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

Abstract

It is expensive to compute residual diffusivity in chaotic incompressible 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 D0 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 D0 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. Keywords: Advection dominated diffusion· residual diffusivity· adaptive basis learning· super-resolution deep neural networks.where T is a scalar function (e.g. temperature or concentration), D 0 > 0 is a constant (the so called molecular diffusivity), v (x, t) is a prescribed incompressible velocity field, D and ∆ are the spatial gradient and Laplacian operators.When the flow is steady, periodic and two dimensional, precise asymptotics of D E are known. A prototypical example is the steady cellular flow [4,5], v = (−H y , H x ), H = sin x sin y, see also [13,19,20] for its application in effective speeds of front propagation. The asymptotics of the effective diffusion along any unit direction in the cellular flow obeys the square root law in the arXiv:1910.00403v1 [physics.comp-ph]

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“…However, when κ is small and the spatial dimension is 3, adaptive FEM can be extremely expensive. For 2D time periodic cellular flows, adaptive basis deep learning is found to improve the accuracy of reduced order modeling [42,43]. Extension of deep basis learning to 3D in the Eulerian setting has not been attempted partly due to the costs of generating sufficient amount of training data.…”

“…For parametric PDEs, a deep operator network (DeepONet) learns operators accurately and efficiently from a relatively small dataset based on the universal approximation theorem of operators [40]; a Fourier neural operator method [34] directly learns the mapping from functional parametric dependence to the solutions (of a family of PDEs). Deep basis learning is studied in [43] to improve proper orthogonal decomposition for residual diffusivity in chaotic flows [42], among other works on reduced order modeling [29,62,8]. In [70,1], weak adversarial network methods are studied for weak solutions and inverse problems, see also related studies on PDE recovery from data via DNN [37,36,49,66] among others.…”