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Physics Attention Transformer Surrogate for Rapid Vertical Instability Growth Rate Prediction: Alcator C-Mod to SPARC

This paper demonstrates that a Physics Attention Transformer surrogate model can rapidly and accurately predict vertical instability growth rates and perturbed current densities for Alcator C-Mod and SPARC equilibria, outperforming traditional operator-based machine learning models while meeting the latency requirements for real-time plasma control.

Original authors: Arunav Kumar, Cesar Clauser, Theodore Golfinopoulos, Cristina Rea, Francesco Capersene, Dan Boyer, SPARC Team, Alcator C-Mod Team

Published 2026-08-26

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CC BY 4.0

Original authors: Arunav Kumar, Cesar Clauser, Theodore Golfinopoulos, Cristina Rea, Francesco Capersene, Dan Boyer, SPARC Team, Alcator C-Mod Team

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to build a practical fusion reactor, scientists face a persistent and formidable challenge: keeping a superheated cloud of plasma, which is hotter than the center of the sun, suspended in mid-air without it touching the walls of its container. This is not a simple matter of holding a balloon; the plasma is a swirling, electrically charged fluid that naturally wants to collapse or shoot out of its magnetic cage. In the most advanced designs, the plasma is stretched into a tall, thin shape to maximize efficiency, but this very shape makes it unstable. If the plasma column tilts even slightly, it can grow unstable and crash into the vessel walls in a fraction of a second, potentially damaging the machine and ending the reaction. To prevent this, control systems must detect the earliest signs of trouble and push the plasma back into place. However, knowing that the plasma has moved is not enough; the system must also know how fast that movement is accelerating. If the instability grows too quickly, standard control actions are too slow to stop it, and the machine must prepare for a controlled shutdown to avoid a disaster.

The problem is that the most accurate way to calculate this growth speed takes far too long. The current gold-standard computer models, which simulate how the plasma interacts with the metal walls and magnetic coils, require minutes to run a single calculation. In a real-time control system that needs to make decisions every few milliseconds, waiting minutes is impossible. For years, scientists have relied on faster, simplified models that treat the plasma as a solid, rigid object. While these are quick, they often miss the subtle, fluid-like shifts in the plasma's internal currents that actually determine how fast an instability will grow, especially in the complex, high-performance shapes planned for future reactors. A team of researchers at MIT and Commonwealth Fusion Systems has now developed a new approach that bridges this gap. They created a sophisticated artificial intelligence tool that learns from the slow, accurate models to predict the instability speed almost instantly, while also mapping out exactly where the plasma is shifting.

The researchers trained this new tool, which they call a Physics Attention Transformer, on a massive library of data generated by the slow, high-fidelity models. This library included thousands of scenarios from the Alcator C-Mod experiment, a real fusion device that has been shut down, as well as thousands of synthetic scenarios designed for SPARC, a compact, high-field fusion reactor currently under construction. The goal was to teach the AI to look at the state of the plasma—the shape of the magnetic fields, the position of the plasma, and the currents in the surrounding coils—and instantly predict two things: how fast the instability is growing, and a detailed map of how the electrical currents inside the plasma are changing. The AI does not just guess a single number; it reconstructs the entire two-dimensional pattern of the disturbance, allowing operators to see not just that the plasma is wobbling, but exactly how it is wobbling.

When tested on data it had never seen before, the new tool proved remarkably accurate. For the C-Mod experiments, it predicted the growth speed with an average error of just 5.4 units per second. For the more challenging, synthetic SPARC scenarios, the error was 12.7 units per second, which is still a significant improvement over the simplified models that often miss the mark by a wide margin. Crucially, the tool also recreated the spatial maps of the disturbance with an error of only about 5 percent. This level of detail is vital because different shapes of instability require different control responses. The researchers compared their new tool against other advanced machine learning methods and found that it outperformed them, particularly in capturing the complex spatial structures that simpler models miss. The tool achieved this speed by learning to compress the vast amount of data describing the plasma's shape into a smaller set of key features, much like how a human might summarize a complex scene by focusing on the most important elements rather than every single detail.

Speed was the primary driver for this development, and the results show a dramatic leap forward. While the traditional, high-fidelity models take about three minutes to calculate a single scenario, the new AI tool can perform the same calculation in roughly 35 milliseconds on a standard computer processor. This represents a speedup of over 5,000 times, bringing the calculation time down to a scale that fits comfortably within the rapid decision-making loops of a fusion reactor's control system. Even when the researchers tested the tool on the more extreme conditions expected in the SPARC reactor—where the plasma is stretched even thinner and the magnetic fields are stronger—it maintained its accuracy. The tool was able to predict the behavior of these high-performance shapes with a level of precision that suggests it could be used to monitor the reactor's stability in real time, alerting controllers to potential problems before they become critical.

