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-possendess. 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; Reynold-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：不确定性量化在工程推断中至关重要。常见的推断任务是从少量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。通过共享多个不同但相关的物理系统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变量几何结构，这阻碍了标准多系统贝叶斯不确定性量化方法的应用。本文提出用于贝叶斯反演的几何自编码器框架，该框架通过学习物理响应的几何感知生成模型，为贝叶斯反演提供信息丰富的几何条件先验。遵循"先学习，后观测"范式，GABI从大量不同几何结构的系统数据集中提取信息（无需控制偏微分方程、边界条件或观测过程的知识），形成丰富的隐式先验。在推断阶段，该先验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。我们提出的框架与架构无关。通过创造性运用近似贝叶斯计算采样方法，实现了利用现代GPU硬件的高效计算。我们在以下场景测试该方法：矩形域上的稳态热传导；翼型周围的雷诺平均纳维-斯托克斯流动；三维车身上的亥姆霍兹共振与声源定位；地形上的雷诺平均空气流动。研究发现：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；在复杂几何结构的挑战性问题中，不确定性量化结果具有良好的校准性和鲁棒性。