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

2025-09-28

AI Summary

Inverse problems governed by partial differential equations (PDEs) on complex geometric domains are inherently ill-posed, and existing methods suffer from sparse data, irregular meshing, and inadequate uncertainty quantification. To address these challenges, we propose GeoFM—a generative operator learning framework based on geometric function flow matching. Its core innovation lies in a geometric function autoencoder (GeoFAE) supporting unstructured meshes—incorporating a Perceiver module—coupled with a latent-space diffusion model optimized via flow matching, enabling posterior sampling for continuous physical field reconstruction and calibration. Evaluated on five PDE-based inverse problem benchmarks, GeoFM consistently outperforms state-of-the-art operator learning and diffusion-based approaches, achieving superior reconstruction accuracy, rapid inference, and principled uncertainty estimation.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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 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 on forward modeling. Here, we introduce {em GeoFunFlow}, a geometric diffusion model framework for inverse problems on complex geometries. GeoFunFlow combines a novel geometric function 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 physical fields, while the diffusion model enables posterior sampling from sparse and noisy data. Across five 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 model baselines.

Problem

• Solving inverse PDE problems on irregular geometric domains
• Addressing ill-posedness and data sparsity in complex geometries
• Enabling efficient posterior sampling from sparse noisy data

Innovation

• Geometric autoencoder processes unstructured meshes continuously
• Latent diffusion model enables posterior sampling efficiently
• Rectified flow training achieves accurate uncertainty quantification