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