Active Subspace-Guided Free-Form Deformation with Sinkhorn Autoencoders for Reduced-Order Modelling of Parametrised Shape Problems
Abstract
Geometrically parametrised PDEs arise in shape optimisation, where solving a PDE over many deformed domains at manageable cost is a central challenge. Free-form Deformation (FFD) parametrises shape variations through a lattice of control points, but the resulting parameter space is typically high-dimensional, with most directions barely affecting the quantity of interest (QoI). We address this with a two-stage reduction pipeline. An Active Subspace (AS)-guided structured control-point selection identifies the low-dimensional subspace of FFD weights driving the QoI. A Sinkhorn Autoencoder (SAE) is then trained on the reduced weights, learning their distribution in a compact latent space by minimising the Wasserstein distance between aggregated posterior and prior. We compare three non-intrusive Reduced-order Models (ROMs), mapping full FFD weights, AS-reduced weights, or SAE latent codes to the QoI, each using Random Forests, Gaussian Processes, and K-Nearest Neighbours. We test the methodology on a non-linear heat equation with Gaussian random Robin boundary data on a deformed Stanford Bunny, and on drag prediction for the DTC Hull bulb under FFD deformations. In both cases, the generative model captures the bulk of the QoI distribution over a comparable support, thinning the shoulder opposite the design objective, while its extremes in the design-relevant direction match the FFD sample within numerical and sampling uncertainty. ROMs learned in the SAE latent space yield smaller test errors than those on full FFD weights in most regressor-metric combinations, whereas AS-reduced weights give no comparable gain. Only the latent representation attains a positive predictive coefficient Q2.
Related
- Multivariate stability of powered triangular recurrences and parametrised Eulerian polynomials
- An asymptotic-preserving reduced-order method for parametrised rarefied gas flow by proper generalised decomposition
- Exponential Convergence of the Sinkhorn Algorithm for the Schrödinger Bridge with Regime Switching
- Wasserstein mirror gradient flow as the limit of the Sinkhorn algorithm
- Tight Nonasymptotic Local Convergence of Sinkhorn-Knopp
BibTeX
@misc{padula2026active,
title = {{Active Subspace-Guided Free-Form Deformation with Sinkhorn Autoencoders for Reduced-Order Modelling of Parametrised Shape Problems}},
author = {G. Padula and C. Giovannini and G. Rozza and A. Dashtimanesh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32373},
eprint = {2609.32373},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32373},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Active Subspace-Guided Free-Form Deformation with Sinkhorn Autoencoders for Reduced-Order Modelling of Parametrised Shape Problems
AU - G. Padula
AU - C. Giovannini
AU - G. Rozza
AU - A. Dashtimanesh
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32373
UR - https://arxiv.org/abs/2609.32373
AB - Geometrically parametrised PDEs arise in shape optimisation, where solving a PDE over many deformed domains at manageable cost is a central challenge. Free-form Deformation (FFD) parametrises shape variations through a lattice of control points, but the resulting parameter space is typically high-dimensional, with most directions barely affecting the quantity of interest (QoI). We address this with a two-stage reduction pipeline. An Active Subspace (AS)-guided structured control-point selection identifies the low-dimensional subspace of FFD weights driving the QoI. A Sinkhorn Autoencoder (SAE) is then trained on the reduced weights, learning their distribution in a compact latent space by minimising the Wasserstein distance between aggregated posterior and prior. We compare three non-intrusive Reduced-order Models (ROMs), mapping full FFD weights, AS-reduced weights, or SAE latent codes to the QoI, each using Random Forests, Gaussian Processes, and K-Nearest Neighbours. We test the methodology on a non-linear heat equation with Gaussian random Robin boundary data on a deformed Stanford Bunny, and on drag prediction for the DTC Hull bulb under FFD deformations. In both cases, the generative model captures the bulk of the QoI distribution over a comparable support, thinning the shoulder opposite the design objective, while its extremes in the design-relevant direction match the FFD sample within numerical and sampling uncertainty. ROMs learned in the SAE latent space yield smaller test errors than those on full FFD weights in most regressor-metric combinations, whereas AS-reduced weights give no comparable gain. Only the latent representation attains a positive predictive coefficient Q^2.
ER -