Source-linked AI summary
Towards Stellarator Geometry Optimisation for Nuclear Fusion
Tobias Weißberg, Moritz Heep, Zorah Lähner, Florian Bernard
TL;DR
Stellarator optimisation relies on geometry representations whose global Fourier coefficients make local edits difficult. This paper combines local gradient-based refinement with a nonlinear 2D latent representation, producing a submitted score of 0.9743 that topped the May 2026 leaderboard snapshot.
Problem
Fourier surface representations are smooth and simulation-compatible but make local geometric edits difficult because individual coefficient changes affect the boundary globally.
Method
The paper replaces CMA-ES with parallel finite-difference L-BFGS refinement and constructs a nonlinear 2D latent space using MDS and mean-value-coordinate interpolation.
Results
A submitted entry scored 0.9743 and topped the leaderboard snapshot, while refining a sampled blended geometry reached 0.9763 and exceeded every entry in that snapshot.
Takeaways & Limitations
Refining existing top-five geometries and sampling their induced latent subspace revealed improvements beyond the original solutions.
Takeaways & Limitations
The refinement is local by construction, evaluation covered only one of three tasks, and leaderboard comparisons use a May 20, 2026 snapshot that has since advanced.
Abstract
from arXiv · showhide
We present a local geometry refinement method and a 2D latent representation that took us to the top of the ConStellaration leaderboard on the geometric task in May 2026.
1 Background on Stellarator Optimisation
Stellarator optimisation designs plasma-boundary shapes whose magnetic fields confine extremely hot plasma. ConStellaration standardises this geometric task, while Fourier surface representations offer compactness but impose global and non-unique parameterisations.
- 1 Background on Stellarator Optimisation: Stellarators confine plasma through magnetic-field shape alone, making plasma-boundary optimisation central because the boundary determines the internal magnetic field.Evaluating a boundary requires numerically solving the magnetohydrodynamic equilibrium.
- 1 Background on Stellarator Optimisation: ConStellaration provides approximately 158k high-fidelity simulations and standardised benchmarks, including a geometric task with four boundary and equilibrium metrics.The geometric task minimises elongation while constraining aspect ratio, average triangularity, and rotational transform.
- 1 Background on Stellarator Optimisation: Only rotational transform among the four geometric-task metrics requires equilibrium evaluation; elongation, aspect ratio, and average triangularity are boundary-surface properties.Average triangularity must be ≤−0.5.
- Shape Representation.: Truncated double Fourier series represent toroidal surfaces compactly and smoothly, and are directly consumed by available physics simulators.Truncation also supplies a natural coarse-to-fine hierarchy.
- Shape Representation.: Fourier coefficients parameterise surfaces globally and non-uniquely, so local edits require coordinated spectral changes and equivalent geometries can have distinct coefficients.Spectral condensation seeks canonical representations but uses non-convex minimisation and need not vary continuously with geometry.
2 Our Geometry Optimisation Method
The paper refines existing top-ranked ConStellaration geometries with local gradient-based optimisation. Applied to the previous top-five entries, this refinement improved every geometry and substantially widened the lead over the prior solutions.
- 2 Our Geometry Optimisation Method: The method replaces CMA-ES with local gradient-based L-BFGS inside an augmented-Lagrangian loop for refining existing leaderboard solutions.Gradients across the 306-dimensional input space use parallel forward two-point finite differences through VMEC++.
- 2 Our Geometry Optimisation Method: Each optimisation run uses 96 concurrent VMEC++ instances per gradient and takes about one hour.
- 2 Our Geometry Optimisation Method: The local refinement improves all previous top-five geometries.Two near-duplicate submissions were merged into one for this comparison.
- 2 Our Geometry Optimisation Method: 115%: four of five refined geometries improve by at least this fraction of the distance between the previous best and fifth-ranked solutions.
2D Latent Space Model.
A two-dimensional latent slice embeds benchmark and refined geometries while preserving their coefficient-space relationships and enabling smooth interpolation. Its landscape exposes feasible high-scoring regions, including a blended geometry that surpasses the known solutions.
- 2D Latent Space Model.: MDS embeds solutions in two dimensions, while mean value coordinates map latent positions back to Fourier coefficients through a closed polygon of ten embedded solutions.The map reproduces each solution at its own position and traces a curved slice through coefficient space.
- 2D Latent Space Model.: The latent feasible region is connected but contains infeasible pockets, while the objective varies smoothly and its best values form a band beyond the refined solutions.The triangularity metric is described as discontinuous.
- Our Novel Solution.: 0.9752: a novel blended geometry is feasible and exceeds every known solution in the benchmark snapshot.It was found by sampling the nonlinear latent subspace generated by leaderboard entries and refinements.
- Our Novel Solution.: All 11 plotted latent-space points are feasible, with white points denoting original benchmark entries and red points denoting refined solutions.
- Our Novel Solution.: 0.9763: refining the blended geometry beats the benchmark by a margin more than 50% larger than that of the original refined solutions.
3 Summary and Conclusion
The paper combines local refinement with a low-dimensional latent representation to improve ConStellaration geometric solutions. Refinement improved all top-five entries, while latent-space sampling produced a geometry that topped the leaderboard; the method remains local and evaluation covered only one task.
- The proposed refinement method improves existing ConStellaration geometric solutions, while the latent representation supports visualisation and sampling.The method targets leaderboard solutions; the representation is based on MDS and mean-value coordinates.
- Refinement improved all top-five entries, and latent-space sampling revealed a better geometry that was subsequently refined.The refined geometry scored 0.9763, exceeding every entry in the authors’ snapshot.
- The submitted entry scored 0.9743 and topped the ConStellaration leaderboard in the geometric task.
- The refinement method is local by construction, inherits its starting basin, and requires solutions worth refining.The authors state that this locality prevented independent discovery of the best known solution.
- The evaluation was restricted to one of ConStellaration’s three benchmark tasks.The authors identify extending coverage to the remaining tasks as a natural continuation.