However, the researchers are careful to note that this tool is a surrogate, or a stand-in, for the complex physics models, not a replacement for the fundamental laws of physics. It learns from the existing models to mimic their results, so its accuracy is tied to the accuracy of the data it was trained on. The study also highlights that the tool is not perfect in every situation. It performs best when the plasma conditions are similar to those it has seen during training. When faced with extreme conditions that are very different from the training data, such as certain high-performance shapes in the SPARC design, the tool's predictions become less certain. To handle this, the researchers built in a system that measures its own confidence. If the tool detects a situation that is very different from its training, it can flag the prediction as uncertain, prompting the control system to rely on slower, more traditional methods or to take a more conservative approach to safety.

The implications of this work extend beyond just faster calculations. By providing a fast and detailed view of plasma stability, the tool could allow future reactors to operate closer to their performance limits without risking a crash. It could act as a constant monitor, watching the "headroom" available for stability and warning operators if the plasma is getting too close to the edge of safety. This would enable more aggressive shaping of the plasma to generate more power, while still maintaining a safety margin. The researchers envision a future where this tool runs alongside the main control systems, offering a rapid assessment of stability that helps guide the reactor's shape and position. While the tool itself is a significant step forward, the researchers emphasize that it will be part of a larger safety architecture, working in tandem with other monitoring systems to ensure the safe and stable operation of fusion reactors.

The study also looked at how well the tool could transfer its knowledge from one machine to another. They trained the AI on data from the C-Mod experiment and tested it on the synthetic SPARC scenarios without showing it any SPARC data first. The results were mixed: while the tool could make rough estimates, the errors were larger, and it frequently flagged the SPARC conditions as uncertain. This suggests that while the tool can learn general principles of plasma behavior, it still needs specific data from the target machine to achieve the highest level of precision. This finding points to a practical path forward: using the tool for initial screening and monitoring, while continuing to generate high-fidelity data for the most critical and extreme operating conditions. The researchers conclude that this approach offers a powerful new capability for fusion research, turning a calculation that once took minutes into a real-time insight that can help keep the promise of fusion energy alive.

Technical Summary: Physics Attention Transformer Surrogate for Rapid Vertical Instability Growth Rate Prediction

Problem Statement
Vertical instability ( $n=0$ axisymmetric mode) represents a critical operational limit for highly elongated tokamaks. Effective plasma control requires not only detecting plasma displacement but also knowing the instantaneous growth rate ( $\gamma$ ) to distinguish routine feedback from states requiring enhanced monitoring or disruption mitigation. High-fidelity non-rigid free-boundary solvers, such as the MEQ-FGE-L code, accurately compute these growth rates and associated perturbed current densities by coupling plasma response to external circuits. However, these solvers require minutes per equilibrium, rendering them incompatible with control cycle timescales (tens of milliseconds). Conversely, reduced rigid-body models (e.g., RZIP) are fast but fail to capture non-rigid current redistribution, leading to significant errors in growth rate prediction, particularly for shaped plasmas near stability limits. There is a need for a surrogate model that bridges this gap, offering solver-level fidelity at control-relevant latency while preserving the spatial structure of the instability.

Methodology
The authors propose a Physics Attention Transformer (PAT) surrogate trained on labels generated by MEQ-FGE-L. The architecture is designed to map equilibrium representations to both a scalar growth rate ( $\hat{\gamma}$ ) and a two-dimensional perturbed toroidal current density ( $\hat{\delta j_\phi}$ ).

