Neural-Network Equilibrium Solvers
Neural networks learn to solve the Grad-Shafranov equation orders of magnitude faster than iterative solvers, enabling near-instant magnetohydrodynamic equilibrium estimates.
Why equilibrium is expensive
A tokamak plasma sits in a force balance between the plasma pressure gradient and the magnetic force, described in axisymmetric geometry by the Grad-Shafranov equation, a nonlinear elliptic partial differential equation for the poloidal flux function psi. Solving it requires iterating a fixed-point loop because the source terms (pressure and current profiles) depend on psi itself. Traditional codes converge in seconds to minutes, which is too slow for many control and design loops.
The learned map
A neural network can be trained to approximate the map from boundary conditions, coil currents, and profile parameters to the full flux map psi(R,z). Once trained on a dataset of converged equilibria, inference is a single forward pass costing microseconds to milliseconds. The network learns the smooth dependence of the solution on inputs rather than re-deriving it each time.
Architectures that work
- Fully connected networks mapping a handful of scalar inputs to flux at fixed grid points
- Convolutional decoders that emit a 2D flux image from a latent vector
- Physics-informed networks that add the Grad-Shafranov residual to the loss so predictions obey the PDE
The physics-informed variant penalizes the differential-operator residual evaluated on the network output by automatic differentiation, so the model is not merely interpolating training data but is anchored to the governing equation. This improves extrapolation and reduces the number of labeled equilibria needed.
Where it fits
Fast equilibrium surrogates feed real-time shape control, between-shot scenario planning, and large design scans. For a negative-triangularity spherical tokamak like the Hyperion breeder concept, a surrogate lets engineers sweep coil configurations that hold the -0.30 triangularity target without launching a full solver for every candidate. The surrogate is a design accelerator, always cross-checked against a first-principles solver before any decision.
The main caution is domain coverage: a network is reliable only inside the region of input space its training set spanned. Out-of-distribution coil currents or profile shapes can yield confident but wrong flux maps, so surrogates are paired with uncertainty estimates and validated against physics-constrained solvers.