Reduced Order Modeling of One-Dimensional Conservative PDEs via the Cumulative Distribution Transform

What the math gives to ML

The paper provides a concrete coordinate system in which one-dimensional mass-preserving transport becomes approximately linear: each density is represented by its monotone quantile map relative to a fixed reference density. In these coordinates, translations are affine shifts and transport-dominated solution manifolds can have dramatically smaller linear dimension than in Eulerian coordinates, including an exactly two-dimensional representation for linear transport. This suggests replacing raw spatial fields with monotone transport-map latents for neural PDE surrogates, while using a monotonicity-preserving decoder to guarantee nonnegative, mass-conserving reconstructions. The most direct experiments are transport-heavy sequence prediction and reduced latent dynamics, comparing CDT latents against standard autoencoder or Eulerian POD latents at equal latent dimension.