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.

翻译：我们提出轨道空间几何概率路径（OGPP），一种面向粒子系统的粒子原生流匹配生成建模框架。OGPP的提出基于两个洞察：（i）粒子定义具有置换对称性，因此匿名索引会放大每个索引的目标方差，并产生弯曲且难以学习的流；（ii）粒子存在于物理空间中，因此流终端速度具有物理意义，可编码几何属性（如表面法向量）。OGPP实例化三个关键组件：（1）概率路径终端的轨道空间规范化，（2）用于角色特化的粒子索引嵌入，（3）具有弧长感知终端速度的几何概率路径，该路径通过流生成法向量作为副产品。我们在极小曲面基准上评估OGPP，单次推理步骤将度量误差降低两个数量级；在ShapeNet上，OGPP以5倍更少步骤达到与当前最优相当的性能，并以26倍更少参数和5倍更少步骤达到与DiT-3D可比的飞机EMD；在单形状编码中，OGPP在完全三维空间中运行，生成的法向量和重建结果与六维生成器竞争力相当。