Abstract:Uncertainty Quantification (UQ) is paramount for inference in engineering. A common inference task is to recover full-field information of physical systems from a small number of noisy observations, a usually highly ill-posed problem. Sharing information from multiple distinct yet related physical systems can alleviate this ill-posedness. Critically, engineering systems often have complicated variable geometries prohibiting the use of standard multi-system Bayesian UQ. In this work, we introduce Geometric Autoencoders for Bayesian Inversion (GABI), a framework for learning geometry-aware generative models of physical responses that serve as highly informative geometry-conditioned priors for Bayesian inversion. Following a ''learn first, observe later'' paradigm, GABI distills information from large datasets of systems with varying geometries, without requiring knowledge of governing PDEs, boundary conditions, or observation processes, into a rich latent prior. At inference time, this prior is seamlessly combined with the likelihood of a specific observation process, yielding a geometry-adapted posterior distribution. Our proposed framework is architecture-agnostic. A creative use of Approximate Bayesian Computation (ABC) sampling yields an efficient implementation that utilizes modern GPU hardware. We test our method on: steady-state heat over rectangular domains; Reynolds-Averaged Navier-Stokes (RANS) flow around airfoils; Helmholtz resonance and source localization on 3D car bodies; RANS airflow over terrain. We find: the predictive accuracy to be comparable to deterministic supervised learning approaches in the restricted setting where supervised learning is applicable; UQ to be well calibrated and robust on challenging problems with complex geometries.
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG); Computational Physics (physics.comp-ph); Data Analysis, Statistics and Probability (physics.data-an) |
| Cite as: | arXiv:2509.19929 [stat.ML] |
| (or arXiv:2509.19929v4 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2509.19929 arXiv-issued DOI via DataCite |
Submission history
From: Arnaud Vadeboncoeur [view email]
[v1]
Wed, 24 Sep 2025 09:38:11 UTC (43,529 KB)
[v2]
Thu, 26 Feb 2026 15:21:16 UTC (44,691 KB)
[v3]
Fri, 27 Feb 2026 10:01:22 UTC (44,691 KB)
[v4]
Wed, 13 May 2026 12:33:14 UTC (44,691 KB)