Plasma Surrogate Modelling using Fourier Neural Operators

TL;DR

This work demonstrates that Fourier Neural Operators (FNOs) can serve as rapid, data-efficient surrogates for plasma evolution in Tokamaks, trained on reduced MHD simulations and validated against real-time MAST camera data. A novel multi-variable FNO is introduced to jointly model correlated fields such as density, temperature, and electric potential, enabling physically consistent cross-variable dynamics and impressive speedups (up to ~$6$ orders of magnitude) over traditional solvers while achieving low MSEs in the normalised domain. The study conducts extensive ablations on step size, Fourier modes, and training data, and shows zero-shot super-resolution capabilities and limited extrapolation, highlighting both the practical potential for real-time monitoring and the boundaries of autoregressive forecasts. The camera-FNO models further illustrate real-time predictive capability for plasma evolution around key tokamak components, offering a pathway toward integrated predictive control in fusion devices. Practical impact includes faster iteration over control strategies, potential real-time diagnostics, and a framework for extending surrogate models to 3D with physics-informed enhancements and active learning.

Abstract

Predicting plasma evolution within a Tokamak reactor is crucial to realizing the goal of sustainable fusion. Capabilities in forecasting the spatio-temporal evolution of plasma rapidly and accurately allow us to quickly iterate over design and control strategies on current Tokamak devices and future reactors. Modelling plasma evolution using numerical solvers is often expensive, consuming many hours on supercomputers, and hence, we need alternative inexpensive surrogate models. We demonstrate accurate predictions of plasma evolution both in simulation and experimental domains using deep learning-based surrogate modelling tools, viz., Fourier Neural Operators (FNO). We show that FNO has a speedup of six orders of magnitude over traditional solvers in predicting the plasma dynamics simulated from magnetohydrodynamic models, while maintaining a high accuracy (MSE in the normalised domain $\approx$ $10^{-5}$). Our modified version of the FNO is capable of solving multi-variable Partial Differential Equations (PDE), and can capture the dependence among the different variables in a single model. FNOs can also predict plasma evolution on real-world experimental data observed by the cameras positioned within the MAST Tokamak, i.e., cameras looking across the central solenoid and the divertor in the Tokamak. We show that FNOs are able to accurately forecast the evolution of plasma and have the potential to be deployed for real-time monitoring. We also illustrate their capability in forecasting the plasma shape, the locations of interactions of the plasma with the central solenoid and the divertor for the full (available) duration of the plasma shot within MAST. The FNO offers a viable alternative for surrogate modelling as it is quick to train and infer, and requires fewer data points, while being able to do zero-shot super-resolution and getting high-fidelity solutions.

Figures (36)

• Figure 1: (a): Fast camera image of a MAST plasma during an Edge-Localised-Mode, where plasma filaments are ejected outwards due to MHD instabilities at the plasma edge (figure reproduced from McArdle_MAST_2010). (b): The evolution of the filamentary structures during the multi-mode ELM simulation (Figure reproduced from Smith_2020) imaged with a synthetic fast camera diagnostic (time given in ms). Full-scale MHD simulations as shown in this figure demonstrate the structural similarities of MHD observed in simulations as well as within the experiment diagnostically captured using the visible camera.
• Figure 2: Evolution of multiple blobs inside a slab for the first 500 time-steps of the numerical simulation (the full run is 2000 steps), taken at intervals of 100 steps (about 15$\mu s$). The top row shows the density, the middle row the temperature, and the bottom row the vorticity.
• Figure 3: Views of the fast cameras on MAST in the absence of plasma, obtained using illuminations on the 3D CAD model of MAST. Camera shown in figure (a), looks across the central solenoid, while the camera in figure (b) shows the view at the divertor.
• Figure 4: Modified architecture of a single Fourier Layer. FNO is constructed by stacking multiple Fourier layers on top of each other, sandwiched by a lifting and projection operation.
• Figure 5: Schematic representation of the two kinds of training configurations deployed within this study. For the simulations, we use the autoregressive configuration (left). We only use the initial set of field values to propagate towards the desired time step, where the intermediary model output is fed back into the model as inputs to further the time evolution. For the experimental data, we use a sequential time window-based configuration. Here, the model output is not factored back into the model, but we perform a fixed window mapping across time. As we go across the time duration of the data, the window is slid further by one-time instance (right). In the example above, we have T_in=3, step=1, T_out=8, the time focus of each forward propagation of the model is given along the $x$-axis while the evolution across the full-time domain is given along the $y$-axis. Each cell block represents the 2D field value at a certain instance of time. The input and output sizes shown in this diagram are representative only and vary across each experiment.