Example-driven Parametrisations for Bayesian Shape Optimisation

arXiv:2610.11984 · cs.LG · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Example-driven Parametrisations for Bayesian Shape Optimisation".

Jane: The gist The authors propose Bayesian Optimisation by Local Deformation (BOLD), a method that learns a linear,

Tom: First, who's behind it and why it matters.

Paper summary: Tom: We've covered the basic idea of "Example-driven Parametrisations for Bayesian Shape Optimisation." The main thesis here is that instead of guessing the right way to parameterise your shape search space, you can learn it from a dataset of existing designs. The authors claim this leads to better sample efficiency for shape design in areas like aerofoils, wings, and radio-frequency cavities.

Jane: They’re essentially saying that when you look at how shapes deform from a central template shape, applying principal component analysis to those deformations gives you a set of coordinates that describe the most important ways the shape changes.

Lu: The paper points out they use two hundred ninety-nine designs as their dataset to figure this out, and they calculate things like the correspondence average and Pareto extrema for these designs <ref:2610.11984#pg2>. Specifically, they find objectives where one objective is between two and eighty-nine another ratio is one to seventy-eight and the lowest value for Bp divided by Et is between five to forty-three <ref:2610.11984#pg2>.

Meng: That's a lot of specific numbers they're using to define what a good shape looks like in their space. So what does this mean practically for someone who actually has these expensive simulations running?

Lalam: It means the search isn't just a random walk anymore; it’s guided by the data, focusing on the deformations that matter most. The result is a linear search space with constraints added to keep those decoded shapes valid.

Tom: Exactly. So, what are they actually proposing as the core technique here? It’s this combination of PCA for learning the parameterisation and then using a TuRBO-style trust region centered on the best shape found so far to explore beyond just what's in their training data.

Jane: That trust region part is interesting because it means they aren't locked into a fixed box, which addresses one of those big challenges where hand-crafted parametrisations often have infeasible regions that you can't escape.

Lu: They also explicitly mention using linear no-fold constraints during the acquisition phase to keep the shapes valid and prevent them from folding in ways that aren't supposed to happen.

Meng: So, if I’m an engineer looking at this, is this just a clever way to make the initial setup easier, or does it actually lead to designs I wouldn't have found otherwise?

Lalam: The paper shows improved sample efficiency over hand-crafted parametrisations and the ability to explore beyond those hand-crafted baselines.

Tom: Right. So, we’ve seen how they learn the parameterisation from existing shapes and how they use that linear space to search more effectively in aerodynamics, wings, and RF cavities. But before we get into what this means for the future of shape design, let's talk about what the authors themselves are flagging as difficult.

Jane: The paper highlights two major challenges in shape optimisation pipelines right now: first, decoding a point in the search space gives you a new encoding that isn't guaranteed to be a valid shape, and second, acquisition optimisation needs a bounded domain because those hand-crafted parametrisations have those infeasible regions.

Lu: They also note that whitened PCA coordinates still require some kind of bounded box size which can be problematic for the search.

Meng: It sounds like the main roadblock right now is the difficulty in reliably translating a coordinate from their learned space back into a real, physical shape that doesn't self-intersect.

Lalam: And they also mention that the number of dimensions you keep matters; very low-dimensional spaces, like one to three modes, just can't represent good aerofoils and perform worst in every single space studied.

Conclusion: Tom: So we’ve gone through the core ideas of "Example-driven Parametrisations for Bayesian Shape Optimisation." To wrap up the main points, this work focuses on learning a linear, interpretable search space directly from existing shapes using PCA on their deformations. The authors show this approach improves sample efficiency across 2D aerofoils, three dee wings, and three dee RF cavities compared to hand-crafted methods <ref:2610.11984#pg1>.

Jane: The title itself suggests the key innovation is using examples to drive the parameterisation process instead of relying on expert knowledge alone for those complex shape descriptions. It moves away from defining the search space upfront.

Lu: The implication here is that you can bypass the incredibly complex, problem-specific construction of user-defined parametrisations, which are often just based on a single CAD model or a specific family like CST for aerofoils.

Meng: For practical application, this means an engineer doesn't have to spend months building that initial search space from scratch; they can start with a collection of existing designs and let the AI figure out the structure of that space.

Lalam: And for the broader field, it shows how learning a linear structure through data analysis like PCA can unlock exploration beyond what traditional methods allow, especially in complex areas like RF cavities where they found features that hand-crafted models couldn't express.

Tom: So, this paper suggests that example-driven learning of parameterisation is a viable path for shape design when the search space itself is the hard part to define and is non-differentiable. We saw how it beats hand-designed CAD models in some cases and performs well where other methods struggle with feasibility.

