Riemannian Variational Flow Matching for Material and Protein Design

• URL: http://arxiv.org/abs/2502.12981v2
• Date: Thu, 02 Oct 2025 17:48:51 GMT
• Title: Riemannian Variational Flow Matching for Material and Protein Design
• Authors: Olga Zaghen, Floor Eijkelboom, Alison Pouplin, Cong Liu, Max Welling, Jan-Willem van de Meent, Erik J. Bekkers,
• Abstract summary: In Euclidean space, predicting endpoints (VFM), velocities (FM), or noise (diffusion) are largely equivalent due to affines.<n>On curved manifold, this equivalence breaks down, and we hypothesize that endpoint prediction provides a stronger learning signal.<n>Building on this insight, we derive a variational flow matching objective.<n> 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.
• Score: 37.328940532069424
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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