We propose Pullback Flow Matching (PFM), a novel framework for generative modeling on data manifolds. Unlike existing methods that assume or learn restrictive closed-form manifold mappings for training Riemannian Flow Matching (RFM) models, PFM leverages pullback geometry and isometric learning to preserve the underlying manifold's geometry while enabling efficient generation and precise interpolation in latent space. This approach not only facilitates closed-form mappings on the data manifold but also allows for designable latent spaces, using assumed metrics on both data and latent manifolds. By enhancing isometric learning through Neural ODEs and proposing a scalable training objective, we achieve a latent space more suitable for interpolation, leading to improved manifold learning and generative performance. We demonstrate PFM's effectiveness through applications in synthetic data, protein dynamics and protein sequence data, generating novel proteins with specific properties. This method shows strong potential for drug discovery and materials science, where generating novel samples with specific properties is of great interest.

本文提出了一种新颖的生成建模框架——回拉流匹配（PFM），解决了现有方法在训练Riemannian流匹配模型时对流形映射的严格假设问题。PFM通过利用回拉几何和等距学习，优化了潜在空间中的高效生成与精确插值，显著提升了流形学习和生成性能，尤其在药物发现和材料科学中显示出广泛的应用潜力。

数据流形上的回拉流匹配