Example-driven Parametrisations for Bayesian Shape Optimisation

Gabriel Diaz-Aylwin; Joseph Neighbor; Abiel Malkani Talwar; Rui-Yang Zhang; Henry B. Moss

cs.LGOct 8, 2026

Example-driven Parametrisations for Bayesian Shape Optimisation

Authors: Gabriel Diaz-Aylwin, Joseph Neighbor, Abiel Malkani Talwar, Rui-Yang Zhang, Henry B. Moss

Organizations: Lancaster University

Abstract

Bayesian optimisation is the natural tool for shape design when objectives are expensive and non-differentiable, but it needs a compact yet expressive parameterisation of the search space. Hand-crafting one is a complex endeavour requiring domain expertise, and often yields implicit infeasible regions, artificial bounds, and coupled, unordered coordinates. We instead learn the parameterisation from a collection of existing designs, applying principal component analysis to the deformations between shapes. The result is a linear, interpretable search space in which the number of components explicitly trades expressivity against dimensionality. Across aerofoils, wings, and radio-frequency cavities, spanning 2D geometry to 3D aerodynamics and electromagnetics, we show improved sample efficiency and the ability to explore beyond the confines of hand-crafted baselines.

Figures & tables

Figure 1: BOLD’s RF cavity designs, read left to right. (a) Some example cavities from our dataset, shaded by their normal displacement field from the correspondence mean (red outwards, blue inwards). These, in essence, are the fields we run PCA on. (b) The correspondence mean of the dataset (grey). We further draw $y=0$ cross-sections along the straight line in shape coordinates interpolating from the mean to three designs on BOLD’s Pareto front. (c) The Pareto designs with their surface $|\mathbf{E}|$ : the two ends of the front, which minimise (i) $E_{p}/E_{t}$ (objective 1) and (iii) $B_{p}/E_{t}$ (objective 2), and (ii) the design whose ratio $B_{p}/E_{p}$ matches a target specification.

Panel of Figure 2 — Figure 2: We plot an objective function landscape (lift-to-drag ratio) across the leading components for a variety of shape representations. In black, we denote the region in which decoding linear combinations of the leading modes yields an invalid aerofoil. Note that the landscape is jagged for the SDFs (a), has clear infeasible regions (black) for the expert-designed CST parametrisation (b) and DFs (c), but is well-behaved for the SVFs (d), since these preserve injectivity by construction. In (c), we further plot DF with our no-fold constraints (white lines), which approximate the infeasible region by linear inequalities and the DFs’ infeasible boundary (red).

Panel of Figure 3 — Figure 3: Two representations of the deformation between a pair of aerofoils. Points of the same colour correspond. (a) A stationary velocity field (grey arrows), defined over the whole plane, whose flow carries one aerofoil onto the other. (b) A displacement field, which moves each point directly to its corresponding point (lines).

Panel of Figure 4 — Figure 4: Structured grids used to build correspondences. (a) A wing sampled at common spanwise sections, each sampled uniformly in arc length. (b) An RF cavity gridded by level sets of the harmonic coordinate $h$ (rings, coloured by $h$ ) and arc length along each ring. In both cases, points with the same grid index correspond across all shapes in the dataset.

Figure 5: Aerodynamic shape optimisation. (a) For 2D aerofoils, BOLD recovers most of the performance of the best hand-tuned parametrisation. (b) BOLD comes close to the best CST airfoil without needing a hand-crafted design. (c) 3D wing optimisation: VF-PCA and BOLD (first 15 modes) and a hand-designed sectional CST parameterisation. BOLD and VF-PCA are statistically indistinguishable and outperform CST, which is also sensitive to the choice of Matérn kernel or SWWL surrogate. BOLD shows little dependence on the kernel.

Panel of Figure 6 — Figure 6: RF cavity optimisation. (a) Best in-band $E_{p}/E_{t}$ (lower is better) and (c) in-band hypervolume against evaluations, showing medians, quartiles and the 30 seeds. BOLD outperforms CAD BO in every seed. (b) The dataset and the single-objective optima projected onto their two leading principal components. BOLD is able to explore far beyond the initial shapes. (d) Pooled in-band Pareto fronts, with the dataset for reference. The front found by BOLD using EHVI dominates both baselines.

Figure 7: The nine-parameter CAD model of the RF-dipole cavity used to generate the initial shape dataset and BO baselines. Cavity length $L$ , the pole width $W_{\mathrm{pole}}$ and length $L_{\mathrm{pole}}$ , the transverse radii $Rc_{x}$ and $Rc_{y}$ , and the inner and outer blend radii are labelled. Red arrows mark where the peak surface fields occur: $E_{p}$ on the pole tips and $B_{p}$ at the pole–body junction. The green dashed line marks the 100 mm aperture, which must be preserved by all designs.

