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

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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

CategoryMethodPhys. loss constraintVariable-count feature inputObject/subblock encodingShared recursive composition
Physical consistency constraintsPINN (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 representationDeepSets (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 4: Overall TFCN framework.
Figure 5: Multi-path latent-space composition.
DatasetMethodKK -IDKK -OOD
RMSEMAERMSEMAE
Temp-BCUNet9.99±0.279.99\pm 0.276.83±0.106.83\pm 0.1046.91±1.5646.91\pm 1.5633.52±1.2233.52\pm 1.22
DeepONet12.94±0.2412.94\pm 0.249.18±0.129.18\pm 0.1238.95±1.1938.95\pm 1.1929.13±0.6829.13\pm 0.68
GINO3.09±0.153.09\pm 0.152.06±0.042.06\pm 0.0412.23±0.5512.23\pm 0.555.42±0.325.42\pm 0.32
TFCN-SP4.88±0.764.88\pm 0.762.97±0.422.97\pm 0.4215.66±2.0015.66\pm 2.009.12±2.379.12\pm 2.37
TFCN-TP1.45±0.13\boldsymbol{1.45\pm 0.13}0.95±0.07\boldsymbol{0.95\pm 0.07}4.21±0.15\boldsymbol{4.21\pm 0.15}2.56±0.09\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- KK extrapolation error.
Figure 7: Training coverage and composition depth.
DatasetMethodKK -IDKK -OODOOD/ID
RMSEMAERMSEMAE
Temp-BCTFCN-SP4.882.9715.669.123.209
w/o Locality4.352.5318.4910.294.254
w/o Composition18.6211.5854.0340.112.902
RadFlux-BCTFCN-SP10.537.3223.0615.452.190
w/o Locality10.007.1225.2915.702.529
Table 3: Structural ablation results.
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 15/1515/15 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

