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 solution accuracy. This decomposition, combined with the constructive nature of the proofs, allows 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.

翻译：POD-DL-ROMs作为一种极具通用性的策略，近期被提出用于构建非线性参数化偏微分方程的精确且可靠的降阶模型。该方法结合了：(i) 通过本征正交分解（POD）进行的初步降维以提高效率，(ii) 进一步将POD空间维度缩减至少量潜在坐标的自编码器架构，以及(iii) 用于学习潜在坐标随输入参数和时间变量演化映射的密集神经网络。本文旨在通过严格的误差分析论证POD-DL-ROMs的卓越逼近能力，揭示训练数据生成所需的采样、POD空间维度以及底层神经网络的复杂度如何影响解精度。结合证明的构造性特点，该分解使我们能够制定控制感兴趣解场逼近相对误差的实用准则，并推导出一般性的误差估计。此外，理论分析表明，POD-DL-ROMs在模型复杂度方面优于多种基于深度学习的技术。最后，我们通过从解析定义的参数依赖算子到多种参数化PDE的数值实验验证了上述结论。