Least-elongated stellarator plasma boundary

Find the outer shape of a stellarator plasma whose cross-sections are as round as possible while meeting fixed targets on compactness, triangularity and the twist of the magnetic field lines. It is the entry-level problem of the ConStellaration benchmark, posed to lower the entry barrier to stellarator design for optimization and machine-learning researchers.

Task

A candidate is a stellarator-symmetric plasma boundary, written as a truncated Fourier series in cylindrical coordinates $(R,\phi,Z)$:

$$R(\theta,\phi)=\sum_{m,n}R_{mn}\cos(m\theta-N_{\mathrm{fp}}n\phi),\qquad Z(\theta,\phi)=\sum_{m,n}Z_{mn}\sin(m\theta-N_{\mathrm{fp}}n\phi),$$

where $\theta$ and $\phi$ are poloidal and toroidal angles and $N_{\mathrm{fp}}$, the number of field periods, is also chosen by the designer. The forward model computes the vacuum ideal-MHD equilibrium inside the boundary with VMEC++ and returns the metrics used below. The problem is

$$\min\ \epsilon_{\max}\quad\text{subject to}\quad A\le 4,\qquad \bar\delta\le-0.5,\qquad \lvert\tilde\iota\rvert\ge 0.3,$$

where $\epsilon_{\max}$ is the largest elongation among the boundary cross-sections, $A$ the aspect ratio (major over minor radius), $\bar\delta$ the triangularity averaged over the two symmetric cross-sections $\phi=0$ and $\phi=\pi/N_{\mathrm{fp}}$, and $\tilde\iota$ the edge rotational transform divided by $N_{\mathrm{fp}}$. A design is feasible when every normalized constraint violation $(c_i-c_i^*)/\lvert c_i^*\rvert$ is at most $0.01$. The score is

$$s=\min\Big(1,\ \max\Big(0,\ 1-\frac{\epsilon_{\max}-1}{9}\Big)\Big)$$

for a feasible design and $0$ otherwise; higher is better, and circular cross-sections would score 1.

Background

Stellarators confine a fusion plasma with external coils alone, which shape twisted magnetic surfaces, so they need no large plasma current and avoid current-driven disruptions; the price is a hard, high-dimensional shape-design problem. ConStellaration (Cadena et al., NeurIPS 2025) was released by Proxima Fusion with a dataset of about 158,000 quasi-isodynamic-like boundaries and their equilibria, and three optimization problems of increasing difficulty. This first one is purely geometric: no confinement or stability property is imposed, so it isolates the optimizer and the choice of boundary parameterization, and it shares its forward model with the two physics problems.

Data and evaluation

• Instances: one fixed problem. The dataset and all public leaderboard submissions (results dataset) can serve as starting points.
• Evaluation code: constellaration==0.2.6, problems.GeometricalProblem().evaluate(boundary), which runs VMEC++ at its high-fidelity setting. The leaderboard scores every submission with exactly this release; later releases give different numbers. A submission is one boundary as SurfaceRZFourier JSON.
• Hardware: none assumed; the score does not depend on run time.

What is known

• Paper baseline (Cadena et al., 2025): an augmented-Lagrangian method with the NGOpt optimizer, seeded from a rotating ellipse with three field periods and Fourier modes up to 4, reached $\epsilon_{\max}=1.27$ and a score of 0.969 after 18 hours on 96 vCPUs. SciPy's trust-constr and COBYQA found no feasible design.
• Weißberg et al. (2026) refined the top leaderboard designs with an augmented-Lagrangian loop around L-BFGS, using finite-difference gradients through VMEC++; one refined design scored 0.9763 in their study, and their submitted entry (0.97440 on the leaderboard) took first place in May 2026.
• Best on the leaderboard on 2026-10-20: 0.98048 (user canyon-ant, September 2026), which corresponds to $\epsilon_{\max}\approx1.176$.

What would count as progress

• A boundary scoring above 0.98048 under constellaration==0.2.6, submitted as JSON; the check is a single forward-model evaluation that anyone can rerun.
• A method that reaches at least 0.98 from a rotating-ellipse start, without seeding from published leaderboard designs, with its code.
• A study of how the best achievable $\epsilon_{\max}$ depends on $N_{\mathrm{fp}}$, with the best boundary for each value.

References

1. S. A. Cadena, A. Merlo, E. Laude, et al. ConStellaration: A dataset of QI-like stellarator plasma boundaries and optimization benchmarks. NeurIPS 2025 Datasets and Benchmarks Track. arXiv:2506.19583
2. J. Schilling. The numerics of VMEC++. 2025. arXiv:2502.04374
3. T. Weißberg, M. Heep, Z. Lähner, F. Bernard. Towards stellarator geometry optimisation for nuclear fusion. 2026. arXiv:2608.28224
4. Proxima Fusion. ConStellaration code, release v0.2.6. GitHub

Source

Imported from ConStellaration geometrical problem, curated by Proxima Fusion. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.