FUSE: A Flow-based Mapping Between Shapes

We introduce a novel neural representation for maps between3D shapes based on flow-matching models, which is computationally ef-ficient and supports cross-representation shape matching without large-scale training or data-driven procedures. 3D shapes are represented asthe probability distribution induced by a continuous and invertible flowmapping from a fixed anchor distribution. Given a source and a targetshape, the composition of the inverse flow (source to anchor) with theforward flow (anchor to target), we map points between the two surfaces.By encoding the shapes with a pointwise task-tailored embedding, thisconstruction provides an invertible and modality-agnostic representationof maps between shapes across point clouds, meshes, signed distancefields (SDFs), and volumetric data. The resulting representation con-sistently achieves high coverage and accuracy across diverse benchmarksand challenging settings in shape matching. Beyond shape matching, ourframework shows promising results in other tasks, including UV mappingand registration of raw point cloud scans of human bodies.