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.

翻译：不确定性量化（UQ）对工程中的推断至关重要。一项常见的推断任务是从少量含噪观测中恢复物理系统的全场信息，这通常是一个高度不适定问题。共享来自多个不同但相关物理系统的信息可以缓解这种不适定性。关键在于，工程系统往往具有复杂的可变几何结构，这阻碍了标准的多系统贝叶斯UQ的应用。在本工作中，我们提出了用于贝叶斯反演的几何自动编码器（GABI），这是一个学习物理响应的几何感知生成模型的框架，该模型作为贝叶斯反演中信息量丰富的几何条件先验。遵循“先学习，后观测”的范式，GABI从具有不同几何结构的大量系统数据集中提取信息，无需了解控制偏微分方程、边界条件或观测过程，并将其浓缩为一个丰富的潜在先验。在推断时，该先验与特定观测过程的似然无缝结合，生成几何自适应的后验分布。我们提出的框架与架构无关。创造性地使用近似贝叶斯计算（ABC）采样方法，结合现代GPU硬件，实现了高效计算。我们在以下案例上测试了该方法：矩形域上的稳态热传导；翼型周围的雷诺平均纳维-斯托克斯（RANS）流动；三维车身上的亥姆霍兹共振与声源定位；地形上的RANS气流。我们发现：在监督学习适用的受限场景中，预测精度与确定性监督学习方法相当；UQ在具有复杂几何结构的挑战性问题中校准良好且鲁棒。