Many generative modeling problems in scientific computing and geometry require samples to satisfy hard constraints, such as incompressibility or unit norm. Existing score-based diffusion approaches often enforce these constraints in ambient space using projection or penalty terms. We propose chart-based score diffusion, a framework that evolves stochastic dynamics in chart coordinates and maps trajectories to ambient space through constraint-preserving chart transforms. We derive the reverse-time formulation in chart space and characterize the relation between chart-density and manifold-density scores, which yields an induced-volume correction applied consistently in denoising score matching and reverse-time sampling. To handle non-global coordinates, we introduce atlas-based modeling, where an atlas (a set of overlapping local charts) is combined with hard assignment, soft weighting, or learned routing; the learned variant uses entropy and balance regularization to stabilize chart utilization. We evaluate the method on 2D incompressible flows (stream-function charts) and \(S^2\)-valued fields (stereographic atlases), and report reproducible benchmark protocols with constraint diagnostics, distributional metrics, and targeted ablations.
Chart-Invariant Few-Step Diffusion for Manifold Fields Research Pinned
Project summary
with Yexiang Xue. We propose chart-based score diffusion, a framework that evolves stochastic dynamics in chart coordinates and maps trajectories to ambient space through constraint-preserving chart transforms. We also introduce atlas-based modeling, where an atlas (a set of overlapping local charts) is combined with hard assignment, soft weighting, or learned routing; the learned variant uses entropy and balance regularization to stabilize chart utilization.
Timeline
January 2026 - Present
Status
In progress
Stack
Python