用几何扩散模型解决复杂形状下的反问题，高效且能量化不确定性。
GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometries
- 基于几何函数自编码器与修正流扩散模型，处理不规则网格上的物理场重建。
- 在五个基准测试中达到顶尖重建精度，支持稀疏噪声数据下的后验采样。
- 适合需要高精度反演和不确定性估计的科学计算领域研究者使用。
由偏微分方程（PDE）支配的反问题是科学与工程中的核心挑战，其难点在于病态性、数据稀疏性以及不规则几何结构带来的复杂性。传统PDE约束优化方法计算成本高，尤其在需重复后验采样时更为明显。基于学习的方法提升了效率与可扩展性，但多数仅适用于规则域或聚焦于前向建模。本文提出GeoFunFlow——一种面向复杂几何反问题的几何扩散模型框架。该框架结合新型几何函数自编码器（GeoFAE）与基于修正流训练的潜在扩散模型。GeoFAE采用Perceiver模块处理不同规模的非结构化网格，并生成物理场的连续重构；扩散模型则可在稀疏噪声数据下实现后验采样。在五个基准测试中，GeoFunFlow实现了复杂几何上的最先进重建精度，提供校准的不确定性量化，并在推理效率上优于现有操作符学习与扩散模型基线。
原文摘要 · Abstract (English)
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.
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