Vector Field Synthesis with Sparse Streamlines Using Diffusion Model

Nguyen K. Phan; Ricardo Morales; Sebastian D. Espriella; Guoning Chen

cs.CVApr 10, 2026

Vector Field Synthesis with Sparse Streamlines Using Diffusion Model

Authors: Nguyen K. Phan, Ricardo Morales, Sebastian D. Espriella, Guoning Chen

Organizations: University of Houston

Abstract

We present a novel diffusion-based framework for synthesizing 2D vector fields from sparse, coherent inputs (i.e., streamlines) while maintaining physical plausibility. Our method employs a conditional denoising diffusion probabilistic model with classifier-free guidance, enabling progressive reconstruction that preserves both geometric and physical constraints. Experimental results demonstrate our method's ability to synthesize plausible vector fields that adhere to physical laws while maintaining fidelity to sparse input observations, outperforming traditional optimization-based approaches in terms of flexibility and physical consistency.

Figures & tables

Figure 1: Overview of our vector field diffusion training pipeline. The process begins with noise addition following a forward diffusion process with $\beta_{t}$ schedule, combined with streamline input values. Both conditional and unconditional paths through parallel UNet2D networks are modeled, with the outputs combined using classifier-free guidance (CFG). This dual-path architecture enables the model to balance fidelity to known streamline regions while generating physically plausible completions in unknown regions.

Figure 2: Our vector field diffusion inference pipeline. The process begins with random noise in unknown regions while preserving known vector data from streamline observations. Through iterative denoising steps guided by CFG, the model produces complete vector fields while maintaining physical consistency.

Figure 3: Qualitative comparison across all three datasets: (A) Synthetic, (B) Red Sea, and (C) CFD, with two examples per dataset. Top row: Ground truth vector fields with sparse streamline inputs overlaid that the model hasn’t seen. Middle row: Vector fields synthesized by our diffusion model. Bottom row: Vector fields synthesized by the baseline optimization method. Our approach consistently outperforms the baseline, better capturing complex flow patterns and maintaining physical consistency throughout the domain.

Figure 4: Vector field synthesis from hand-drawn streamlines. Each column shows a different input pattern: a simple vortex (left), a simple saddle (middle), and a complex S-shaped streamline (right). Top row: Results from our diffusion model with low CFG values and different random seeds, exhibiting more varied fields despite simple input streamlines. Note how the first column shows a vortex with near-parallel streamlines and the second column displays a more complex saddle pattern. Middle row: Results from our diffusion model with CFG $\geq$ 1, showing standard expected features that perfectly adhere to the input streamlines: a large simple vortex (left) and large simple saddle (middle). Bottom row: Results from the SVD method, which produces single, simple flows with physical properties concentrated around constraints for each set of constraints. The thick black lines with arrows are the hand-drawn input streamlines. The dashed black arrow signifies sparser sampling of a long hand-drawn streamline to smooth out artifacts like jitter or sharp angles from the drawing process. The thin white lines are streamlines depicting the synthesized flow.

Dataset	Method	MSE $\downarrow$	Angular (°) $\downarrow$	Physics $\downarrow$
Synthetic	SVD	0.005075	12.14	0.711280
Synthetic	Ours	0.000610	3.49	0.389157
Red Sea	SVD	0.014640	37.16	0.653732
Red Sea	Ours	0.007494	25.61	0.425823
CFD	SVD	0.144679	39.39	0.619762
CFD	Ours	0.038301	13.51	0.288306

Table 1: Quantitative comparison across all datasets. Lower values indicate better performance.

Physics-Informed Diffusion Models

Physics-informed diffusion models use physical laws and measurements to guide generative sampling, reconstruction, and forecasting.

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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.

Abstract

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