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 non-intrusive 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 speed-up of $7 \times 10^{6}$ because our framework is not bound by the Courant-Friedrichs-Lewy 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 (two-order of magnitude lower than the maximum value of the finite element results) in the worst-case 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 t-Distributed Stochastic Neighbor Embedding (t-SNE) could indicate which method will perform better prior to employing any specific compression strategy.

翻译：多孔媒体的自然相融合是一个高度非线性多物理问题,与许多工程应用(例如,$\mathrm{CO_2}美元固存过程)相关。在这里,我们展示了一种非侵入性减少的自然相融合模式,在使用深层连动自动对齐器进行压缩和重建的多孔媒体中,在绘制相应非线性方程(PDEs)的部分差分方程参数(PDEs)方面,这是一个高度非线性多物理问题。为了衡量我们的方法,我们还描述了依赖正或超异性分解(POD)和ANNNes的线性调整和重建进程。我们通过三个基准问题对不同模式进行全面的比较。简化的订单模型、线性和非线性基化方法比定值模型(RBF)要快得多,因为我们的框架不受Colant-Frichtal-Lewy;因此,在任何给定值的直径直径直径直径性方方方阵度(Oral-lickal roupal road) 分析结果中,在任何特定的直径直径直径直径性模型中,可以显示的平方值中,在最差的平方值中,在最深的平方值模型中可以显示的平方值中表现。