POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear parametrized partial differential equations, combining (i) a preliminary dimensionality reduction obtained through proper orthogonal decomposition (POD) for the sake of efficiency, (ii) an autoencoder architecture that further reduces the dimensionality of the POD space to a handful of latent coordinates, and (iii) a dense neural network to learn the map that describes the dynamics of the latent coordinates as a function of the input parameters and the time variable. Within this work, we aim at justifying the outstanding approximation capabilities of POD-DL-ROMs by means of a thorough error analysis, showing how the sampling required to generate training data, the dimension of the POD space, and the complexity of the underlying neural networks, impact on the solutions us to formulate practical criteria to control the relative error in the approximation of the solution field of interest, and derive general error estimates. Furthermore, we show that, from a theoretical point of view, POD-DL-ROMs outperform several deep learning-based techniques in terms of model complexity. Finally, we validate our findings by means of suitable numerical experiments, ranging from parameter-dependent operators analytically defined to several parametrized PDEs.

POD-DL-ROM 最近被提出作为一种极其通用的策略，为非线性参数化偏微分方程构建准确可靠的降阶模型 (ROM)，结合(i)通过适当的正交分解 (POD) 获得的初步降维为了提高效率，(ii)一种自动编码器架构，可进一步将 POD 空间的维数减少到几个潜在坐标，以及(iii)一个密集的神经网络，用于学习将潜在坐标的动态描述为以下函数的映射：输入参数和时间变量。在这项工作中，我们的目标是通过彻底的误差分析来证明 POD-DL-ROM 的出色逼近能力，展示生成训练数据所需的采样方式、POD 空间的维度以及底层神经网络的复杂性网络，对解决方案的影响，我们制定实用标准来控制感兴趣的解决方案领域的近似中的相对误差，并得出一般误差估计。此外，我们表明，从理论角度来看，POD-DL-ROM 在模型复杂性方面优于多种基于深度学习的技术。最后，我们通过适当的数值实验验证了我们的发现，范围从分析定义的参数相关算子到几个参数化偏微分方程。