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Example-driven Parametrisations for Bayesian Shape Optimisation
Gabriel Diaz-Aylwin, Joseph Neighbor, Abiel Malkani Talwar, Rui-Yang Zhang, Henry B. Moss
TL;DR
Shape optimisation needs compact search spaces for Bayesian optimisation, but hand-crafted parameterisations can require expertise and constrain feasible designs. BOLD learns a linear search space from example deformations and reports improved sample efficiency across aerofoils, wings, and RF cavities compared with hand-crafted baselines.
Problem
Method
BOLD applies PCA to shape displacements from a template, adds linear no-fold constraints, and searches around the best design found so far.
Source: §4 · p. 4, §4.1 · p. 5, §4.2 · p. 5Results
BOLD is more sample-efficient than hand-crafted parametrisations across aerofoils, wings, and cavities, and significantly improves on a carefully hand-crafted RF-Dipole baseline.
Source: §1 · p. 2, §5.3 · p. 8Takeaways & Limitations
BOLD matched the custom CST parametrisation for aerofoils, beat it for wings, and produced RF-Dipole features that the hand-designed CAD model cannot represent.
Source: §6 · p. 8The template fixes topology, exploration is limited to modes in the examples, and local no-fold constraints cannot rule out global self-intersection.
Source: §6 · pp. 8–9
Abstract
from arXiv · showhide
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.
1 INTRODUCTION
Shape optimisation offers costly, often gradient-free objectives for Bayesian optimisation, but applying it requires a compact search space of valid shapes. BOLD learns a linear space from example deformations and reports improved sample efficiency across shape tasks, including gains over hand-crafted baselines.
1 INTRODUCTION
- Shape evaluations require costly simulations or experiments, while black-box solvers frequently lack gradients and realistic meshes may contain 10^5–10^6 vertices.Bayesian optimisation is a sample-efficient strategy for expensive black-box objectives.Source: §1 · p. 1
- Expert parametrisations require domain expertise, often create infeasible regions and artificial bounds, and do not order coordinates by importance.Grid-based PCA can produce invalid shapes, while deep generative models require large datasets and yield nonlinear, uninterpretable latent spaces.Source: §1 · p. 1
- BOLD applies PCA to example shapes’ displacements from a template, producing an interpretable linear search space with no-fold constraints and a component-count trade-off between expressivity and dimensionality.Its trust region is centred on the best design so far and is not clipped to a global box; Figure 1 shows example deformations and Pareto designs.Source: §1 · pp. 1–2, Figure 1 · p. 2
- BOLD is more sample-efficient than hand-crafted parametrisations across aerofoils, wings, and cavities, and significantly improves on a carefully designed RF-Dipole baseline.It also finds designs outside the support of those baselines.Source: §1 · p. 2
2 BACKGROUND
The background introduces Bayesian optimisation and reviews shape representations and prior structure-optimisation approaches. It frames RF-Dipole design as a demanding application because its objectives require expensive 3D electromagnetic simulations and existing parameterisations restrict feasible or expressible designs.
2.1 Bayesian Optimisation
- Bayesian optimisation uses a probabilistic surrogate and acquisition function to choose evaluations; effective acquisition search needs low-dimensional, bounded domains of valid designs.Trust-region and multi-objective variants address high-dimensional and Pareto-front searches, while cheap constraints can be imposed during acquisition.Source: §2.1 · pp. 2–3
2.2 Shape Representations
- Expert-defined representations such as CST are compact but problem-specific, grid-based characteristic and signed distance functions are not closed under linear combinations, and deformation-based methods use correspondences to map shapes from a template.PCA can be applied to displacements or to smooth velocity fields with invertible flows; these deformation parametrisations were established in medical imaging but had not, to the authors’ knowledge, been considered for BO.Source: §2.2 · p. 3
2.3 Bayesian Optimisation over Structures
- Prior shape BO commonly uses expert-defined boxes, grid-function PCA, or deep generative latents (G´omez-Bombarelli et al., 2018; Moss et al., 2025; Fan et al., 2026).The deep latent approaches may require large training sets and can yield geometrically uninterpretable distances and invalid shapes (Bodin et al., 2025).Source: §2.3 · p. 3
2.4 Radio-Frequency Cavity Design
- RF-Dipole design minimises normalised peak electric and magnetic fields subject to 394 MHz resonance, but requires costly 3D electromagnetic simulation and lacks rotational symmetry.Existing CAD parametrisations can fail for some combinations and restrict expressible designs; the passage reports prior cavity optimisation methods taking about 50,000 evaluations, with full-fidelity solves taking an hour of compute.Source: §2.4 · p. 3
3 THE PERILS OF SHAPE PARAMETRISATIONS
Shape optimisation needs compact coordinates for expensive black-box search, but common encodings can produce invalid shapes, jagged objectives, and problematic bounds. BOLD uses deformation-based PCA and local search to address these issues.
