- The paper introduces PNOT, a model that integrates structured boundary encoding and a gradient-constrained Sobolev loss for accurate temperature field reconstruction.
- It employs a novel hybrid attention mechanism combining global slice-based and local heat graph propagation to capture both long-range and fine spatial dependencies.
- Experimental results on FEM-simulated data demonstrate a 61.9% error reduction, robust generalization, and enhanced performance under diverse heat-flux scenarios.
The modeling of temperature fields in plasma-facing components of nuclear fusion devices is pivotal for operational safety and device longevity. In particular, the tungsten monoblock divertor on EAST faces highly localized thermal loading, necessitating rapid temperature field reconstruction for real-time monitoring and control. FEM approaches, while accurate, are computationally prohibitive for online inference. Machine learning surrogates, including PINNs and neural operators, have shown promise; however, prior approaches either compress spatial boundary conditions, lack explicit local heat propagation, or fail to enforce gradient regularization. This work introduces a Physics-aware Neural Operator Transformer (PNOT) for spatiotemporal PDE operator learning, mapping boundary heat flux and global operating conditions to temperature fields.
Model Architecture
PNOT encodes both global and boundary conditions as structured token sequences: global tokens are learned via an embedding of operating parameters, while boundary heat flux samples form a local graph, with nodes representing spatial sampling points and edges modeling geometric proximity and heat flux differences. Edge-aware bias is incorporated in the graph attention to retain spatial correlations of the boundary, improving the physical fidelity of downstream temperature predictions.
Query Point Embedding and Attention
Spatial query coordinates are embedded into position tokens. Cross-attention fuses query features with boundary and global condition tokens, constructing a hybrid memory representation. This enables condition-aware temperature estimation at arbitrary spatial-temporal query locations.
Physics Enhancement Modules
PNOT introduces two complementary mechanisms: (a) a global slice-based physics attention mechanism, grouping query points into latent states by physical similarity and facilitating global dependencies through slice attention/desslice operations; (b) a local Heat Graph Propagation module, constructing a KNN graph over query coordinates and propagating distance-weighted feature differences via Laplacian-style diffusion. Local spatial gradients are preserved through gated residual message passing.
Gradient-Constrained Sobolev Loss
Physical consistency is enforced by a graph-based Sobolev regularization term, penalizing predicted-versus-ground-truth directional gradient discrepancies along KNN graph edges. This joint optimization of solution values and spatial gradients regularizes local smoothness and improves out-of-distribution generalization. The training objective combines mean squared error with the Sobolev loss weighted by a tradeoff hyperparameter.
Experimental Validation
Dataset and Metrics
The evaluation leverages a finite-element-generated dataset simulating transient temperature fields in the EAST divertor under ten heat-source power levels (1–10 MW) and diverse boundary heat-flux configurations. Each sample contains thousands of spatial nodes and temporal snapshots, with boundary heat flux discretized across 53 points. Metrics include relative L2, rMAE, rRMSE, and MAE.
PNOT achieves the lowest error rates across all metrics compared to a comprehensive suite of operator learning baselines (DeepONet, LNO, WNO, FNO, GNOT, Transolver, DPOT, TNO, RIGNO, etc.). For example, the reported relative L2 error is $0.0008$, outperforming the next-best model by a significant margin. Notably, PNOT demonstrates superior preservation of temperature gradients and robust generalization under unseen heat-flux scenarios.
Ablation Results
Component analysis confirms additive benefits from structured boundary encoding, Sobolev loss, and physics-aware modules. The combination produces a 61.9% error reduction versus baseline, indicating synergistic gains in both accuracy and physical consistency. Experiments optimizing block depth and KNN neighborhood size show optimal performance at three stacked PNOT blocks and K=8 neighbors, with performance degradation observed beyond these thresholds.
Visual and Error Analysis
Temporal and spatial reconstructions exhibit high fidelity to FEM solutions, with absolute errors concentrated near boundaries and regions of steep temperature gradients. Variability across parameter yi configurations is accurately captured, affirming model robustness to complex control parameter distributions.
Practical and Theoretical Implications
PNOT represents an advance in the application of neural operators to engineering PDEs by explicitly encoding structured boundary relationships and enforcing local/gradient-level physical priors. The model's speed, accuracy, and generalization facilitate integration into real-time fusion device monitoring and feedback control systems, with the potential for deployment in more complex or noisy experimental regimes. Theoretically, the adoption of slice attention and graph-based diffusion modules demonstrates the importance of incorporating both global latent state and local spatial coupling in operator learning architectures.
Limitations include reliance on finite-element simulated data and two-dimensional modeling assumptions; extension to real-world three-dimensional measurements and full-coupled multi-physics is needed for thorough applicability. Generalization across diverse device architectures has yet to be established.
Conclusion
The presented work introduces a Physics-aware Neural Operator Transformer for efficient temperature field reconstruction in the EAST tungsten monoblock divertor. By integrating structured boundary representation, physics-guided attention, spatial heat diffusion modeling, and gradient-constrained optimization, the model outperforms established neural operator baselines in accuracy and physical consistency. The architecture is validated for fast inference and strong generalization to unseen heat-flux and control parameter conditions. Future work will focus on validating the approach against empirical measurements and complex, high-dimensional scenarios, broadening its impact on real-world fusion engineering and operator learning for scientific PDEs.
Citation: "Temperature Field Reconstruction of Tungsten Monoblock Divertor on EAST using Physics-aware Neural Operator Transformer" (2606.31574)