We present a non-intrusive model reduction framework for linear poroelasticity problems in heterogeneous porous media using proper orthogonal decomposition (POD) and neural networks, based on the usual offline-online paradigm. As the conductivity of porous media can be highly heterogeneous and span several orders of magnitude, we utilize the interior penalty discontinuous Galerkin (DG) method as a full order solver to handle discontinuity and ensure local mass conservation during the offline stage. We then use POD as a data compression tool and compare the nested POD technique, in which time and uncertain parameter domains are compressed consecutively, to the classical POD method in which all domains are compressed simultaneously. The neural networks are finally trained to map the set of uncertain parameters, which could correspond to material properties, boundary conditions, or geometric characteristics, to the collection of coefficients calculated from an $L^2$ projection over the reduced basis. We then perform a non-intrusive evaluation of the neural networks to obtain coefficients corresponding to new values of the uncertain parameters during the online stage. We show that our framework provides reasonable approximations of the DG solution, but it is significantly faster. Moreover, the reduced order framework can capture sharp discontinuities of both displacement and pressure fields resulting from the heterogeneity in the media conductivity, which is generally challenging for intrusive reduced order methods. The sources of error are presented, showing that the nested POD technique is computationally advantageous and still provides comparable accuracy to the classical POD method. We also explore the effect of different choices of the hyperparameters of the neural network on the framework performance.

中文翻译：

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基于不连续Galerkin逼近的非侵入式非均质多孔弹性体的降阶建模

我们基于常规的离线在线范例，使用适当的正交分解（POD）和神经网络，为非均质多孔介质中的线性孔隙弹性问题提供了一种非侵入式模型简化框架。由于多孔介质的电导率可能高度不均一且跨越几个数量级，因此我们利用内部罚分不连续伽勒金（DG）方法作为全阶求解器来处理不连续性并确保离线阶段的局部质量守恒。然后，我们将POD用作数据压缩工具，并将嵌套POD技术（其中时间和不确定参数域被连续压缩）与经典POD方法（其中所有域都被同时压缩）进行比较。最终训练了神经网络以映射不确定参数集，它可以对应于材料属性，边界条件或几何特征，也可以对应于在缩减的基础上根据$ L ^ 2 $投影计算得出的系数集合。然后，我们对神经网络进行非侵入式评估，以获得与在线阶段不确定参数的新值相对应的系数。我们证明了我们的框架为DG解决方案提供了合理的近似值，但是速度明显更快。此外，降阶框架可以捕获由介质电导率异质性引起的位移场和压力场的急剧不连续性，这通常对侵入式降阶方法具有挑战性。列出了错误的来源，说明嵌套POD技术在计算上具有优势，并且仍可提供与经典POD方法相当的准确性。我们还探讨了神经网络超参数的不同选择对框架性能的影响。