Vector Field Synthesis with Sparse Streamlines Using Diffusion Model

We present a generative modeling framework for synthesizing physically feasible two-dimensional incompressible flows under arbitrary obstacle geometries and boundary conditions. Whereas existing diffusion-based flow generators either ignore physical constraints, impose soft penalties that do not guarantee feasibility, or specialize to fixed geometries, our approach integrates three complementary components: (1) a boundary-conditioned diffusion model operating on velocity fields; (2) a physics-informed training objective incorporating a divergence penalty; and (3) a projection-constrained reverse diffusion process that enforces exact incompressibility through a geometry-aware Helmholtz-Hodge operator. We derive the method as a discrete approximation to constrained Langevin sampling on the manifold of divergence-free vector fields, providing a connection between modern diffusion models and geometric constraint enforcement in incompressible flow spaces. Experiments on analytic Navier-Stokes data and obstacle-bounded flow configurations demonstrate significantly improved divergence, spectral accuracy, vorticity statistics, and boundary consistency relative to unconstrained, projection-only, and penalty-only baselines. Our formulation unifies soft and hard physical structure within diffusion models and provides a foundation for generative modeling of incompressible fields in robotics, graphics, and scientific computing.

Guiding Vector Fields (GVFs) are a powerful tool for robotic path following. However, classical methods assume smooth, ordered curves and fail when paths are unordered, multi-branch, or generated by probabilistic models. We propose a unified framework, termed the Score-Induced Guiding Vector Field (SGVF), which leverages score-based generative modeling to construct vector fields directly from data distributions. SGVF learns tangent fields from point clouds with unit-norm, orthogonality, and directional-consistency losses, ensuring geometric fidelity and control feasibility. This approach removes the reliance on ad-hoc path segmentation and enables guidance along complex topologies such as branching and pseudo-manifolds. The study establishes a correspondence between score vanishing in diffusion models and GVF singularities and highlights representational capacity near sharp path curvatures. Experiments on robotic navigation in planar environments demonstrate that SGVF achieves reliable path following in scenarios where classical GVFs fail, underscoring its potential as a bridge between generative modeling and geometric control. Code and experiment video are available at https://github.com/czr-gif/Guiding-Vector-Field-Generation-via-Score-based-Diffusion-Model.

Generating and predicting spatiotemporal physical fields from scarce measurements is challenging, as observations are insufficient to characterize a distribution over complete fields. This limits conventional data-driven diffusion models that rely on full-field datasets. We introduce PhysDEM, a physics-defined diffusion framework that combines governing equations with spatially sparse observations to generate multiple plausible fields. First, we construct a Gibbs target by reweighting a measurement-conditioned Gaussian reference with PDE residual energy. Second, we derive an exact conditional-mean identity that reduces denoising to supervised learning of the standardized energy-induced mean correction. Third, a physics-displacement probability flow cancels Gaussian reference terms and enables amortized sampling with changing measurements through Gaussian conditioning, without retraining. Experiments on synthetic PDE systems and real-world-informed applications demonstrate that PhysDEM supports coherent field recovery and efficient sampling while maintaining stable diagnostics under tested noise levels, illustrating its practical value for field assessment. To our knowledge, PhysDEM is the first physics-defined diffusion model enabling amortized spatiotemporal field inference without preassembled full-field datasets.