Real-time virtual circuits for plasma shape control via neural network emulators

This paper presents a neural network-based approach that generates real-time, state-aware virtual circuits from a library of over one million simulated equilibria, enabling accurate and robust independent control of coupled plasma shape parameters for the MAST Upgrade tokamak.

The Gist

Imagine a Tokamak (a type of fusion reactor) as a giant, invisible balloon made of super-hot gas (plasma) floating inside a magnetic cage. To keep this balloon from popping or drifting away, scientists use powerful magnets (coils) to squeeze and shape it.

The problem is that these magnets are like a tangled web of strings. If you pull one string to move the balloon up, it might accidentally squish it sideways or stretch it in a way you didn't want. This is called "coupling."

The Old Way: The Static Map

To fix this, scientists used to create a "cheat sheet" called a Virtual Circuit (VC). Think of this like a pre-drawn map for a specific moment in time.

• How it worked: Before an experiment, they would calculate exactly how to pull the strings to move the balloon in a straight line, assuming the balloon stays in one specific shape.
• The flaw: If the balloon starts to wobble, change size, or drift away from that exact spot, the old map becomes useless. The instructions no longer match reality. To fix this, scientists had to manually draw new maps for every tiny step of the journey, which was slow, tedious, and required an expert to constantly tweak the plan.

The New Way: The Smart GPS

This paper introduces a new, smarter way to control the balloon using Neural Networks (a type of AI).

Instead of using a static, pre-drawn map, the researchers built a digital twin of the plasma.

1. The Library: They created a massive library of over one million simulated plasma shapes. Imagine taking a photo of the balloon in every possible position, size, and wobble it could ever have.
2. The Brain: They trained an AI (a neural network) to look at the current state of the magnets and instantly predict what the balloon's shape will be.
3. The Magic Trick: Because this AI is built with math that allows for instant "reverse engineering" (called differentiable functions), it can instantly answer the question: "If I want the balloon to move 5 millimeters to the right, exactly how much do I need to tweak each of the 10 magnets?"

Why This is a Big Deal

• Real-Time Awareness: The old method was like driving with a map from yesterday. This new method is like having a live GPS that recalculates the best route every millisecond as the road (the plasma) changes.
• Unraveling the Knots: The AI is so good at this that it can figure out the perfect combination of magnet adjustments to move the balloon in one direction without accidentally messing up the other directions. It effectively "untangles" the knots in the control system instantly.
• Speed: Calculating these instructions the old way took seconds (too slow for real-time control). The AI does it in microseconds.

The Results

The researchers tested this "Smart GPS" on the MAST-U fusion machine.

• Accuracy: For the main body of the plasma, the AI was incredibly accurate, making tiny errors (less than 5%).
• The Tricky Parts: It was slightly less perfect at controlling the very tips of the plasma (where it touches the reactor walls), with errors up to 15%. The paper notes this isn't because the AI is bad, but because those specific parts are naturally very hard to control independently, even for the best human experts.
• Reliability: By using a "team" of eight slightly different AI models (an ensemble) instead of just one, they made the system even more robust and reliable.

The Bottom Line

This paper proves that we can replace slow, manual, pre-calculated maps with a fast, intelligent, self-updating system. This allows the fusion reactor to maintain its shape perfectly, even as the plasma evolves rapidly, paving the way for more stable and efficient fusion energy experiments. The method is designed specifically for the MAST-U machine but is built to work on any similar fusion reactor in the future.

Technical Summary

Technical Summary: Real-time virtual circuits for plasma shape control via neural network emulators

Problem Statement
Reliable plasma position and shape control in tokamaks, such as the MAST Upgrade (MAST-U), requires the accurate real-time regulation of strongly coupled shape parameters. Current control systems utilize "virtual circuits" (VCs)—linearised mappings that relate small changes in poloidal field (PF) coil currents to variations in specific shape parameters around a reference Grad–Shafranov (GS) equilibrium. These VCs effectively decouple the control problem, allowing for independent regulation of parameters like X-point locations and divertor strike points.

However, calculating VCs in real-time using numerical solvers is computationally prohibitive due to latency constraints. Consequently, existing systems rely on precomputed schedules of VCs derived from a limited set of reference equilibria sampled along a desired trajectory. This approach has two primary limitations:

1. Local Validity: As the plasma evolves away from the specific reference equilibrium, the linearisation degrades, leading to unwanted coupling between shape parameters and degraded controller performance.
2. Manual Complexity: The workflow is highly manual, requiring expert intervention to select reference equilibria and tune control phases, which complicates the design of robust strategies for rapidly evolving scenarios.

