VEQ: a fast parametric Grad--Shafranov solver for fixed-boundary tokamak equilibria with flexible source profiles

The paper introduces VEQ, a fast parametric solver for fixed-boundary tokamak equilibria that utilizes flexible source profiles and harmonic expansions to achieve low-latency, high-accuracy equilibrium reconstruction suitable for repeated queries in transport-geometry coupling workflows.

The Gist

Imagine a tokamak (a doughnut-shaped nuclear fusion reactor) as a giant, invisible balloon filled with super-hot plasma. To keep this balloon from popping or collapsing, scientists need to know exactly how the magnetic "skin" holding it together is shaped and how the forces inside are balanced. This is called finding the "equilibrium."

Usually, calculating this equilibrium is like trying to solve a massive, 3D jigsaw puzzle where every piece changes shape as you touch it. It's accurate, but it takes a long time—too long if you need to check the shape hundreds of times a second to control the reactor.

This paper introduces VEQ (Veloce EQuilibrium), a new tool designed to be the "fast-forward" button for these calculations. Here is how it works, using simple analogies:

1. The "Shape Shifter" vs. The "Pixel Painter"

Traditional methods are like a Pixel Painter. They try to map the entire magnetic field by calculating the value at millions of tiny grid points (pixels). It's detailed, but heavy and slow.

VEQ is more like a Shape Shifter. Instead of painting every pixel, it describes the plasma shape using a flexible, mathematical "skeleton" made of a few key knobs and dials (called parameters).

• Think of the plasma shape as a piece of clay.
• Traditional solvers try to sculpt every tiny bump on the clay.
• VEQ uses a set of pre-defined "stretching" and "bending" tools (mathematical harmonics and polynomials). You just turn a few dials to stretch the clay into the right shape. This makes the calculation incredibly fast because there are far fewer dials to turn than pixels to paint.

2. The "Universal Translator"

One of the biggest headaches in fusion research is that different computer programs speak different languages. One program might give you the "pressure," another the "current," and another the "safety factor" (a measure of stability).

VEQ acts as a Universal Translator. The paper shows that VEQ has six different "input routes" (like different USB ports). You can plug in data from any of these different sources, and VEQ translates them all into its own internal language to solve the problem.

• The Claim: The authors tested this by feeding the same perfect data into all six ports. They found that no matter which port you used, VEQ produced the exact same result. This proves the translator works perfectly and doesn't introduce errors just because you used a different input cable.

3. The "Speed vs. Accuracy" Trade-off

The paper doesn't just say "it's fast"; it explores the trade-off between speed and precision, like choosing between a sketch and a photograph.

• The Sketch (Low Parameters): You use very few dials. It's instant (milliseconds) and good enough for a quick look, but it might miss tiny details.
• The Photo (High Parameters): You use many dials. It takes a bit longer (still very fast, around 19 milliseconds for complex shapes) but captures the shape with high precision.
• The Result: The authors tested this on three different types of plasma shapes (a standard "D" shape, a high-performance "H-mode," and a complex shape with a "X-point"). They found that even with a small number of dials, VEQ could recreate the complex shapes with an error so small it's almost invisible (less than 0.2% of the size of the reactor).

4. The "Stress Test" (Where the cracks show)

The authors were honest about the limits. They checked where the "force balance" (the tension holding the plasma) was perfect and where it wasn't.

• The Interior: In the middle of the plasma, VEQ is excellent. The forces are balanced almost perfectly.
• The Edge: Near the very outer skin (the boundary), the errors are slightly higher. This is because VEQ uses a smooth, flexible skeleton, but real plasma boundaries can be jagged or have sharp corners (like an X-point).
• The Takeaway: VEQ is great for the "middle" of the reactor. If you need to know exactly what's happening at the very edge, you might need to double-check with a slower, more detailed tool. But for most control tasks, VEQ is fast and accurate enough.

5. The "Transport" Test

Finally, they tested if the small errors in VEQ's shape would cause big problems if you tried to use it to predict how heat moves through the plasma (a "transport" test).

• The Result: The errors were tiny (less than 1%). It's like if you measured a room with a slightly bent ruler; the mistake in the room's size wouldn't change your decision on what size rug to buy.

Summary

VEQ is a new, super-fast calculator for fusion reactors. Instead of mapping every single point, it uses a flexible mathematical skeleton to describe the plasma shape.

• It's fast: It can solve complex shapes in milliseconds.
• It's flexible: It accepts data from many different sources.
• It's reliable: It works well for the interior of the reactor and is accurate enough for most control systems, provided you keep an eye on the very outer edges.

The authors conclude that VEQ is perfect for systems that need to ask "What does the plasma look like right now?" over and over again, such as in real-time reactor control or running thousands of simulations to find the best operating conditions.

