Riemannian Variational Flow Matching for Material and Protein Design
This work addresses generative modeling challenges in domains like materials science and biology, offering a geometric extension that improves structure capture, though it is incremental as it builds on existing flow matching methods.
The paper tackled the problem of generative modeling on curved manifolds by proposing Riemannian Gaussian Variational Flow Matching (RG-VFM), which directly minimizes geodesic distances to improve learning signals, resulting in better performance on synthetic benchmarks and real-world material and protein generation tasks over baselines.
We present Riemannian Gaussian Variational Flow Matching (RG-VFM), a geometric extension of Variational Flow Matching (VFM) for generative modeling on manifolds. In Euclidean space, predicting endpoints (VFM), velocities (FM), or noise (diffusion) are largely equivalent due to affine interpolations. On curved manifolds this equivalence breaks down, and we hypothesize that endpoint prediction provides a stronger learning signal by directly minimizing geodesic distances. Building on this insight, we derive a variational flow matching objective based on Riemannian Gaussian distributions, applicable to manifolds with closed-form geodesics. We formally analyze its relationship to Riemannian Flow Matching (RFM), exposing that the RFM objective lacks a curvature-dependent penalty - encoded via Jacobi fields - that is naturally present in RG-VFM. Experiments on synthetic spherical and hyperbolic benchmarks, as well as real-world tasks in material and protein generation, demonstrate that RG-VFM more effectively captures manifold structure and improves downstream performance over Euclidean and velocity-based baselines.