Title: GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometries
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: operator learning, flow matching, diffusion models, partial differential equations, inverse problems

Abstract: Inverse problems governed by partial differential equations (PDEs) are crucial in science and engineering. They are particularly challenging due to ill-posedness, data sparsity, and the added complexity of irregular geometries. Classical PDE-constrained optimization methods are accurate and robust but are computationally expensive, especially when repeated posterior sampling is required. Learning-based approaches improve efficiency and scalability, yet most are designed for regular domains or focus primarily on forward modeling.
To fill this gap, we introduce *GeoFunFlow*, a geometric diffusion framework for inverse problems on complex geometries. GeoFunflow combines a novel geometric functional autoencoder (GeoFAE) and a latent diffusion model trained via rectified flow. GeoFAE employs a Perceiver module to process unstructured meshes of varying sizes and produces continuous reconstructions of solution fields, while the diffusion model enables posterior sampling from sparse and noisy data. Across  five standard benchmarks, GeoFunflow achieves state-of-the-art reconstruction accuracy over complex geometries, provides calibrated uncertainty quantification, and delivers efficient inference compared to operator-learning and diffusion baselines.

TL;DR: A geometric diffusion framework that solves PDE inverse problems on complex geometries