Panel of Figure 8 — Figure 8: The lowest- $E_{p}/E_{t}$ design of the single-objective sweep ( $E_{p}/E_{t}$ 2.73, $B_{p}/E_{t}$ 13.65). As in Figure 1 , we overlay its $|\textbf{E}|$ field. Below, we add its two most outlying shape coordinates, shaded by their normal displacement field from the correspondence mean (red outwards, blue inwards).

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9: SVF registration between two aerofoils. Top left: source and target after chord normalisation and arc-length resampling. Bottom left: the learnt velocity field with the source, target and deformed source. Right: the loss on a log scale, with dashed lines marking transitions between coarse-to-fine stages.

Figure 10: Mean shape of 100 normalised aerofoils (red), computed by registering the pointwise mean to the dataset, with the individual aerofoils in blue.

Figure 11: Final best objective after

T=300

evaluations as the search box grows from

\pm 1\sigma

\pm 5\sigma

. Widening the box from

\pm 1\sigma

\pm 3\sigma

markedly improves the final objective in every space and with both optimisers, since larger ranges admit more extreme shapes that lie outside the bulk of the dataset. Beyond

\pm 3\sigma

the gains saturate: the PCA spaces show no further improvement, with a slightly lower mean and larger spread at

\pm 5\sigma

as the search volume grows, whereas CST with TuRBO continues to improve up to

\pm 5\sigma

Panel of Figure 12 — Figure 12: Left: Best-so-far objective against the number of evaluations for Matérn + LogEI with an unbounded (may leave the box) TuRBO trust region on 9D DP-PCA, for search boxes of half-width $\pm 1\sigma$ to $\pm 5\sigma$ . Lines show the median over 50 seeds and shaded bands the interquartile range; each run uses 50 shared initial aerofoils and 300 evaluations in total. As expected, unbounded TuRBO is largely agnostic to the box size as its trust region is not clipped to the box, so the box serves mainly to set the coordinate scale and starting domain and does not limit where the search can go. Right: Share of runs whose best solution lies outside the search box, as a function of the box half-width. Runs use Matérn + LogEI with an unbounded TuRBO trust region on 9D DP-PCA, with 50 seeds, 50 initial and 300 total evaluations per run. We see that a tight box excludes better solutions and constrains the search.

Figure 13: Best aerofoils found by Matérn+TuRBO under search boxes of

\pm 1\sigma

\pm 2\sigma

and

\pm 5\sigma

. The best objective value is given in parentheses above each panel. Widening the box improves the best objective (

189.2\to 226.2\to 242.2

) - the winning shapes are far from the typical atlas aerofoil. As in the earlier ablations, the best foils once again require quite extreme, large coordinates in the latent space, often far from the centre of the box, so the achievable objective depends strongly on how large a region we allow.

Figure 14: Final best objective after

T=300

evaluations as a function of the number of retained dimensions, all in a

\pm 3\sigma

box. Boxes show the median and interquartile range over 50 seeds, whiskers the range of non-outlying values, triangles the mean and circles outliers. Very low-dimensional spaces (1–3 modes) cannot represent good aerofoils and perform worst in every space. Performance then rises sharply and peaks at intermediate dimensions (5–9 modes for SDF-PCA and DP-PCA), after which it declines slowly as the larger search space becomes harder to optimise, while DP-PCA remains the strongest space at its optimum. CST-PCA gains up to 7–9 dimensions and, with TuRBO, continues to improve up to 15. TuRBO improves the final objective in most settings, most visibly at higher dimensions, where it limits the loss of performance seen with global Matérn + LogEI.

Parameter	Value/Unit
Frequency $f_{0}$	$394\text{\,}\mathrm{MHz}$
Transverse voltage $V_{t}$	$3.5\text{\,}\mathrm{MV}$
$E_{p}/E_{t}$	4.895
$B_{p}/E_{t}$	$8.702\text{\,}\mathrm{mT}\text{\,}{\mathrm{}}^{-1}$
Aperture diameter	$100\text{\,}\mathrm{mm}$

Appendix

Table 1: Design specification for the 394 MHz RFD cavity ( Abdul Khalek et al., 2022 )

Figure 15: The 20 shape modes of the cavity dataset, each drawn as the wall displacement it produces on the mean design per one dataset standard deviation, normal to the wall (red outwards, blue inwards; each panel on its own colour scale), with the correspondence grid.

Figure 16: Shape coordinates of the dataset and of the pooled Pareto fronts of BOLD and the fixed-TR ablation.

Figure 17: Comparison of PALACE finite-element solver with CAD solver (CST Microwave Studio) for a total of 130 solves. Optima from the single objective optimisation runs minimising

E_{p}/E_{t}

are shown in blue while Pareto-optimal solutions from the multiobjective runs are shown in orange.

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