Diffusion / Transport

Definition

This motif focuses on how probability mass, geometry, or signal energy moves over time. The transfer question is whether the AI objective should be phrased as a transport problem rather than a pointwise regression problem.

ISOM treats diffusion and transport as claims about paths, not just endpoints. A valid transfer should specify what moves, what is conserved, what boundary conditions apply, and how intermediate states can be evaluated.

Mathematical Structure

The mathematical core is an evolution equation over states or distributions, often with drift, diffusion, transport cost, or conservation terms.

Physics Side

In physics, diffusion and transport describe how local interactions accumulate into global motion, spread, and equilibration.

AI Side

In AI, the same structure appears in denoising diffusion, representation alignment, uncertainty propagation, and flow-based synthesis. A transport view can reveal the right intermediate variables and the right stability tests.

This motif is relevant for denoising models, domain adaptation, representation alignment, and uncertainty propagation. It is most useful when the intermediate trajectory carries operational meaning rather than serving only as a sampling trick.

Failure Modes

Not every smooth interpolation is a transport process. If the model ignores boundary conditions, geometry, or mass accounting, a transport analogy can become decorative rather than useful.

A smooth latent interpolation can look like transport without obeying transport constraints. The transfer fails if the path ignores geometry, violates mass or probability accounting, or produces intermediate states that cannot be interpreted.

Open Questions

Which latent spaces admit meaningful transport metrics, and how should transport costs be chosen when semantic distance and geometric distance disagree?