Riemannian Variational Flow Matching for Material and Protein Design

ICLR 2026 Conference Submission8821 Authors

Keywords: Flow matching, variational inference, riemannian manifolds, material generation, metal-organic framework, protein backbone generation

TL;DR: We derive a variational objective for flow matching on manifolds with closed-form geodesics and test it on material and protein backbone generation.

Abstract: We present Riemannian Gaussian Variational Flow Matching (RG-VFM), a geometric extension of Variational Flow Matching (VFM) for generative modeling on manifolds. Motivated by the benefits of VFM, we derive a variational flow matching objective for manifolds with closed-form geodesics based on Riemannian Gaussian distributions. Crucially, in Euclidean space, predicting endpoints (VFM), velocities (FM), or noise (diffusion) is largely equivalent due to affine interpolations. However, on curved manifolds this equivalence breaks down. For this reason, we formally analyze the relationship between our model and Riemannian Flow Matching (RFM), revealing that the RFM objective lacks a curvature-dependent penalty -- encoded via Jacobi fields -- that is naturally present in RG-VFM. Based on this relationship, we hypothesize that endpoint prediction provides a stronger learning signal by directly minimizing geodesic distances. 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.

Primary Area: generative models

Submission Number: 8821