Computing Residual Diffusivity by Adaptive Basis Learning via Super-Resolution Deep Neural Networks

TL;DR

This paper introduces a novel super-resolution deep neural network approach to enhance residual diffusivity computations in chaotic flows, significantly improving accuracy over traditional POD methods by modeling internal layer sharpening effects.

Contribution

It develops a super-resolution GAN-based method to adaptively improve POD basis functions for better residual diffusivity predictions in advection-diffusion problems.

Findings

Super-resolution GAN improves residual diffusivity prediction accuracy.

The method effectively models internal layer sharpening effects.

Numerical experiments validate the enhanced performance.

Abstract

It is expensive to compute residual diffusivity in chaotic in-compressible flows by solving advection-diffusion equation due to the formation of sharp internal layers in the advection dominated regime. Proper orthogonal decomposition (POD) is a classical method to construct a small number of adaptive orthogonal basis vectors for low cost computation based on snapshots of fully resolved solutions at a particular molecular diffusivity $D_{0}^{*}$. The quality of POD basis deteriorates if it is applied to $D_0\ll D_{0}^{*}$. To improve POD, we adapt a super-resolution generative adversarial deep neural network (SRGAN) to train a nonlinear mapping based on snapshot data at two values of $D_{0}^{*}$. The mapping models the sharpening effect on internal layers as $D_0$ becomes smaller. We show through numerical experiments that after applying such a mapping to snapshots, the prediction accuracy of residual diffusivity improves considerably that of the standard POD.