AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-20
Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures
Parametrized PDEs with density-valued solutions are hard to approximate with classical linear reduced-order models, especially in transport-dominated regimes. The authors propose an optimal-transport-based reduced-order method that represents each density by the Kantorovich potential carrying a fixed reference density to the target, embeds these potentials in a Hilbert space as transport signatures, and uses a neural network to learn the parameter-to-coefficient map over a low-rank skeleton decomposition. Reconstruction by pushing the reference density forward preserves mass and positivity by construction, with a proven mean-squared Wasserstein error bound, and on a density-valued continuity equation and Burgers equation the signatures yield substantially lower-rank structure than the original density snapshots.