SymbolMeaning
Ωh\Omega_{h}Discrete grid
GGCoordinate condition
C\mathcal{C}Heat-source components
B\mathcal{B}Boundary segments
F\mathcal{F}Factors
fif_{i}Factor representation
Appendix
Table 1: Summary of principal notation used in TFCN.
Network typeOutput size
Input spatial fieldsN×64×64×42N\times 64\times 64\times 42
Encoder blocks ×4\times 4N×{32,64,128,256}×{64,32,16,8}2N\times\{32,64,128,256\}\times\{64,32,16,8\}^{2}
BottleneckN×512×4×4N\times 512\times 4\times 4
Up-conv + skip decoder blocks ×4\times 4N×{256,128,64,32}×{8,16,32,64}2N\times\{256,128,64,32\}\times\{8,16,32,64\}^{2}
Conv2dResBlock + 3×33\times 3 ConvN×1×64×64N\times 1\times 64\times 64
Output temperature fieldN×64×64×1N\times 64\times 64\times 1
Appendix
Table 2: Detailed UNet architecture.
Network typeOutput size
Branch input fieldsN×64×64×39N\times 64\times 64\times 39
Flatten branch inputN×159744N\times 159744
Branch MLP: Linear + GELU ×2\times 2 + LinearN×64N\times 64
Trunk coordinate inputN×64×64×2N\times 64\times 64\times 2
Trunk MLP: Linear + GELU ×2\times 2 + LinearN×64×64×64N\times 64\times 64\times 64
Branch broadcast + branch–trunk productN×64×64×64N\times 64\times 64\times 64
Appendix
Table 3: Detailed DeepONet architecture.
Network stageOutput size and configuration
Input spatial fieldsN×64×64×9N\times 64\times 64\times 9
Normalized spatial coordinates1×4096×21\times 4096\times 2 (shared geometry)
Input GNO integral transformN×16×16×9N\times 16\times 16\times 9 ; radius 0.110.11 ; nonlinear MLP 265→64→64→9265\rightarrow 64\rightarrow 64\rightarrow 9
Channel liftingN×48×16×16N\times 48\times 16\times 16 ; 9→96→489\rightarrow 96\rightarrow 48
FNO blocks ×4\times 4 [ 28 ]N×48×16×16N\times 48\times 16\times 16 ; nominal modes (16,16)(16,16) ; spectral weights 48×48×16×948\times 48\times 16\times 9
Channel MLP in each FNO blockN×48×16×16N\times 48\times 16\times 16 ; 48→24→4848\rightarrow 24\rightarrow 48
Appendix
Table 4: Detailed GINO architecture.
Network typeOutput size
Inputs
Input: geometry gridN×64×64×2N\times 64\times 64\times 2
Input: component factorsSDF N×64×64×20N\times 64\times 64\times 20 ; value N×20N\times 20
Input: boundary factorsSDF N×64×64×2N\times 64\times 64\times 2 ; value N×2N\times 2
Encoder
Bucket features and encoder tokens4×(N×64×64×32)4\times(N\times 64\times 64\times 32)
Appendix
Table 5: Detailed TFCN architecture.
Figure 11: Temperature-field sample distribution diagnostics.
ItemSetting
OptimizerAdamW
Training epochs1500
Batch size32
Learning rate8×10−48\times 10^{-4} peak; 1×10−51\times 10^{-5} final
Warm-up and decay60 epochs, linear; cosine decay
Weight decay1×10−41\times 10^{-4}
Appendix
Table 6: Training and evaluation configuration.
SymbolPhysical meaningDefinition
ETmax⁡meanE_{T_{\max}}^{\mathrm{mean}}Mean peak-temperature deviation (K)1N∑i=1N∣T^max⁡,i−Tmax⁡,i∣\displaystyle\frac{1}{N}\sum_{i=1}^{N}\left|\hat{T}_{\max,i}-T_{\max,i}\right|
ETmax⁡P95E_{T_{\max}}^{\mathrm{P95}}Weighted P95 peak-temperature deviation (K)Q0.95(w) ⁣({∣T^max⁡,i−Tmax⁡,i∣}i=1N)\displaystyle Q_{0.95}^{(w)}\!\left(\left\{\left|\hat{T}_{\max,i}-T_{\max,i}\right|\right\}_{i=1}^{N}\right)
EhotrelE_{\mathrm{hot}}^{\mathrm{rel}}Mean relative hot-spot location deviation1N∑i=1N∥x^max⁡,i−xmax⁡,i∥2LΩ\displaystyle\frac{1}{N}\sum_{i=1}^{N}\frac{\left\|\hat{\boldsymbol{x}}_{\max,i}-\boldsymbol{x}_{\max,i}\right\|_{2}}{L_{\Omega}}
RMSEB\mathrm{RMSE}_{B}Boundary-band temperature RMSE (K)[∑i=1N∑x∈ΩB,i(T^i(x)−Ti(x))2∑i=1N∣ΩB,i∣]1/2\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.
Indicator ↓\downarrowGINOTFCN-SPTFCN-TP
Temperature BC
ETmax⁡meanE_{T_{\max}}^{\mathrm{mean}} (K)3.14 / 8.032.12 / 7.671.34 / 2.92
ETmax⁡P95E_{T_{\max}}^{\mathrm{P95}} (K)9.09 / 26.416.83 / 28.444.13 / 8.91
EhotrelE_{\mathrm{hot}}^{\mathrm{rel}}0.118 / 0.0780.088 / 0.0900.058 / 0.068
RMSEB\mathrm{RMSE}_{B} (K)5.16 / 14.081.81 / 5.821.22 / 3.70
RMSE\mathrm{RMSE} (worst case, K)22.29 / 77.0056.22 / 82.538.35 / 45.80
Appendix
Table 8: Physics-oriented thermal indicators and supplementary RMSE statistics under ID/OOD evaluation.
MethodBoundary conditionSingle config.5,000 sequential5,000 parallel/batched
FEniCSxNon-radiative288 s16.7 d6.25 h ∗
FEniCSxRadiative320 s18.5 d6.94 h ∗
TFCNNon-radiative7 ms35 s †1.8 s ‡
TFCNRadiative7 ms35 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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