Pullback Flow Matching (PFM) is a novel framework for generative modeling on data manifolds. Unlike traditional Riemannian Flow Matching (RFM) models that assume or learn a restricted closed manifold mapping, PFM utilizes pullback geometry and equidistant learning to preserve the geometry of the underlying manifold while enabling efficient generation and precise interpolation in the latent space. This approach not only facilitates closed mapping on the data manifold, but also allows for a designable latent space using assumed metrics on both the data and latent manifolds. By improving equidistant learning with Neural ODEs and proposing scalable training objectives, we achieve a latent space more suitable for interpolation, resulting in improved manifold learning and generation performance. We demonstrate the effectiveness of PFM through applications to synthetic data, protein dynamics, and protein sequence data, generating novel proteins with specific properties. This method has strong potential in drug discovery and materials science, where generating novel samples with specific properties is important.