Jane: It's about shifting the complexity from manual construction of geometry descriptions to learning those descriptions from data, which seems like a big step for general shape optimisations.

Lu: The limitations they mentioned are pretty clear: they still have to deal with decoding validity and managing the bounded box size for acquisition optimisation, which points toward future work needing better constraint handling.

Meng: So, it’s not a complete solution yet; there’s still work needed on making the constraints robust enough so that those decoded shapes are always physically sensible without constant manual checks.

Lalam: The paper shows that even with these limitations, this method provides a much more efficient and flexible way to explore the design space for various applications.

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

Lancaster University

cs.LG

Submitted: 2026-10-08

Updated: 2026-10-08

Code: https://github.com/peterdsharpe/AeroSandbox

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 81/100

The gist: The gist The authors propose Bayesian Optimisation by Local Deformation (BOLD), a method that learns a linear, interpretable search space from existing designs using Principal Component Analysis to

Key concepts

Deformation-based representations: This approach describes a shape not by its absolute parameters, but by how it deforms from a simple template shape. This is done using displacement fields. It's cheap to compute and allows for handling validity through linear constraints, making it useful for complex 3D shapes.
Principal Component Analysis (PCA) in Shape Design: PCA is used to analyze the differences (deformations) between a collection of existing shapes. By finding the principal components of these deformations, the authors learn a low-dimensional, linear search space. The coordinates in this new space are ordered by how much variance they explain, making the search more structured.
Trust Region Search (TuRBO-style): The optimization process uses a trust region centered around the best shape found so far. This method allows the optimizer to explore areas outside the initial training data's support. It is effective for searching complex spaces where hand-crafted bounds might be infeasible or restrictive.
Linear No-Fold Constraints: These are constraints added during optimization to ensure that shapes remain valid, specifically preventing self-intersections or 'folded' geometries. They enforce that the orientation of edges on the correspondence grid remains consistent with the original template shape.

Summary

The gist The authors propose Bayesian Optimisation by Local Deformation (BOLD), a method that learns a linear, interpretable search space from existing designs using Principal Component Analysis to improve sample efficiency for shape design in fields like aerofoils, wings, and radio-frequency cavities.

How it works

BOLD learns the parameterisation by applying PCA to the deformations between shapes, resulting in a linear search space where coordinates are ordered by explained variance The search is conducted using a TuRBO-style trust region centred on the best design so far, which allows it to explore beyond the confines of hand-crafted baselines This approach is more sample-efficient than BO over hand-crafted parametrisations and can find designs outside their support

Shape Representations

The paper reviews three representation families for shape analysis:

: User-defined parametrisations describe shapes by parameters chosen by an expert, which are compact but problem-specific. 2.

: Implicit field representations encode a shape by its characteristic function (CF) or signed distance function (SDF), which can be discretized on a fixed grid. 3.

: Deformation-based representations represent each shape by the transformation that maps a shared template onto it, such as displacement fields, which are cheap to compute and handle validity separately through linear constraints. 4.

Challenges in Shape Parametrisations

The authors identify two pathological issues with current pipelines for shape optimisation:

: Decoding a point in the search space returns a new encoding, but these encodings are not guaranteed to yield a valid shape, and implicit representations also make the objective harder to model. 5.

: Acquisition optimisation needs a bounded domain, as hand-crafted parametrisations suffer from infeasible regions or self-intersecting geometries, and whitened PCA coordinates still require a bounded box size which is problematic. 6.

BOLD Methodology

BOLD addresses these issues by:

Representing each shape by its displacement from a template shape and applying PCA to these displacements to get a low-dimensional linear search space. 7.
Adding linear no-fold constraints to keep decoded shapes valid, preventing folded shapes by requiring that each edge e of the correspondence grid keeps its orientation on the template. 8.
Searching this space with a trust region centred on the best shape found so far, which avoids fixing a global box and allows for exploration beyond training data. 9.

Experimental Results

BOLD was evaluated on three problems: 2D aerofoils, 3D wings, and 3D RF cavities. For the RF cavity design, BOLD outperformed BO over a hand-designed CAD model and produced features that the CAD model could not represent. In 3D wing optimisation, BOLD was found to be statistically indistinguishable from VF-PCA and outperformed CST. For aerofoils, BOLD recovered most of the performance of the best hand-crafted parametrisation and beat it for wings. Furthermore, BOLD's optima for RF cavities were physically sensible and relied on features that the hand-crafted parametrisation could not express. 3.

Sensitivity to Search Space

The effectiveness of the search space size is significant; widening the box from ±1σ to ±3σ 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 ±3σ the gains saturate, with PCA spaces showing no further improvement, while CST with TuRBO continues to improve up to ±5σ. The search box size also affects how often the best solution lies outside the box. 4.