3 THE PERILS OF SHAPE PARAMETRISATIONS
- Data-driven shape encodings can be reduced with PCA, but box width remains problematic: smaller boxes can exclude potentially good designs, while c = 3 puts most volume where learned representations need not produce sensible geometry.The whitened examples have mean squared norm p; a uniform draw from [−c, c]^p has expected squared norm pc^2/3.Source: §3 · p. 3, §3.2 · p. 4
3.1 Representations Can Generate Invalid Shapes and Non-smooth Objectives
- Decoding can produce invalid shapes, and implicit representations can make objectives jagged when small coordinate changes alter shape topology.BOLD instead uses deformations of the current best shape and linear constraints so feasible points decode to valid shapes.Source: §3.1 · p. 4
3.2 Search Spaces Need Bounds
- A fixed global box can either include large regions beyond the examples or exclude potentially good designs; BOLD searches in an adaptively sized box around the best shape found so far.This keeps the search near shapes the representation describes well while allowing it to move beyond the examples over successive iterations.Source: §3.2 · p. 4
4 BO BY LOCAL DEFORMATIONS
BOLD learns a linear shape search space from displacements relative to a template, constrains decoded shapes to remain valid, and searches around the current best design. Its correspondence construction uses geometry-specific structure for aerofoils, wings, and RF cavities.
4 BO BY LOCAL DEFORMATIONS
- BOLD represents each shape by displacement from a template, applies PCA to obtain a low-dimensional linear search space, and adds linear no-fold constraints to keep decoded shapes valid.The number of retained components trades expressivity against dimensionality.Source: §4 · p. 4
4.1 A Search Space of Deformations
- PCA orders displacement coordinates by explained variance, while linear no-fold inequalities preserve edge orientation and constrain valid decoded shapes.The margin δ is initialised at 0.2 mm and can be increased adaptively if mesh failures occur despite the constraints holding.Source: §4.1 · p. 5
4.2 Bound-free Search by Re-centring
- The trust region is centred on the incumbent, expands after three consecutive improvements, and contracts after five consecutive failures; clipping to a global box is optional.Unclipped search suits trusted numerical simulators, while global bounds can suit unreliable objectives away from the examples.Source: §4.2 · p. 5
4.3 Constructing Correspondences
- Correspondences should match equivalent geometric features; aerofoils use leading and trailing edges and arc-length samples, while wings use common spanwise sections sampled as aerofoils.Points sharing an index correspond across shapes.Source: §4.3 · pp. 5–6, Figure 4 · p. 6
RF cavities.
- RF cavities are sampled on harmonic-coordinate level-set rings, with points placed uniformly in arc length and ring locations aligned to prominent features such as barrel ends.The number of rings per segment is proportional to its mean length across the dataset.Source: RF cavities · p. 6, Figure 4 · p. 6
5 EXPERIMENTAL RESULTS
The experiments evaluate BOLD on 2D aerofoils, 3D wings, and 3D RF cavities against hand-crafted or CAD-parameter baselines. Results show strong comparisons across the tested problems, with RF-cavity designs also exploring beyond the initial dataset.
5 EXPERIMENTAL RESULTS
- Across three problems, methods share initial designs, evaluation budgets, acquisition functions, and trust-region logic, with performance reported over 30 seeds.Source: §5 · p. 6
5.1 2D Aerofoils
- BOLD recovers most of the best hand-tuned parametrisation’s performance and comes close to the best CST airfoil without a hand-crafted design.Source: Figure 5 · p. 7
5.2 3D Wings
- On 3D wings, BOLD and VF-PCA are statistically indistinguishable and outperform the sectional CST baseline.CST is also sensitive to the Matérn kernel or SWWL surrogate, while BOLD shows little dependence on the kernel.Source: Figure 5 · p. 7
5.3 RF Cavities
- BOLD’s worst of 30 seeds beats CAD BO on both problem types; removing the global box adds a further consistent improvement and reaches beyond the dataset.Every design on the pooled Pareto front has at least one coordinate outside the dataset’s range.Source: §5.3 · p. 7
- RF optima use geometry the hand-crafted parametrisation cannot express, including outward-bulging pole tips and lobed end-caps.Pole bulges add surface area without reducing the aperture; lobed end-caps add end volume and allow frequency tuning without changing the poles.Source: §5.3 · pp. 7–8
- Across aerofoils, wings, and RF cavities, BOLD approaches strong aerofoil baselines, matches VF-PCA on wings while beating CST, and beats CAD BO in every seed.The RF comparison covers both single- and multi-objective problems.Source: Figure 5 · p. 7, §5.3 · p. 7, Figure 6 · p. 8
6 DISCUSSION
BOLD learns a linear shape search space from existing designs and matched or outperformed the stated hand-crafted baselines. Its scope is bounded by template topology and example-supported deformation modes, and its local no-fold constraints do not rule out global self-intersection.
