Mapping of coherent structures in parameterized flows by learning optimal transportation with Gaussian models

TL;DR

This paper introduces a model-independent interpolation method for parameterized advection-dominated flows using optimal transportation of Gaussian models, enabling effective tracking of coherent structures.

Contribution

It proposes a novel interpolation technique based on optimal transportation maps between Gaussian models, applicable to various flow problems.

Findings

Effective in recasting self-similar solutions via optimal transportation

Numerical examples demonstrate the method's strengths and limitations

Potential extension to more complex flow problems

Abstract

We present a general (i.e., independent of the underlying model) interpolation technique based on optimal transportation of Gaussian models for parametric advection-dominated problems. The approach relies on a scalar testing function to identify the coherent structure we wish to track; a maximum likelihood estimator to identify a Gaussian model of the coherent structure; and a nonlinear interpolation strategy that relies on optimal transportation maps between Gaussian distributions. We show that well-known self-similar solutions can be recast in the frame of optimal transportation by appropriate rescaling; we further present several numerical examples to motivate our proposal and to assess strengths and limitations; finally, we discuss an extension to deal with more complex problems.