Sensitivity to PCA Components

The number of retained dimensions impacts performance; 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. 5.

Experimental Details

The experiments used a budget of T = 300 evaluations for each seed, with initial designs drawn without replacement from dataset designs lying inside the search box. The default surrogate is a GP with an ARD Matérn-5/2 kernel, and the acquisition function is multiplied by the resulting probability of feasibility for constrained BO. 6.

Conclusion

BOLD matched the custom CST parametrisation for aerofoils and beat it for wings, and for the RF-dipole cavity, it beat BO over a hand-designed CAD model while producing features the CAD model cannot represent. 7. The topology is fixed by the template, and constraints that guarantee injectivity are left for future work. 8.

References

Abdul Khalek, R., Accardi, A., Adam, J., Adamiak, D., Akers, W., Albaladejo, M., Al-Bataineh, A., Alexeev, M., Ameli, F., Antonioli, P., et al. (2022). Science requirements and detector concepts for the electron-ion collider: Eic yellow report. Nuclear Physics A, 1026:1–902

Arsigny, V., Commowick, O., Pennec, X., and Ayache, N. (2006). A Log-Euclidean Framework for Statistics on Diffeomorphisms. In Medical Image Computing and Computer-Assisted Intervention – MICCAI 2006, volume 4190 of Lecture Notes in Computer Science, pages 924–931

Beg, M. F., Miller, M. I., Trouv´e, A., and Younes, L. (2005). Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 61(2):139–157

Bendsoe, M. P. and Sigmund, O. (2013). Topology Optimization: Theory, Methods, and Applications. Springer Science & Business Media

Bodin, E., Stere, A., Margineantu, D., Ek, C., and Moss, H. (2025). Linear combinations of latents in generative models: Subspaces and beyond. In International Conference on Learning Representations, volume 2025, pages 8014–8043

Chen, R., Zhang, J., Liang, Y., Luo, G., Li, W., Liu, J., Li, X., Long, X., Feng, J., and Tan, P. (2025). Dora: Sampling and benchmarking for 3D shape variational auto-encoders. In 2025 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 16251–16261

Chhantyal, B., Khanal, S., and Maharjan, S. (2025). Comparison of particle swarm optimization and Bayesian optimization algorithms for the shape optimization of the star crash box. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 239(10-11):4595–4607

Cootes, T. F., Taylor, C. J., Cooper, D. H., and Graham, J. (1995). Active shape models-their training and application. Computer Vision and Image Understanding, 61(1):38–59

Dassault Syst´emes (2025). CST Studio Suite. https://www.3ds.com/products/simulia/cst-studiosuite

Daulton, S., Eriksson, D., Balandat, M., and Bakshy, E. (2022). Multi-objective Bayesian optimization over high-dimensional search spaces.

Improvements for AI systems

textbf Learned Parametrization for Expensive Shape Design (BOLD): The system can learn a linear, interpretable search space from existing designs by applying principal component analysis to the deformations between shapes, allowing it to explore beyond hand-crafted baselines with improved sample efficiency and the ability to explore beyond the confines of hand-crafted baselines.
textbf Adaptive Search Strategy: Bound-Free Exploration: The system implements a strategy where we focus on an adaptively-sized box around the best shape found so far, moving beyond fixed global bounds, which allows the search to move far from the training data while ensuring proposals are near a shape that has already been evaluated.
textbf Robust Validity Guarantee: No-Fold Constraints: The system incorporates linear no-fold constraints to ensure decoded shapes remain valid, preventing issues like self-intersection, by requiring that each edge e of the correspondence grid keeps its orientation on the template, which reduces to linear inequalities Az ≥ b imposed directly during acquisition optimisation.
textbf Enhanced Aerodynamic Design: Superior Airfoil Optimization: The system can match or beat hand-crafted parametrisations for 2D aerofoils, as demonstrated by recovering most of the performance of the best hand-tuned parametrisation and finding optima that require quite extreme, large coordinates in the latent space.
textbf Complex Electromagnetic Design: RF Cavity Optimization: The system can optimize complex 3D RF-dipole cavities by learning a search space directly from examples, achieving results that outperform BO over a hand-designed CAD model and identifying key geometric changes, which will guide future CAD parametrisations of RF cavities.
textbf Flexible Dimensionality Scaling: Sensitivity to PCA Components: The system can dynamically adjust its search complexity by showing that performance peaks at an intermediate dimension (around 9 for SDF-PCA and DP-PCA) rather than increasing with dimensionality, optimizing the trade-off between expressivity against dimensionality.

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.