6 DISCUSSION
- BOLD matched custom CST for aerofoils, beat it for wings, and beat BO over hand-designed CAD for the RF-dipole cavity.For the cavity, it also avoided a classifier for failed designs and produced features the CAD model cannot represent.Source: §6 · p. 8
- The template fixes topology, and the search can leave the dataset only along modes present in the examples; correspondences are needed for each shape family.Building correspondences was simple for the families considered.Source: §6 · p. 8
- Neighbour-only no-fold constraints cannot rule out global self-intersection, although none occurred in the experiments.Guarantees of injectivity and automatic global-box selection are left for future work.Source: §6 · p. 9
A SHAPE DEFORMATIONS VIA STATIONARY VECTOR FIELDS
The appendix describes stationary-velocity-field registration used to establish shape correspondences, detailing its B-spline velocity representation, registration objective, coarse-to-fine optimisation, and computational cost.
A SHAPE DEFORMATIONS VIA STATIONARY VECTOR FIELDS
- SVF registration requires a finite-dimensional velocity-field parameterisation, a data term comparing the deformed template with the target, and a regulariser.The appendix uses these choices to describe the registration procedure.Source: A SHAPE DEFORMATIONS VIA STATIONARY… · p. 11
A.1 Velocity field parametrisation
- Tensor-product cubic B-splines on a regular control lattice represent the velocity field, with compact support limiting each point's velocity to 4^d coefficients.The extension to d dimensions uses dS^d coefficients in total.Source: §A.1 · p. 11
A.2 Registration objective
- The registration objective deforms template points by integrating the velocity field, scores agreement with the target using correspondence distances or symmetric Chamfer distance, and penalises rough velocity fields.When correspondences are available, the data term is the sum of squared distances; otherwise it uses the symmetric Chamfer distance, while the regulariser approximates Dirichlet energy.Source: §A.2 · p. 11
A.3 Optimisation
- A coarse-to-fine schedule starts with a coarse lattice, few boundary samples, and large λ, then projects the field to finer stages with more samples and smaller λ.Adam followed by L-BFGS is used at each stage; the authors report more stable optimisation and tighter registrations than direct finest-resolution optimisation.Source: §A.3 · p. 12, Figure 9 · p. 12
A.4 Computational cost
- Registration cost sums per-evaluation flow, distance, and regulariser costs across stages, multiplied by the Adam iterations and L-BFGS evaluations.The flow cost is O(QP4^d d), and the Chamfer term costs O(P log P) with KD-trees or O(P^2) by brute force.Source: §A.4 · p. 12
A.5 Aerofoil example
The aerofoil example gives a three-stage SVF registration setup and describes a mean contour formed by registering the pointwise mean to a dataset of 100 normalised aerofoils.
A.5 Aerofoil example
- The three-stage registration used 8 × 8, 16 × 16, and 32 × 32 control lattices, and reached loss below 10^-6 in around 170 optimiser iterations.It used Q = 10 Runge–Kutta steps and boundary resolutions P = 100, 300, and 500, with λ = 5 × 10^-2, 10^-3, and 10^-5.Source: §A.5 · p. 12
- The mean contour is obtained by registering the pointwise mean of the examples to the whole dataset, rather than optimising P free points directly.The objective minimises the total Chamfer distance to the examples.Source: §A.5 · p. 12
- Figure 10 shows the registered mean of 100 normalised aerofoils in red, with the individual aerofoils in blue.Source: §A.5 · p. 12, Figure 10 · p. 13
B EXPERIMENTAL DETAILS
The experimental details specify the aerofoil and wing datasets, their evaluation pipelines, and a common Bayesian-optimisation budget with trust-region search.
