Generative Modeling with Orbit-Space Particle Flow Matching

2026-05-04

AI Summary

Traditional particle-based generative models suffer from high variance, difficulty in learning curved manifolds, and limited exploitation of geometric information due to their neglect of permutation symmetry. This work proposes the Orbit-space Geometric Probability Path (OGPP) framework, which preserves permutation symmetry through orbit-space normalization, incorporates particle index embeddings for role specialization, and introduces an arc-length-aware geometric probability path to jointly generate surface normals. OGPP is the first approach to integrate orbit-space normalization with geometric probability paths, enabling direct generation of high-quality 3D shapes and normals without resorting to high-dimensional representations. Experiments demonstrate that OGPP reduces single-step inference error by two orders of magnitude on minimal surface tasks, achieves state-of-the-art performance on ShapeNet with one-fifth the number of steps, matches DiT-3D in Earth Mover’s Distance for airplanes while using 26× fewer parameters, and produces shape and normal quality comparable to 6D generators in single-shape reconstruction.

Abstract

We present Orbit-Space Geometric Probability Paths (OGPP), a particle-native flow-matching framework for generative modeling of particle systems. OGPP is motivated by two insights: (i) particles are defined up to permutation symmetries, so anonymous indexing inflates per-index target variance and yields curved, hard-to-learn flows; and (ii) particles live in physical space, so the flow terminal velocity has physical meaning and can encode geometric attributes, e.g., surface normals. OGPP instantiates three key components: (1) orbit-space canonicalization of the probability-path terminal endpoint, (2) particle index embeddings for role specialization, and (3) geometric probability paths with arc-length-aware terminal velocities that generate normals as a byproduct of the flow. We evaluate OGPP on minimal-surface benchmarks, where it reduces metric error by up to two orders of magnitude in a single inference step; on ShapeNet, where it matches the state of the art with 5x fewer steps and reaches airplane EMD comparable to DiT-3D with 26x fewer parameters and 5x fewer steps; and on single-shape encoding, where it produces normals and reconstructions competitive with 6D generators while operating entirely in 3D.

Problem

• generative modeling
• particle systems
• permutation symmetry
• geometric attributes
• flow matching

Innovation

• orbit-space canonicalization
• particle flow matching
• geometric probability paths
• permutation symmetry
• surface normals generation