Generative Modeling with Orbit-Space Particle Flow Matching

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; (ii) particles live in physical space, so the flow's 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 5× fewer steps and reaches airplane EMD comparable to DiT-3D with 26× fewer parameters and 5× fewer steps; and on single-shape encoding, where it produces normals and reconstructions competitive with 6D generators while operating entirely in 3D.