose the Tree-Structured Factor Composition Network (TFCN), which decomposes complex spacecraft thermal configurations into reusable local physical factors and employs a tree-structured composition module to learn the global temperature-field response associated with different factor combinations. TFCN is evaluated on two-dimensional steady-state spacecraft thermal-analysis cases with prescribed-temperature and radiative-flux boundary conditions. The model is trained exclusively on configurations containing no more than 15 heat-generating components and evaluated on unseen configurations containing 16-25 components. For the prescribed-temperature and radiative-flux cases, TFCN achieves component-count out-of-distribution RMSE values of 4.21 K and 18.62 K, respectively, representing reductions of 65.6% and 33.6% relative to the strongest baseline. These results demonstrate that TFCN improves the reliability of temperature-field prediction under variations in component count and provides an efficient surrogate for rapid spacecraft thermal-design evaluation and large-scale configuration screening.

Figures & tables

Category	Method	Phys. loss constraint	Variable-count feature input	Object/subblock encoding	Shared recursive composition
Physical consistency constraints	PINN (2019) [ 11 ]	$\checkmark$	$\times$	$\times$	$\times$
	PI-ML (2023) [ 7 ]	$\checkmark$	$\times$	$\times$	$\times$
	PI-DeepONet (2024) [ 12 ]	$\checkmark$	$\times$	$\times$	$\times$
	DSSL (2026) [ 8 ]	$\checkmark$	$\times$	$\times$	$\times$
Spatial/set representation	DeepSets (2017) [ 13 ]	$\times$	$\checkmark$	$\checkmark$	$\times$
	GINO (2023) [ 14 ]	$\times$	$\checkmark$	$\checkmark$	$\times$

Table 1: Literature-level feature matrix for related physical-field surrogate methods.

Figure 1: From a componentized spacecraft layout to the two-dimensional HSL-TFP problem. The underlying problem follows the general heat-source-layout benchmark of Chen et al. [ 21 ] , and its spacecraft-oriented extension follows our previous work [ 8 ] .

Figure 2: Automated data-generation pipeline inherited from our previous work [ 8 ] .

Figure 3: Data construction and evaluation splits.

Figure 5: Multi-path latent-space composition.

Dataset	Method	$K$ -ID		$K$ -OOD
Dataset	Method	RMSE	MAE	RMSE	MAE
Temp-BC	UNet	$9.99\pm 0.27$	$6.83\pm 0.10$	$46.91\pm 1.56$	$33.52\pm 1.22$
	DeepONet	$12.94\pm 0.24$	$9.18\pm 0.12$	$38.95\pm 1.19$	$29.13\pm 0.68$
	GINO	$3.09\pm 0.15$	$2.06\pm 0.04$	$12.23\pm 0.55$	$5.42\pm 0.32$
	TFCN-SP	$4.88\pm 0.76$	$2.97\pm 0.42$	$15.66\pm 2.00$	$9.12\pm 2.37$
	TFCN-TP	$\boldsymbol{1.45\pm 0.13}$	$\boldsymbol{0.95\pm 0.07}$	$\boldsymbol{4.21\pm 0.15}$	$\boldsymbol{2.56\pm 0.09}$

Table 2: Interpolation and high-component-count extrapolation performance. Values are reported as mean

\pm

standard deviation over five independent runs.

Figure 6: Per-

K

extrapolation error.

Figure 7: Training coverage and composition depth.

Dataset	Method	$K$ -ID		$K$ -OOD		OOD/ID
Dataset	Method	RMSE	MAE	RMSE	MAE	OOD/ID
Temp-BC	TFCN-SP	4.88	2.97	15.66	9.12	3.209
	w/o Locality	4.35	2.53	18.49	10.29	4.254
	w/o Composition	18.62	11.58	54.03	40.11	2.902
RadFlux-BC	TFCN-SP	10.53	7.32	23.06	15.45	2.190
	w/o Locality	10.00	7.12	25.29	15.70	2.529

Table 3: Structural ablation results.

Panel of Figure 8 — Figure 8: PHATE diagnostic of intermediate-representation geometry.

Figure 9: MMD diagnostic of the ID/OOD latent-distribution discrepancy.

Figure 10: Single-sample intermediate-representation diagnostic during factor composition. Intermediate panels are decoder projections; only the fully composed $15/15$ panel is the final full-system temperature prediction and uses the Kelvin scale.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$\Omega_{h}$	Discrete grid
$G$	Coordinate condition
$\mathcal{C}$	Heat-source components
$\mathcal{B}$	Boundary segments
$\mathcal{F}$	Factors
$f_{i}$	Factor representation

Appendix

Table 1: Summary of principal notation used in TFCN.

Network type	Output size
Input spatial fields	$N\times 64\times 64\times 42$
Encoder blocks $\times 4$	$N\times\{32,64,128,256\}\times\{64,32,16,8\}^{2}$
Bottleneck	$N\times 512\times 4\times 4$
Up-conv + skip decoder blocks $\times 4$	$N\times\{256,128,64,32\}\times\{8,16,32,64\}^{2}$
Conv2dResBlock + $3\times 3$ Conv	$N\times 1\times 64\times 64$
Output temperature field	$N\times 64\times 64\times 1$

Appendix

Table 2: Detailed UNet architecture.