B EXPERIMENTAL DETAILS
- The aerofoil experiments use 1,657 filtered UIUC designs, chord-normalised and resampled to 400 points, with NeuralFoil evaluation at α = 4° and Re = 10^6.Invalid geometries and failed or non-finite evaluations score zero; the objective is L/D multiplied by analysis confidence.Source: B EXPERIMENTAL DETAILS · p. 13
- The wing experiments use all 776 OptiWing3D designs, lofted from 9 to 27 spanwise stations and evaluated with AeroBuildup at fixed α = 4°.The objective is full-wing L/D; wings failing topology checks or returning non-finite aerodynamics score zero.Source: B EXPERIMENTAL DETAILS · p. 13
- Each method runs 300 evaluations—50 initial designs and 250 BO iterations—with a GP and analytic LogEI as the default surrogate.All runs use a TuRBO trust region centred on the incumbent; Sobol uses scrambled sequences over the same CST box.Source: B EXPERIMENTAL DETAILS · pp. 13–14
C SENSITIVITY TO BOX SIZE
Wider bounded search boxes improve aerofoil objectives up to about ±3σ, with gains then saturating in most settings. With unbounded TuRBO, box size mainly sets coordinate scale and starting domain rather than limiting where the search can go.
C SENSITIVITY TO BOX SIZE
- Widening the box from ±1σ to ±3σ markedly improves the final objective in every space and with both optimisers, while gains then saturate except for CST with TuRBO.At ±5σ, PCA spaces show no further improvement, with slightly lower mean and larger spread.Source: Figure 11 · p. 14, C SENSITIVITY TO BOX SIZE · p. 14
- Unbounded TuRBO is largely agnostic to box size because its trust region is not clipped to the box, which mainly sets coordinate scale and starting domain.Runs also show that a tight box excludes better solutions and constrains the search.Source: C SENSITIVITY TO BOX SIZE · p. 14, Figure 12 · p. 15
- 189.2 → 226.2 → 242.2: the best objective rises as the box widens from ±1σ to ±2σ to ±5σ.The winning aerofoils have extreme latent coordinates, often far from the box centre and atypical of the dataset.Source: Figure 13 · p. 15
D SENSITIVITY TO NUMBER OF PCA COMPONENTS
Aerofoil optimisation performs worst with very few retained modes and peaks at intermediate dimensionality in SDF-PCA and DP-PCA. The optimum depends on the shape space and optimiser, while TuRBO often helps at higher dimensions.
D SENSITIVITY TO NUMBER OF PCA COMPONENTS
- Very low-dimensional spaces with 1–3 modes perform worst, while SDF-PCA and DP-PCA peak at 5–9 modes and DP-PCA remains strongest at its optimum.Performance then declines slowly as the larger search space becomes harder to optimise.Source: Figure 14 · p. 16, D SENSITIVITY TO NUMBER OF PCA… · p. 16
- CST-PCA gains up to 7–9 dimensions and, with TuRBO, continues improving up to 15 dimensions.Source: Figure 14 · p. 16
- TuRBO improves final objective in most settings, most visibly at higher dimensions, where it limits the performance loss seen with global Matérn + LogEI.Source: Figure 14 · p. 16
E RF CAVITY DETAILS
The RF-cavity details include its external design specification, interpretable PCA modes, and optimisation-front coordinates. A PALACE-to-CST solver check reports close agreement for Ep/Et and Frequency, with a local discrepancy at larger Bp/Et excluded from subsequent analysis.
E RF CAVITY DETAILS
- Table 1 gives the external design specification for the 394 MHz RFD cavity.Source: Table 1 · p. 16, E RF CAVITY DETAILS · p. 16
- The leading cavity modes represent interpretable bulk motion of the mean design, while trailing modes refine increasingly subtle features.Figure 15 plots the first 20 PCA modes as normal wall displacements, each scaled by one dataset standard deviation.Source: E RF CAVITY DETAILS · pp. 16–17, Figure 15 · p. 17
- BOLD’s pooled Pareto front extends beyond the fixed-bounds ablation’s 6σ cap, exceeding 25σ for mode 18.Figure 16 compares the dataset coordinates with the pooled fronts of BOLD and the fixed-TR ablation.Source: E RF CAVITY DETAILS · p. 17, Figure 16 · p. 17
- PALACE and CST agree with mean absolute percentage errors below 3% for Ep/Et and Frequency, while larger-Bp/Et discrepancies are excluded from subsequent analysis.The discrepancy is attributed to mesh differences in tightly curved geometries; the paper says optimisation conclusions are unaffected.Source: E RF CAVITY DETAILS · p. 17, Figure 17 · p. 18