Methodology
The authors propose a machine-learning-based framework to replace static, precomputed VC schedules with "state-aware" VCs derived from neural network (NN) emulators. The methodology proceeds through three main stages:

1. Synthetic Equilibrium Library Generation:

   • Instead of relying solely on limited historical discharge data, the authors generated a library of over 1.6 million synthetic MAST-U GS equilibria using the free-boundary equilibrium code FreeGSNKE.
   • A Markov Chain Monte Carlo (MCMC) approach was employed to explore the input space, using 3,999 "seed" equilibria from historic shots as starting points. This ensured dense sampling of desirable operational regions while still covering a broad parameter space, including both limited and diverted configurations.
   • The dataset was filtered to ensure robust calculation of the strike point ($R_{strike}$) by requiring agreement between three distinct calculation methods, resulting in a final dataset of 918,124 equilibria.

2. Neural Network Training:

   • Feedforward neural networks were trained to emulate the mapping from input parameters (PF coil currents, total plasma current, and "Lao85" profile parameters) to seven key output shape parameters ($R_{in}, R_{out}, R_X, Z_X, R_{strike}, R_{nose}, S_{gap}$).
   • The authors explicitly chose to train NNs to predict the shape parameters ($P$) rather than the Jacobian matrices directly. This allows the sensitivity matrix ($S = \partial P / \partial I_{shape}$) to be derived via automatic differentiation or finite differences, avoiding the need for a significantly larger model and dataset required to predict Jacobians directly.
   • An ensemble of eight top-performing models was constructed to reduce variance and improve robustness.

3. Virtual Circuit Derivation:

   • The trained NNs provide differentiable functions. The sensitivity matrix $S$ is computed rapidly using either automatic differentiation (AD) or finite differences (FD) applied to the ensemble output.
   • The VC matrix $V$ is then derived as the pseudoinverse of $S$. Because NN inference is orders of magnitude faster than solving the GS equation, these VCs can be computed in real-time based on the instantaneous plasma state.

Key Results
The authors performed extensive static validation by applying VC-derived current perturbations to a test set of 5,000 equilibria and evaluating the resulting plasma response using FreeGSNKE.

• Accuracy: The ensemble of NN models demonstrated high predictive accuracy for shape parameters, with root mean squared errors (RMSE) in the range of millimeters for core parameters.
• VC Performance:
  • Ensemble vs. Single Model: The ensemble approach consistently outperformed single best-performing models, yielding narrower distributions of realised shifts and lower errors (typically <5% for core parameters).
  • AD vs. FD: Automatic differentiation and finite differences produced comparable results for core shape parameters, with errors in the 1–5% range. This confirms the emulators are sufficiently smooth for AD-based derivative calculation.
  • Comparison to Physics-Based VCs: The emulator-derived VCs closely matched the performance of VCs calculated via finite-difference GS solutions (the physics-based reference). While GS-based VCs remained the most accurate, the emulator-based approach showed only modest degradation in accuracy for core parameters.
• Limitations: Larger deviations were observed for divertor-related parameters, particularly the strike point ($R_{strike}$), with errors reaching up to 15–25%. The authors attribute this to the intrinsic difficulty of decoupling these specific parameters with the available MAST-U coil set, noting that this behavior is also present in the physics-based GS VCs.

Significance and Claims
The paper claims to establish the physical validity of emulated VCs as a scalable and general alternative to precomputed schedules. The primary significance of this work lies in:

1. Real-Time State-Awareness: The approach enables the calculation of VCs directly from the plasma state in real-time, removing the reliance on local linearisation around fixed reference points. This allows the control system to remain effective as the plasma evolves away from nominal conditions.
2. Automation and Robustness: By generating VCs dynamically, the method reduces the need for manual expert intervention in selecting reference equilibria and tuning control phases, simplifying the design of robust controllers for complex scenarios.
3. Machine Agnosticism: While developed for MAST-U, the methodology is device-agnostic. It relies only on a dataset of GS equilibria, which can be generated for any tokamak configuration, making the approach directly transferable to other devices.
4. Interpretability: The method retains the familiar, interpretable VC-based control framework, serving as a minimal extension to existing Plasma Control System (PCS) architectures rather than a complete paradigm shift.

The authors conclude that this neural-network-based approach offers a promising route toward more adaptive and accessible plasma control, balancing high accuracy with the computational speed required for real-time deployment.