Network type	Output size
Branch input fields	$N\times 64\times 64\times 39$
Flatten branch input	$N\times 159744$
Branch MLP: Linear + GELU $\times 2$ + Linear	$N\times 64$
Trunk coordinate input	$N\times 64\times 64\times 2$
Trunk MLP: Linear + GELU $\times 2$ + Linear	$N\times 64\times 64\times 64$
Branch broadcast + branch–trunk product	$N\times 64\times 64\times 64$

Appendix

Table 3: Detailed DeepONet architecture.

Network stage	Output size and configuration
Input spatial fields	$N\times 64\times 64\times 9$
Normalized spatial coordinates	$1\times 4096\times 2$ (shared geometry)
Input GNO integral transform	$N\times 16\times 16\times 9$ ; radius $0.11$ ; nonlinear MLP $265\rightarrow 64\rightarrow 64\rightarrow 9$
Channel lifting	$N\times 48\times 16\times 16$ ; $9\rightarrow 96\rightarrow 48$
FNO blocks $\times 4$ [ 28 ]	$N\times 48\times 16\times 16$ ; nominal modes $(16,16)$ ; spectral weights $48\times 48\times 16\times 9$
Channel MLP in each FNO block	$N\times 48\times 16\times 16$ ; $48\rightarrow 24\rightarrow 48$

Appendix

Table 4: Detailed GINO architecture.

Network type	Output size
Inputs
Input: geometry grid	$N\times 64\times 64\times 2$
Input: component factors	SDF $N\times 64\times 64\times 20$ ; value $N\times 20$
Input: boundary factors	SDF $N\times 64\times 64\times 2$ ; value $N\times 2$
Encoder
Bucket features and encoder tokens	$4\times(N\times 64\times 64\times 32)$

Appendix

Table 5: Detailed TFCN architecture.

Panel of Figure 11 — Figure 11: Temperature-field sample distribution diagnostics.

Item	Setting
Optimizer	AdamW
Training epochs	1500
Batch size	32
Learning rate	$8\times 10^{-4}$ peak; $1\times 10^{-5}$ final
Warm-up and decay	60 epochs, linear; cosine decay
Weight decay	$1\times 10^{-4}$

Appendix

Table 6: Training and evaluation configuration.

Symbol	Physical meaning	Definition
$E_{T_{\max}}^{\mathrm{mean}}$	Mean peak-temperature deviation (K)	$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\left\|\hat{T}_{\max,i}-T_{\max,i}\right\|$
$E_{T_{\max}}^{\mathrm{P95}}$	Weighted P95 peak-temperature deviation (K)	$\displaystyle Q_{0.95}^{(w)}\!\left(\left\{\left\|\hat{T}_{\max,i}-T_{\max,i}\right\|\right\}_{i=1}^{N}\right)$
$E_{\mathrm{hot}}^{\mathrm{rel}}$	Mean relative hot-spot location deviation	$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\frac{\left\\|\hat{\boldsymbol{x}}_{\max,i}-\boldsymbol{x}_{\max,i}\right\\|_{2}}{L_{\Omega}}$
$\mathrm{RMSE}_{B}$	Boundary-band temperature RMSE (K)	$\displaystyle\left[\frac{\sum_{i=1}^{N}\sum_{\boldsymbol{x}\in\Omega_{B,i}}\left(\hat{T}_{i}(\boldsymbol{x})-T_{i}(\boldsymbol{x})\right)^{2}}{\sum_{i=1}^{N}\|\Omega_{B,i}\|}\right]^{1/2}$

Appendix

Table 7: Definitions of the physics-oriented thermal indicators.

Temperature BC
Indicator $\downarrow$	GINO	TFCN-SP	TFCN-TP
$E_{T_{\max}}^{\mathrm{mean}}$ (K)	3.14 / 8.03	2.12 / 7.67	1.34 / 2.92
$E_{T_{\max}}^{\mathrm{P95}}$ (K)	9.09 / 26.41	6.83 / 28.44	4.13 / 8.91
$E_{\mathrm{hot}}^{\mathrm{rel}}$	0.118 / 0.078	0.088 / 0.090	0.058 / 0.068
$\mathrm{RMSE}_{B}$ (K)	5.16 / 14.08	1.81 / 5.82	1.22 / 3.70
$\mathrm{RMSE}$ (worst case, K)	22.29 / 77.00	56.22 / 82.53	8.35 / 45.80

Appendix

Table 8: Physics-oriented thermal indicators and supplementary RMSE statistics under ID/OOD evaluation.

Method	Boundary condition	Single config.	5,000 sequential	5,000 parallel/batched
FEniCSx	Non-radiative	288 s	16.7 d	6.25 h ∗
FEniCSx	Radiative	320 s	18.5 d	6.94 h ∗
TFCN	Non-radiative	7 ms	35 s †	1.8 s ‡
TFCN	Radiative	7 ms	35 s †	1.8 s ‡

Appendix

Table 9: Computational efficiency under two boundary-condition settings.

Figure 12: RadFlux-BC temperature-field prediction.

Figure 13: Supplementary RadFlux-BC temperature-field prediction samples.

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High-density through-substrate vias (TSVs) enable 2.5D/3D heterogeneous integration but introduce significant signal-integrity and thermal-reliability challenges due to electrical coupling, insertion loss, and self-heating. Conventional full-wave finite-element method (FEM) simulations provide high accuracy but become computationally prohibitive for large design-space exploration. This work presents a scalable electro--thermal modeling and optimization framework that combines physics-informed analytical modeling, graph neural network (GNN) surrogates, and full-wave sign-off validation. A multi-conductor analytical model computes broadband S-parameters and effective anisotropic thermal conductivities of TSV arrays, achieving

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