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首页> 外文期刊>Journal of Scientific Computing >A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs
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A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs

机译：基于深度学习的基于深度学习的方法来减少非线性时间依赖参数化PDE的顺序建模

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摘要

Conventional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) may incur in severe limitations when dealing with nonlinear time-dependent parametrized PDEs, as these are strongly anchored to the assumption of modal linear superimposition they are based on. For problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method may yield inefficient reduced order models (ROMs) when very high levels of accuracy are required. To overcome this limitation, in this work, we propose a new nonlinear approach to set ROMs by exploiting deep learning (DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM, both the nonlinear trial manifold (corresponding to the set of basis functions in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to the projection stage in a linear ROM) are learned in a non-intrusive way by relying on DL algorithms; the latter are trained on a set of full order model (FOM) solutions obtained for different parameter values. We show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs. Moreover, we assess its accuracy and efficiency on different parametrized PDE problems. Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to efficiently approximate the solution of parametrized PDEs, especially in cases for which a huge number of POD modes would have been necessary to achieve the same degree of accuracy.
机译：传统的减少阶建模技术，例如降低的基础（RB）方法（依赖于适当的正交分解（POD））可能在处理非线性时间依赖的参数化PDE时产生严重的限制，因为它们强烈锚定到假设模态线性叠加，它们是基于的。对于具有连贯结构的问题，该结构随时间传播，例如传输，波或对流主导的现象，RB方法可以在需要非常高的精度时产生低效减少的订单模型（ROM）。为了克服这一限制，在这项工作中，我们提出了一种通过利用深度学习（DL）算法来设置ROM的新非线性方法。在所得到的非线性ROM中，我们将其称为DL-ROM，非线性试验歧管（对应于线性ROM中的基本函数集）以及非线性减少动态（对应于线性ROM中的投影级） ）通过依赖DL算法以非侵入性的方式学习;后者培训用于针对不同参数值获得的一组全阶模型（FOM）解决方案。我们展示了如何为线性和非线性时间依赖参数化PDE构造DL-ROM。此外，我们评估了对不同参数化PDE问题的准确性和效率。数值结果表明，其尺寸等于PDE解决方案歧管的内在量度的DL-ROM能够有效地近似参数化PDE的溶液，尤其是在需要大量豆荚模式以实现相同的情况下准确度。

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