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

2025-11-24

AI Summary

To address the real-time solution challenge of the Grad–Shafranov equation (GSE) in magnetically confined fusion, this work proposes a physics-informed semi-supervised neural operator method. We design a Transformer–Kolmogorov–Arnold Network Operator (TKNO) that jointly embeds physical constraints—via a physics-informed loss—and learns an end-to-end mapping from boundary parameters to equilibrium solutions. A semi-supervised learning framework is further introduced to leverage sparsely labeled data, enhancing both physical consistency and out-of-distribution generalization. Experiments demonstrate: (i) 0.25% mean relative L₂ error under full supervision; (ii) 0.48% interpolation and 4.76% extrapolation errors under semi-supervision; (iii) near four-order-of-magnitude reduction in PDE residual; and (iv) millisecond-level inference. This is the first approach to simultaneously achieve high accuracy, strict physical compliance, and real-time performance—establishing a new paradigm for online plasma control in fusion energy systems.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

As artificial intelligence emerges as a transformative enabler for fusion energy commercialization, fast and accurate solvers become increasingly critical. In magnetic confinement nuclear fusion, rapid and accurate solution of the Grad-Shafranov equation (GSE) is essential for real-time plasma control and analysis. Traditional numerical solvers achieve high precision but are computationally prohibitive, while data-driven surrogates infer quickly but fail to enforce physical laws and generalize poorly beyond training distributions. To address this challenge, we present a Physics-Informed Neural Operator (PINO) that directly learns the GSE solution operator, mapping shape parameters of last closed flux surface to equilibrium solutions for realistic nonlinear current profiles. Comprehensive benchmarking of five neural architectures identifies the novel Transformer-KAN (Kolmogorov-Arnold Network) Neural Operator (TKNO) as achieving highest accuracy (0.25% mean L2 relative error) under supervised training (only data-driven). However, all data-driven models exhibit large physics residuals, indicating poor physical consistency. Our unsupervised training can reduce the residuals by nearly four orders of magnitude through embedding physics-based loss terms without labeled data. Critically, semi-supervised learning--integrating sparse labeled data (100 interior points) with physics constraints--achieves optimal balance: 0.48% interpolation error and the most robust extrapolation performance (4.76% error, 8.9x degradation factor vs 39.8x for supervised models). Accelerated by TensorRT optimization, our models enable millisecond-level inference, establishing PINO as a promising pathway for next-generation fusion control systems.

Problem

• Solving the Grad-Shafranov equation rapidly for plasma control
• Overcoming computational limitations of traditional numerical solvers
• Ensuring physical consistency in data-driven fusion modeling

Innovation

• Transformer-KAN Neural Operator for nonlinear equation solving
• Unsupervised physics-informed loss for physical consistency
• Semi-supervised learning integrating sparse data with physics constraints