Towards Stellarator Geometry Optimisation for Nuclear Fusion

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.Source: §1 · p. 1
• 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.Source: ConStellaration Benchmark · p. 1
• 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.Source: ConStellaration Benchmark · p. 1

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.Source: Shape Representation · p. 2
• 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.Source: Shape Representation · p. 2

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++.Source: §2 · p. 2
• Each optimisation run uses 96 concurrent VMEC++ instances per gradient and takes about one hour.Source: §2 · p. 2
• The local refinement improves all previous top-five geometries.Two near-duplicate submissions were merged into one for this comparison.Source: §2 · p. 3, Figure 2 · p. 3
• 115%: four of five refined geometries improve by at least this fraction of the distance between the previous best and fifth-ranked solutions.Source: §2 · p. 3

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.Source: 2D Latent Space Model · p. 3
• 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.Source: 2D Latent Space Model · p. 3

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.Source: Our Novel Solution · pp. 3–4
• All 11 plotted latent-space points are feasible, with white points denoting original benchmark entries and red points denoting refined solutions.Source: Figure 3 · p. 4
• 0.9763: refining the blended geometry beats the benchmark by a margin more than 50% larger than that of the original refined solutions.Source: Our Novel Solution · p. 4

• 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.Source: §3 · p. 4
• 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.Source: §3 · p. 4
• The submitted entry scored 0.9743 and topped the ConStellaration leaderboard in the geometric task.Source: §3 · p. 4
• 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.Source: §3 · p. 4
• 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.Source: §3 · p. 4