POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition
POD-DL-ROM: Enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition
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DOI：
10.1016/j.cma.2021.114181
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发表时间：
2021-10-13
影响因子：
7.2
通讯作者：
Manzoni, Andrea
中科院分区：
文献类型：
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作者：
Fresca, Stefania;Manzoni, Andrea
最近已经提出了基于深度学习的降阶模型（DL-ROM），以克服传统降阶模型（ROM）所共有的共同限制，例如，通过适当的正交分解（POD）-当应用于非线性时间依赖的参数化偏微分方程（PDE）。这些可能与（i）需要处理到高维线性近似试验流形上的投影，（ii）昂贵的超约简策略，或（iii）用模式的线性叠加处理物理复杂性的内在困难有关。当采用DL-ROM时，避免了所有这些方面，DL-ROM通过依赖于深度（例如，前馈、卷积、自动编码器）神经网络。虽然在测试时非常有效，但在评估任何新测试参数实例的PDE解决方案时，DL-ROM需要昂贵的训练阶段，因为要估计的网络参数数量非常大。在本文中，我们提出了一种可能的方法来避免DL-ROM的昂贵的训练阶段，通过（i）通过POD执行先验降维，以及（ii）依赖于多保真度预训练阶段，其中可以有效地组合不同的物理模型。所提出的POD-DL-ROM在几个（标量和矢量，线性和非线性）时间相关参数化PDE（例如，线性对流扩散反应，非线性扩散反应，非线性弹性动力学，和Navier-Stokes方程），以显示这种方法的通用性和显着的计算节省。(C)2021由爱思唯尔公司出版
Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional reduced order models (ROMs) - built, e.g., through proper orthogonal decomposition (POD) - when applied to nonlinear time-dependent parametrized partial differential equations (PDEs). These might be related to (i) the need to deal with projections onto high dimensional linear approximating trial manifolds, (ii) expensive hyper-reduction strategies, or (iii) the intrinsic difficulty to handle physical complexity with a linear superimposition of modes. All these aspects are avoided when employing DL-ROMs, which learn in a non-intrusive way both the nonlinear trial manifold and the reduced dynamics, by relying on deep (e.g., feedforward, convolutional, autoencoder) neural networks. Although extremely efficient at testing time, when evaluating the PDE solution for any new testing-parameter instance, DL-ROMs require an expensive training stage, because of the extremely large number of network parameters to be estimated. In this paper we propose a possible way to avoid an expensive training stage of DL-ROMs, by (i) performing a prior dimensionality reduction through POD, and (ii) relying on a multi-fidelity pretraining stage, where different physical models can be efficiently combined. The proposed POD-DL-ROM is tested on several (both scalar and vector, linear and nonlinear) time-dependent parametrized PDEs (such as, e.g., linear advection-diffusion-reaction, nonlinear diffusion-reaction, nonlinear elastodynamics, and Navier-Stokes equations) to show the generality of this approach and its remarkable computational savings. (C) 2021 Published by Elsevier B.V.