Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later

2025-09-24

AI Summary

This paper addresses the severely ill-posed inverse problem of reconstructing full-field physical quantities from sparse, noisy observations in complex-geometry engineering systems. We propose the Geometry-Aware Bayesian Inversion (GABI) framework. Methodologically, GABI introduces a geometry-aware autoencoder as a data-driven generative prior—enabling cross-geometry generalization without requiring knowledge of governing equations, boundary conditions, or observation models. It adopts a “learn-then-observe” paradigm, integrating latent-space prior modeling with Approximate Bayesian Computation (ABC) to support plug-and-play inversion under arbitrary observation configurations. Accelerated on GPUs, GABI achieves supervised-learning-level accuracy across diverse physics—including heat conduction, RANS turbulence, and Helmholtz resonance—while delivering well-calibrated uncertainty quantification and strong robustness. The framework significantly enhances generalizability and reliability of physical-field reconstruction for complex geometries.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Uncertainty Quantification (UQ) is paramount for inference in engineering applications. 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. Critically, engineering systems often have complicated and variable geometries prohibiting the use of standard 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 the 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. The method provides a flexible geometry-aware train-once-use-anywhere foundation model which is independent of any particular observation process.

Problem

• Recovering full-field physical system information from sparse noisy observations
• Handling Bayesian uncertainty quantification for systems with complex variable geometries
• Learning geometry-aware generative models without requiring governing PDE knowledge

Innovation

• Geometry-aware generative models as informative priors
• Architecture-agnostic framework with ABC sampling
• Train-once-use-anywhere foundation model for Bayesian inversion