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

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 sparse data anchors with PDE constraints, physics-anchored training reduces worst-tail errors on shape-driven and joint shifts. Mechanism diagnostics indicate that data anchors suppress branch-scale failures, whereas PDE residuals reduce OOD-induced high-frequency error amplification. Evaluated on EXL-50U discharge inputs against the device's operational equilibrium solver, the model achieves close agreement (RMSE $<$ 1.3\%) with millisecond-level inference. These results provide a practical route toward physically reliable AI surrogates for fusion workflows. Our code will be available at https://github.com/dsqzhou/physics-anchored-gse.

Fusion experiments try to control extremely hot charged gas inside doughnut-shaped machines. To plan and guide an experiment, scientists need fast estimates of where this gas will sit and what shape it will take. Trusted physics programs can do this, but they may be too slow when quick decisions are needed. Faster AI tools are attractive, but we found a risk: an AI model that only learns from past computer-generated examples can look accurate in familiar situations and still give impossible-looking answers when conditions change. We studied how to make this kind of AI tool more reliable. Instead of asking the model only to imitate examples, we also asked it to respect basic physics rules. We compared several designs and ways of teaching the model. The best approach in our study used both examples and physics rules, which helped reduce the worst mistakes in unfamiliar situations. We also compared our model with the planning software used on the EXL-50U fusion device. It gave closely matching results in milliseconds. This points toward faster AI tools for fusion experiments that stay tied to physics rather than only copying past examples.