Speaker
Description
In this work, we propose a reduced-order modeling (ROM) framework for conservation laws that operates in the Cumulative Distribution Transform (CDT) domain. The CDT maps nonnegative, unit-mass states to an $L^2$ Hilbert space, in which pure translations become affine lines and $W_2$ (optimal transport) distances coincide with Euclidean distances. This linearization dramatically improves the approximability of advective dynamics: for linear transport, the solution manifold is exactly two-dimensional in CDT space $(d_2 = 0)$, and for advection–diffusion we show that, once an initial layer is excluded, the CDT trajectory has uniformly bounded curvature, implying quadratic Kolmogorov $n$-width decay with $n$.
Building on these insights, we develop a CDT–POD algorithm: (i) map snapshots to CDT space, (ii) compute a Proper Orthogonal Decomposition (POD) basis in that space, (iii) evolve reduced coefficients via a transform-space surrogate, and (iv) invert the CDT to recover physical states. Numerical experiments on linear advection, advection–diffusion, and both viscous and inviscid Burgers equations show that CDT–POD attains rank-2 exactness for transported signals, robustly resolves moving shocks and contact discontinuities with orders of magnitude fewer modes than Eulerian POD, and remains competitive in smoothly diffusive regimes. The method preserves mass by construction and is compatible with positivity.