2D Change of Variable — Discrete Layers

Interactive · 2D Change of Variable: Discrete Layers from Noise to Data

Start from Gaussian noise and apply a chain of discrete change-of-variable maps. The slider shows the layer index L = 1,2,⋯,100,⋯,∞. At finite L, the transport is written as a composition of maps Ψ₁ ∘ ⋯ ∘ Ψ_L. At L=∞, this becomes the continuity equation ∂pₜ/∂t = −∇·(pₜ v). The deformed background mesh stays visible so you can directly see the Jacobian effect: compressed cells indicate higher density, stretched cells indicate lower density.

Discrete transport layer (1 = first map, 100 = data, ∞ = continuous-limit reference)

L = 1

first map

grid deformation density p_L trajectories particles