GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometries
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916975059402752 |
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| author | Wang, Sifan Wu, Zhikai van Dijk, David Lu, Lu |
| author_facet | Wang, Sifan Wu, Zhikai van Dijk, David Lu, Lu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24117 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometries Wang, Sifan Wu, Zhikai van Dijk, David Lu, Lu Machine Learning Computational Physics 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. |
| title | GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometries |
| topic | Machine Learning Computational Physics |
| url | https://arxiv.org/abs/2509.24117 |