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

arXiv: https://arxiv.org/abs/2509.24117 · HTML: https://mlsanity.com/paper/2509.24117/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 54 of its 79 references that are in the map.
- First posted on arXiv 28 Sep 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 54)

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- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
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- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [High-Quality Prediction Intervals for Deep Learning: A Distribution-Free, Ensembled Approach](https://mlsanity.com/paper/1802.07167/) (2018)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)

## Abstract (arXiv)

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.