Physics-informed Neural Operator Learning for Nonlinear Grad-Shafranov Equation

Siqi Ding, Zitong Zhang, Shi Guoyang, Xingyu Li, Xiang Gu, Yanan Xu, Huasheng Xie, Hanyue Zhao, Yuejiang Shi, Tianyuan Liu
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:25305-25332, 2026.

Abstract

AI for fusion requires bridging a critical “sim-to-real” gap: simulation-trained models must generalize reliably under distribution shifts in safety-critical workflows. Focusing on the nonlinear Grad-Shafranov equation (GSE), we develop and analyze a physics-anchored operator-learning framework for fixed-boundary equilibrium prediction. The framework combines data anchors with PDE residual constraints and uses a physics-motivated Transformer-KAN Neural Operator (TKNO) to capture global elliptic coupling and nonlinear source response. Under multi-parameter distribution shifts, our analysis shows that data-only surrogates can develop severe OOD tails, while physics-only training may converge to incorrect solution branches; by combining data anchors with PDE constraints, physics-anchored training reduces worst-tail errors on shape-driven and joint shifts. Non-causal diagnostics associate data supervision with fewer branch-scale failures and physics-anchored training with lower OOD-induced high-frequency error amplification. Evaluated on EXL-50U discharge inputs against the device’s operational equilibrium solver, the model achieves close agreement (mean relative RMSE of 1.27%) with millisecond-level inference. These results provide a practical route toward physically reliable AI surrogates for fusion workflows. Our Code is available at https://github.com/dsqzhou/physics-anchored-gse

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-ding26w, title = {Physics-informed Neural Operator Learning for Nonlinear Grad-Shafranov Equation}, author = {Ding, Siqi and Zhang, Zitong and Guoyang, Shi and Li, Xingyu and Gu, Xiang and Xu, Yanan and Xie, Huasheng and Zhao, Hanyue and Shi, Yuejiang and Liu, Tianyuan}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {25305--25332}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/ding26w/ding26w.pdf}, url = {https://proceedings.mlr.press/v306/ding26w.html}, abstract = {AI for fusion requires bridging a critical “sim-to-real” gap: simulation-trained models must generalize reliably under distribution shifts in safety-critical workflows. Focusing on the nonlinear Grad-Shafranov equation (GSE), we develop and analyze a physics-anchored operator-learning framework for fixed-boundary equilibrium prediction. The framework combines data anchors with PDE residual constraints and uses a physics-motivated Transformer-KAN Neural Operator (TKNO) to capture global elliptic coupling and nonlinear source response. Under multi-parameter distribution shifts, our analysis shows that data-only surrogates can develop severe OOD tails, while physics-only training may converge to incorrect solution branches; by combining data anchors with PDE constraints, physics-anchored training reduces worst-tail errors on shape-driven and joint shifts. Non-causal diagnostics associate data supervision with fewer branch-scale failures and physics-anchored training with lower OOD-induced high-frequency error amplification. Evaluated on EXL-50U discharge inputs against the device’s operational equilibrium solver, the model achieves close agreement (mean relative RMSE of 1.27%) with millisecond-level inference. These results provide a practical route toward physically reliable AI surrogates for fusion workflows. Our Code is available at https://github.com/dsqzhou/physics-anchored-gse} }
Endnote
%0 Conference Paper %T Physics-informed Neural Operator Learning for Nonlinear Grad-Shafranov Equation %A Siqi Ding %A Zitong Zhang %A Shi Guoyang %A Xingyu Li %A Xiang Gu %A Yanan Xu %A Huasheng Xie %A Hanyue Zhao %A Yuejiang Shi %A Tianyuan Liu %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-ding26w %I PMLR %P 25305--25332 %U https://proceedings.mlr.press/v306/ding26w.html %V 306 %X AI for fusion requires bridging a critical “sim-to-real” gap: simulation-trained models must generalize reliably under distribution shifts in safety-critical workflows. Focusing on the nonlinear Grad-Shafranov equation (GSE), we develop and analyze a physics-anchored operator-learning framework for fixed-boundary equilibrium prediction. The framework combines data anchors with PDE residual constraints and uses a physics-motivated Transformer-KAN Neural Operator (TKNO) to capture global elliptic coupling and nonlinear source response. Under multi-parameter distribution shifts, our analysis shows that data-only surrogates can develop severe OOD tails, while physics-only training may converge to incorrect solution branches; by combining data anchors with PDE constraints, physics-anchored training reduces worst-tail errors on shape-driven and joint shifts. Non-causal diagnostics associate data supervision with fewer branch-scale failures and physics-anchored training with lower OOD-induced high-frequency error amplification. Evaluated on EXL-50U discharge inputs against the device’s operational equilibrium solver, the model achieves close agreement (mean relative RMSE of 1.27%) with millisecond-level inference. These results provide a practical route toward physically reliable AI surrogates for fusion workflows. Our Code is available at https://github.com/dsqzhou/physics-anchored-gse
APA
Ding, S., Zhang, Z., Guoyang, S., Li, X., Gu, X., Xu, Y., Xie, H., Zhao, H., Shi, Y. & Liu, T.. (2026). Physics-informed Neural Operator Learning for Nonlinear Grad-Shafranov Equation. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:25305-25332 Available from https://proceedings.mlr.press/v306/ding26w.html.

Related Material