We present a novel reduced order model (ROM) approach for parameterized time-dependent PDEs based on modern learning. The ROM is suitable for multi-query problems and is nonintrusive. It is divided into two distinct stages: A nonlinear dimensionality reduction stage that handles the spatially distributed degrees of freedom based on convolutional autoencoders, and a parameterized time-stepping stage based on memory aware neural networks (NNs), specifically causal convolutional and long short-term memory NNs. Strategies to ensure generalization and stability are discussed. The methodology is tested on the heat equation, advection equation, and the incompressible Navier-Stokes equations, to show the variety of problems the ROM can handle.

中文翻译：

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使用空间和内存感知的深度学习对参数化时间相关PDE的降阶建模

我们提出了一种基于现代学习的参数化时间相关PDE的新颖的降阶模型（ROM）方法。ROM适用于多查询问题，并且是非侵入式的。它分为两个不同的阶段：一个非线性的降维阶段，该阶段基于卷积自动编码器处理空间分布的自由度；一个基于记忆感知神经网络（NNs）的参数化时间步进阶段，尤其是因果卷积和长短整数。术语记忆神经网络。讨论了确保泛化和稳定性的策略。在热方程，对流方程和不可压缩的Navier-Stokes方程上测试了该方法，以显示ROM可以处理的各种问题。