Nonintrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniques
Abstract
Natural convection in porous media is a highly nonlinear multiphysical problem relevant to many engineering applications (e.g., the process of $\mathrm{CO_2}$ sequestration). Here, we present a nonintrusive reduced order model of natural convection in porous media employing deep convolutional autoencoders for the compression and reconstruction and either radial basis function (RBF) interpolation or artificial neural networks (ANNs) for mapping parameters of partial differential equations (PDEs) on the corresponding nonlinear manifolds. To benchmark our approach, we also describe linear compression and reconstruction processes relying on proper orthogonal decomposition (POD) and ANNs. We present comprehensive comparisons among different models through three benchmark problems. The reduced order models, linear and nonlinear approaches, are much faster than the finite element model, obtaining a maximum speedup of $7 \times 10^{6}$ because our framework is not bound by the CourantFriedrichsLewy condition; hence, it could deliver quantities of interest at any given time contrary to the finite element model. Our model's accuracy still lies within a mean squared error of 0.07 (twoorder of magnitude lower than the maximum value of the finite element results) in the worstcase scenario. We illustrate that, in specific settings, the nonlinear approach outperforms its linear counterpart and vice versa. We hypothesize that a visual comparison between principal component analysis (PCA) or tDistributed Stochastic Neighbor Embedding (tSNE) could indicate which method will perform better prior to employing any specific compression strategy.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.11460
 Bibcode:
 2021arXiv210711460K
 Keywords:

 Computer Science  Computational Engineering;
 Finance;
 and Science;
 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis