MHD-stable QI stellarators: compactness versus simple coils

Map the best trade-off between a compact stellarator and one that can be built with simple coils, among designs that are quasi-isodynamic, stable to pressure-driven instabilities and shaped to limit turbulence. It is the most complex of the three ConStellaration problems and adds the stability and turbulence constraints that reactor designs need.

Task

A submission is a set of stellarator-symmetric plasma boundaries, each given by Fourier coefficients $R_{mn}$, $Z_{mn}$ and a number of field periods $N_{\mathrm{fp}}$ as in the geometric problem. For each boundary the forward model computes the vacuum equilibrium with VMEC++ and two objectives: the normalized magnetic-field gradient scale length $\tilde L_{\nabla B}$, to be maximized, and the aspect ratio $A$, to be minimized. A boundary is feasible when

$$\lvert\tilde\iota\rvert\ge0.25,\qquad \log_{10}\mathit{QI}\le-3.5,\qquad \Delta\le0.25,\qquad W_{\mathrm{MHD}}\ge0,\qquad \langle\chi_{\nabla r}\rangle\le0.9,$$

each up to a normalized violation of $0.01$ (the vacuum-well violation is divided by $0.1$). Here $\tilde\iota$ is the edge rotational transform per field period, $\mathit{QI}$ the quasi-isodynamicity residual, $\Delta$ the mirror ratio on the boundary, $W_{\mathrm{MHD}}$ the vacuum magnetic well (a proxy for ideal-MHD stability), and $\langle\chi_{\nabla r}\rangle$ the flux-surface average, on the surface $r/a=0.7$, of the flux compression $\lVert\nabla r\rVert^2$ in regions of bad curvature (a geometric proxy for turbulent transport). The score is the hypervolume dominated by the feasible boundaries in the plane $(-\tilde L_{\nabla B},A)$ with reference point $(1,20)$:

$$\mathrm{HV}=\operatorname{area}\Big(\bigcup_{i\ \mathrm{feasible}}[-\tilde L_i,\,1]\times[A_i,\,20]\Big).$$

Higher is better and there is no fixed maximum.

Background

Quasi-isodynamic fields remove current-driven disruptions but not pressure-driven instabilities, and turbulence driven by ion temperature gradients limits fusion gain. Compact devices (low $A$) cost less per unit of power, but in quasi-isodynamic designs the gradient scale length is roughly proportional to the aspect ratio, so compactness pushes coils closer to the plasma and makes them harder to build. Cadena et al. (2025) therefore posed a Pareto search rather than a single objective. The paper's own baseline solved a sequence of single-objective problems with $A\le A^*$ for $A^*\in\{6,8,10,12\}$.

Data and evaluation

• Instances: one fixed problem. Starting points: the dataset and every public submission in the results dataset, which stores the boundaries.
• Evaluation code: constellaration==0.2.6, problems.MHDStableQIStellarator().evaluate(boundaries). Release 0.3.0 computes the flux compression with a different Jacobian, which changes feasibility, so it is not comparable; the leaderboard briefly used it from 2026-10-02 to 2026-10-08 and then re-evaluated the affected entries with 0.2.6.
• Hardware: none assumed; the score does not depend on run time. Cost grows with the number of boundaries, one VMEC++ run each.

What is known

• Paper baseline (Cadena et al., 2025): hypervolume 130.0, from a sparse front with points such as $(A,\tilde L_{\nabla B})=(7.93,5.60)$, $(9.98,8.45)$ and $(11.9,11.1)$; each run took 15 to 24 hours on 96 vCPUs.
• The leaderboard has moved fast in 2026: about 181 in June, 205.8 in September, and the best on 2026-10-20 is 248.23 (user Wenyang Li, 2026-10-18), ahead of 245.16 (phanerozoic) and 242.94 (mmdeuss).
• In that best set of 783 boundaries, all feasible, the most compact design has $A\approx6.05$ with $\tilde L_{\nabla B}\approx3.3$; the best $\tilde L_{\nabla B}$ is about 8.2 at $A\le8$ and 11.1 at $A\le10$.

What would count as progress

• A set of boundaries with hypervolume above 248.23 under constellaration==0.2.6; it is checked by re-running the evaluation.
• A feasible boundary with $A<6$, which no public submission has, or a higher $\tilde L_{\nabla B}$ at fixed $A$ (for example at $A=8$).
• Evidence that top designs keep their properties at finite plasma pressure, since the benchmark scores vacuum equilibria only.

References

1. S. A. Cadena 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. Kappel, M. Landreman, D. Malhotra. The magnetic gradient scale length explains why certain plasmas require close external magnetic coils. Plasma Physics and Controlled Fusion 66, 025018 (2024). doi:10.1088/1361-6587/ad1a3e
3. A. G. Goodman et al. Constructing precisely quasi-isodynamic magnetic fields. Journal of Plasma Physics 89, 905890504 (2023). doi:10.1017/S002237782300065X
4. Proxima Fusion (scadena-pf). docs: pin leaderboard benchmark to v0.2.6 (v0.3.0 changes MHD-stable feasibility). ConStellaration pull request #127, October 2026.

Source

Imported from ConStellaration MHD-stable QI 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.