Deep Learning is having a remarkable impact on the design of Reduced Order Models (ROMs) for Partial Differential Equations (PDEs), where it is exploited as a powerful tool for tackling complex problems for which classical methods might fail. In this respect, deep autoencoders play a fundamental role, as they provide an extremely flexible tool for reducing the dimensionality of a given problem by leveraging on the nonlinear capabilities of neural networks. Indeed, starting from this paradigm, several successful approaches have already been developed, which are here referred to as Deep Learning-based ROMs (DL-ROMs). Nevertheless, when it comes to stochastic problems parameterized by random fields, the current understanding of DL-ROMs is mostly based on empirical evidence: in fact, their theoretical analysis is currently limited to the case of PDEs depending on a finite number of (deterministic) parameters. The purpose of this work is to extend the existing literature by providing some theoretical insights about the use of DL-ROMs in the presence of stochasticity generated by random fields. In particular, we derive explicit error bounds that can guide domain practitioners when choosing the latent dimension of deep autoencoders. We evaluate the practical usefulness of our theory by means of numerical experiments, showing how our analysis can significantly impact the performance of DL-ROMs.

深度学习对偏微分方程 (PDE) 的降阶模型 (ROM) 的设计产生了显着影响，它被用作解决经典方法可能无法解决的复杂问题的强大工具。在这方面，深度自动编码器发挥着基础作用，因为它们提供了一种极其灵活的工具，可以通过利用神经网络的非线性功能来降低给定问题的维度。事实上，从这个范式开始，已经开发了几种成功的方法，这些方法在这里被称为基于深度学习的 ROM (DL-ROM)。然而，当涉及到由随机场参数化的随机问题时，当前对 DL-ROM 的理解主要基于经验证据：事实上，他们的理论分析目前仅限于依赖于有限数量（确定性）的偏微分方程的情况参数。这项工作的目的是通过提供一些有关在存在随机场生成的随机性的情况下使用 DL-ROM 的理论见解来扩展现有文献。特别是，我们得出了明确的误差范围，可以在选择深度自动编码器的潜在维度时指导领域从业者。我们通过数值实验评估了我们理论的实际实用性，展示了我们的分析如何显着影响 DL-ROM 的性能。