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）在工程推断中至关重要。一项常见的推断任务是从少量含噪声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。共享多个不同但相关的物理系统间的信息可以缓解这种不适定性。关键在于，工程系统通常具有复杂的可变几何形状，这阻碍了标准多系统贝叶斯UQ方法的应用。在本工作中，我们提出了用于贝叶斯反演的几何自编码器（GABI），这是一个学习物理响应几何感知生成模型的框架，该模型可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”范式，GABI从具有变化几何形状的系统大型数据集中提取信息——无需已知控制偏微分方程、边界条件或观测过程——并将其提炼为丰富的隐式先验。在推断阶段，该先验可与特定观测过程的似然函数无缝结合，产生几何适应的后验分布。我们提出的框架是架构无关的。通过创造性地使用近似贝叶斯计算（ABC）采样，我们实现了一种能充分利用现代GPU硬件的高效实施方案。我们在以下问题上测试了我们的方法：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：在监督学习可适用的受限场景下，其预测精度与确定性监督学习方法相当；在具有复杂几何形状的挑战性问题上，其UQ结果校准良好且稳健。