VIOT · arXiv 2026

A Variational Optimal Transport Operator on Incompressible Flow

Jinjin He, Shenyifan Lu, Sinan Wang, Zhiqi Li, Duowen Chen, Bo Zhu

Georgia Institute of Technology

Paper arXiv Live demo Code Models BibTeX
Green-to-gold fluid on a pale background sequentially forms the fourteen characters of the opening couplet of 春江花月夜.

Incompressible transport · Chinese poetry

Poetry in motion

春江潮水连海平，
海上明月共潮生。

Zhang Ruoxu · A Moonlit Night on the Spring River

A single trained operator transports density through the opening couplet, forming each character through a continuous sequence of fluid motion.

Full GIF

Abstract

We present the Variational Incompressible Optimal Transport (VIOT) operator, a generative neural operator for amortized incompressible density transport. Given a new source-target density pair, VIOT predicts a divergence-free velocity field and generates the full transport trajectory by feed-forward inference, replacing the hour-scale per-pair optimization used by adjoint fluid solvers and differentiable simulation baselines.

The system consists of three components: a stream-function or vector-potential representation that enforces incompressibility by construction, a regularized incompressible transport objective that balances endpoint accuracy and flow smoothness, and a Fourier Neural Operator backbone that amortizes the solve across new pairs and grid resolutions. Together, these components make incompressible transport a reusable neural operator that facilitates various transport processes. Further, the generative capability extends beyond the training distribution, with VIOT producing incompressible transports for user-drawn source-target pairs in a real-time interactive system.

We demonstrate VIOT on 2D and 3D density-transport benchmarks. Both 2D and 3D rollouts complete in seconds per pair, while per-instance baselines in our 2D comparisons optimize each new pair from scratch and require on the order of an hour, a roughly 104× online speedup.

Try it: draw your own transport

Sketch a source and a target density. The trained VIOT operator predicts a divergence-free velocity at every step and transports the source into the target in 50 steps, running entirely in your browser with WebGPU.

ρ0 Sourcedraw here
Draw a digit, letter or shape
ρ1 Targetdraw here
Draw the shape to transport into
VIOT transportstep 0 / 50

The model (65 MB) is downloaded once and cached by your browser.

Examples:

This is the paper's MNIST operator at 256×256 (FNO backbone, 8 layers), trained only on MNIST digit pairs; the spectral weights are stored in 4 bits to keep the download small. Sketches are recentred, rescaled to fit inside the frame (thin strokes are drawn bolder rather than enlarged past it) and blurred to match the training densities, so position and size do not matter. Press Continue chain to transport the result into a new target.

From density pairs to fluid motion

VIOT pipeline: current density, target density and time enter an FNO; a predicted potential defines a divergence-free velocity, which advects the density.
The operator receives the current density, target density, and time. It predicts a stream function in 2D or a vector potential in 3D, from which a divergence-free velocity is constructed. Repeated operator queries and numerical advection produce the transport trajectory.

Transport in 2D and 3D

VIOT transports 2D digits and silhouettes, 3D letters, human poses, and a sphere into an airplane; red boxes mark target keyframes.
Examples generated by trained VIOT operators: 2D MNIST digits and MPEG-7 silhouettes, a 3D “SMOKE” sequence, a 3D human-pose sequence, and sphere-to-airplane transport. Red boxes mark target keyframes; insets show the ground truth.

Long chains, varied shapes, and user-drawn inputs. Select an image to view it at full resolution.

3D volumetric fonts

Seven rows of 3D fluid letters and digits form GRAPH, SHAPE, FLUID, SIGAS, THANK, 2026S, and 12345.
Seven chains generated by the same trained 3D operator. Red boxes mark target keyframes; the intervening cells show intermediate transport frames.

3D human poses

Five rows of smoke-like human bodies move through sequences of five target poses, with intermediate transport frames between them.
Five chains, each passing through five randomly selected human poses with one trained 3D operator. Two intermediate frames lie between adjacent target keyframes.

From spheres to airplanes

Seven columns show spheres deforming into different 3D airplanes through six sampled transport frames.
Seven sphere-to-airplane rollouts from the same trained 3D operator. Each column is one rollout, shown from top to bottom; bottom-row insets show the target airplanes.

Long chains of 2D glyphs

A 2D fluid density sequentially forms the letters in FLUIDOT OPERATOR.
A single trained 2D operator successively forms the glyphs in “FLUIDOT OPERATOR.” Marked cells are target keyframes; unmarked cells show the transport between them.

A hundred silhouette transitions

A ten-by-ten grid of transported MPEG-7 silhouettes, with a small target silhouette inset in each tile.
One hundred random MPEG-7 transitions. Each tile shows the final density after five consecutive transport segments, using one trained operator without retraining between segments. Dark insets show the target silhouettes.

Transport from interactive sketches

The interactive interface shows a painted source density, a painted target density, and the resulting VIOT transport.
The user draws source and target densities, and VIOT produces a 50-step incompressible transport. These sketches fall outside the MNIST training distribution and require no per-pair optimization. Try it live above.

BibTeX

@misc{he2026variational,
  title={A Variational Optimal Transport Operator on Incompressible Flow},
  author={Jinjin He and Shenyifan Lu and Sinan Wang and Zhiqi Li and Duowen Chen and Bo Zhu},
  year={2026},
  eprint={2609.13729},
  archivePrefix={arXiv},
  primaryClass={cs.LG},
  url={https://arxiv.org/abs/2609.13729}
}