GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometries

Inverse problems governed by partial differential equations (PDEs) arise widely in science and engineering, but are often ill-posed and limited by sparse, noisy observations. In many applications, measurements reveal only part of the physical state, while the domain geometry may also be unknown even though it shapes the observed response. Joint field and geometry inference across varying computational domains and discretizations remains challenging, whereas many existing machine learning approaches are designed for known geometries and deterministic field reconstruction. Here, we introduce GeoFunFlow, a probabilistic framework that unifies field reconstruction on known domains and joint field and geometry inference on unknown domains. GeoFunFlow combines a geometric function autoencoder (GeoFAE) with flow matching in the latent space to model a joint distribution over physical fields and geometries. GeoFAE establishes a common representation across spatial discretizations that captures the relationship between physical fields and domain geometries, with unknown geometry represented by a signed distance function. The resulting representation allows observations to guide both field reconstruction and geometry recovery, while latent rectified flow enables efficient conditional sampling and spatially resolved uncertainty quantification. A calibration procedure further provides geometry uncertainty estimates with interpretable empirical coverage. Across seven benchmarks spanning porous media flow, fluid mechanics, and optical tomography, GeoFunFlow accurately recovers fields and geometries across complex, variable, and unknown domains while quantifying spatially resolved conditional uncertainty.