Input Representation: The model ingests local mesh fields (from EFIT), global scalar parameters (e.g., $I_p$ , $q_{95}$ , $\kappa$ ), active coil currents, and precomputed electromagnetic matrices ( $M_{ee}, M_{ey}, R_e$ ).
Architecture:
- Tokenization: Mesh features are compressed into $M=64$ learned mesh tokens via soft assignment, reducing computational cost while retaining spatial resolution.
- Electromagnetic Embeddings: Crucially, the model appends projected electromagnetic matrix tokens (including the plasma current response operator $\Xi_{Iy}$ ) to the token set. This allows the attention mechanism to explicitly access plasma-to-conductor coupling information without inferring it solely from geometry.
- Attention Blocks: Six identical attention blocks process the augmented token set. The mesh tokens exchange global information with the electromagnetic embeddings and are then broadcast back to the grid for spatial reconstruction.
- Outputs: Cross-attention heads predict the growth rate and the conductor-space eigenvector. A pointwise decoder reconstructs the 2D perturbed current density field.
Deployment Bottleneck Resolution: The operator $\Xi_{Iy}$ typically requires a slow perturbed Grad-Shafranov solve ( $\sim$ 3 min). To enable real-time inference, a secondary MLP surrogate ( $\hat{\Xi}_{Iy}$ ) predicts this operator from scalar inputs in $\sim$ 1 ms. The full inference pipeline (EFIT $\to$ $\hat{\Xi}_{Iy}$ $\to$ PAT) runs in under 40 ms on a single CPU core.
Training Objective: The model is trained on a composite loss function comprising data fidelity (growth rate and spatial field), eigenvalue consistency (enforcing $A\hat{v} = \hat{\gamma}\hat{v}$ ), boundary condition constraints, and eigenvector normalization.

Key Contributions

Architecture: Introduction of a Physics Attention Transformer that integrates learned mesh tokens with explicit electromagnetic matrix embeddings, enabling the model to capture non-rigid plasma responses and long-range coupling.
Dual-Output Capability: The model simultaneously predicts the scalar growth rate and the spatial structure of the unstable eigenmode (perturbed current density), providing information necessary for shape control and stability margin monitoring.
Real-Time Feasibility: By replacing the expensive Grad-Shafranov solve with a lightweight surrogate for $\Xi_{Iy}$ , the authors achieve a latency of $\sim$ 35 ms on a single CPU core (5,100 $\times$ speedup over MEQ-FGE-L), making it viable for control cycle integration.
Cross-Device Transfer: The study evaluates transfer from Alcator C-Mod experimental data to synthetic SPARC scenarios, demonstrating the necessity of joint training or synthetic data for high-fidelity performance in reactor-relevant regimes.

Results

Growth Rate Prediction:
- C-Mod: Achieved a Mean Absolute Error (MAE) of 5.4 s $^{-1}$ and $R^2 = 0.997$ on held-out experimental equilibria.
- SPARC: Achieved an MAE of 12.7 s $^{-1}$ and $R^2 = 0.94$ on the synthetic SPARC test set. Errors were higher for high-elongation double-null cases.
- Comparison: PAT significantly outperformed a scalar-input MLP baseline (94.2% of test cases had lower error) and reduced error by 24% compared to Fourier Neural Operators (FNO) and 35% compared to DeepONet.
Spatial Reconstruction: The model reproduced normalized spatial eigenfunctions with relative errors near 5% ( $\epsilon_{L2} \approx 0.05$ ) and a Dice coefficient of 0.91 on the combined test set.
Latency: Inference times were 35 ms (1 CPU), 8.5 ms (8 CPUs), and 1.18 ms (V100 GPU, batch=1).
Uncertainty & Generalization:
- Ensemble uncertainty correlates strongly with prediction error and Mahalanobis distance from the training centroid.
- Zero-shot transfer from C-Mod to SPARC resulted in degraded performance ( $R^2=0.78$ , MAE=18.7 s $^{-1}$ ) and high Out-of-Distribution (OOD) flag rates (67%), indicating that synthetic SPARC data is required for precise predictions in that regime.
Operational Metrics: At a critical threshold of $\gamma = 100$ s $^{-1}$ , the model achieved a false negative rate of 2.2% (C-Mod) and 5.8% (SPARC), with high uncertainty successfully flagging many of the missed cases.

Significance and Claims
The paper claims that PAT can reproduce high-fidelity MEQ-FGE-L outputs at control-relevant latencies, offering a practical tool for growth rate headroom monitoring and proximity-aware shape control. The authors emphasize that the surrogate is not a replacement for nonlinear MHD simulations or a standalone safety barrier but serves as a fast advisory layer within a "defense-in-depth" architecture.

The work highlights that while scalar descriptors are insufficient for capturing the spatial complexity of vertical instability, mesh-resolved attention mechanisms combined with explicit electromagnetic embeddings can effectively learn the non-rigid plasma response. The authors conclude that for future devices like SPARC, targeted labeling of high-elongation and double-null scenarios is necessary to minimize extrapolation errors, and that uncertainty quantification is essential for triaging predictions before they influence control actions.

Technical Summary: Physics Attention Transformer Surrogate for Rapid Vertical Instability Growth Rate Prediction

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