Model reduction of traveling-wave problems via Radon cumulative distribution transform

Traveling-wave problems, due to their sizable Kolmogorov n-width, have brought critical challenges to conventional model reduction techniques. This study aims to provide insights into this problem by exploiting the Radon cumulative distribution transform (R-CDT) [Kolouri, Park, and Rohde, IEEE Trans. Image Process. 25, 920 (2016)IIPRE41057-714910.1109/TIP.2015.2509419], which has emerged in the sector of computer vision science. The core lies in the unique property of the nonlinear invertible R-CDT that renders both traveling and scaling components into amplitude modulations. In contrast to the physical space, a substantial model reduction is achieved in the R-CDT space while sustaining high accuracy. The method is parameter-free and data-driven and lends itself to problems regardless of the dimensions or boundary conditions. Examples start with a one-dimensional Burgers' equation subject to nonperiodic boundary conditions, where both traveling and diffusion dominate the physics. In higher-dimensional problems, we show the model reduction of traveling Gaussian solitons. In addition to foreseeable motions, the proposed method is capable of handling random traveling with a nondifferentiable trajectory.