Technical Summary

Technical Summary: VEQ – A Fast Parametric Grad–Shafranov Solver for Fixed-Boundary Tokamak Equilibria

Problem Statement
Integrated modeling workflows for tokamaks (e.g., IMAS, RAPTOR, SuperCode) require repeated queries of equilibrium geometry, metric coefficients, and diagnostic profiles. These workflows differ from one-time high-fidelity reconstructions; they demand a representation that is compact enough for parameter scans and control-oriented iterations yet retains sufficient two-dimensional shaping and force-balance information to be useful beyond simple analytic closures. Existing tools often fall into two extremes: low-order engineering closures (e.g., Miller models) are fast but lack residual diagnostics, while full equilibrium solvers (e.g., CHEASE, EFIT, DESC) provide high-fidelity native outputs but are often too computationally expensive or infrastructure-heavy for inner-loop use. There is a gap for a continuous, fixed-boundary representation that preserves useful 2D shaping and residual diagnostics while remaining inexpensive to regenerate.

Methodology
The paper introduces VEQ (Veloce EQuilibrium), a compact parametric framework designed to fill this intermediate regime. The specific implementation benchmarked, VEQPy, is an axisymmetric, fixed-boundary Grad–Shafranov solver.

• Parametric Representation: The equilibrium is defined by a parametric flux-surface map $(\rho, \theta) \to [R(\rho, \theta), Z(\rho, \theta)]$. The geometry is parameterized using MXH-type flux-surface harmonics (distorted poloidal angles) and shifted-Chebyshev polynomials for radial profiles. The active unknowns are coefficients of these harmonics and profiles, rather than nodal values of a grid-based poloidal flux map.
• Variational Formulation: The solver enforces a variationally induced projected Grad–Shafranov residual. Instead of solving the strong-form equation pointwise, the method restricts admissible variations to those generated by the active coefficients of the parametric representation. This results in a finite-dimensional system where the shape block is a variationally structured projection (Petrov–Galerkin condition) and profile blocks are closed by moment projections.
• Flexible Input Routes: To ensure compatibility with diverse inputs from analytic studies, reconstruction files, and control models, VEQ supports six input routes (PF, PP, PI, PJ1, PJ2, PQ). These routes accept pressure gradients, toroidal-field functions, poloidal-flux gradients, enclosed toroidal current, current density, or safety-factor profiles. A route-specific closure maps these heterogeneous inputs to a common normalized source representation, allowing all routes to feed into the same finite-dimensional residual operator.
• Implementation: The solver uses a nonlinear solver (SciPy's Powell hybrid method) with an accelerated backend (Numba). It separates setup (grid-dependent precomputation, operator data) from the solve phase to enable low-latency repeated queries.

Key Contributions

1. Finite-Dimensional Parametric Representation: Definition of a compact MXH–Chebyshev flux-surface representation and the derivation of the variationally induced projected Grad–Shafranov residual, including convective variations associated with flux-surface motion.
2. Route-Specific Source Closures: Introduction of a framework that maps heterogeneous 1D inputs (pressure, current, safety factor, etc.) to a common residual system, distinguishing input-route consistency from geometric approximation error.
3. Dual Diagnostic Approach: Reporting both the projected residual norm (minimized by the solver) and sampled pointwise strong-form Grad–Shafranov residuals. This distinguishes the convergence of the reduced system from pointwise force-balance screening.
4. Benchmarking: Comprehensive testing on Solov'ev, CHEASE, and EFIT-derived G-EQDSK cases, quantifying input-route consistency, flux-surface reconstruction error, accuracy–cost trade-offs, and repeated-solve latency.

Results

• Route Consistency: Controlled tests using smooth, mutually compatible inputs generated from a common reference equilibrium demonstrated that all six input routes converge to the same fixed-boundary parametric equilibrium, with differences attributable only to numerical recovery paths (e.g., integration/differentiation steps).
• Accuracy and Latency: The solver was tested on three representative cases: a D-shaped Solov'ev case, an H-mode CHEASE case, and an X-point EFIT case.
  • D-shaped: Achieved a minor-radius-normalized shape error of $1.4 \times 10^{-3}$ with 9 active parameters and a solve-only median time of 1.6 ms.
  • H-mode: Achieved an error of $1.1 \times 10^{-3}$ with 65 parameters and a median time of 19 ms.
  • X-point: Achieved an error of $1.9 \times 10^{-3}$ with 94 parameters and a median time of 15 ms.
• Residual Diagnostics: Pointwise strong-form diagnostics revealed that enriching the active representation primarily improves interior force balance. In H-mode and X-point cases, global RMS and maximum residual values remained dominated by near-boundary contributions, indicating that the fixed-boundary ansatz cannot fully compensate for boundary-fit mismatches or singular structures (like X-points) through interior shape variations alone.
• Transport Coupling: In an isolated 1D transport-geometry coupling test, the temperature-profile response to VEQ geometry errors remained below approximately 1%.

Significance and Claims
The paper claims that VEQ is suitable for repeated equilibrium-geometry queries in integrated modeling and control workflows, provided that pointwise diagnostics are retained to screen cases requiring boundary refinement, local correction, or higher-fidelity solvers.

The authors position VEQ as complementary to high-fidelity reconstruction solvers, not a universal replacement. Its primary value lies in providing a continuous, compact representation with explicit projected residuals and sampled diagnostics that is inexpensive to regenerate. The paper explicitly states that the timing results reflect "solve-only" latency after setup and should not be interpreted as a replacement for full pipeline timings that include preprocessing, I/O, and discretization. The method is intended to bridge the gap between analytic closures and full solver-native pipelines, offering a balance of geometric richness and computational efficiency.