In recent years, deep learning has gained increasing popularity in the fields of Partial Differential Equations (PDEs) and Reduced Order Modeling (ROM), providing domain practitioners with new powerful data-driven techniques such as Physics-Informed Neural Networks (PINNs), Neural Operators, Deep Operator Networks (DeepONets) and Deep-Learning based ROMs (DL-ROMs). In this context, deep autoencoders based on Convolutional Neural Networks (CNNs) have proven extremely effective, outperforming established techniques, such as the reduced basis method, when dealing with complex nonlinear problems. However, despite the empirical success of CNN-based autoencoders, there are only a few theoretical results supporting these architectures, usually stated in the form of universal approximation theorems. In particular, although the existing literature provides users with guidelines for designing convolutional autoencoders, the subsequent challenge of learning the latent features has been barely investigated. Furthermore, many practical questions remain unanswered, e.g., the number of snapshots needed for convergence or the neural network training strategy. In this work, using recent techniques from sparse high-dimensional function approximation, we fill some of these gaps by providing a new practical existence theorem for CNN-based autoencoders when the parameter-to-solution map is holomorphic. This regularity assumption arises in many relevant classes of parametric PDEs, such as the parametric diffusion equation, for which we discuss an explicit application of our general theory.

翻译：近年来，深度学习在偏微分方程（PDEs）和降阶建模（ROM）领域日益普及，为领域从业者提供了强大的数据驱动新技术，如物理信息神经网络（PINNs）、神经算子、深度算子网络（DeepONets）以及基于深度学习的降阶模型（DL-ROMs）。在这一背景下，基于卷积神经网络（CNN）的深度自编码器已被证明极为有效，在处理复杂非线性问题时表现优于已确立的技术（如降基方法）。然而，尽管基于CNN的自编码器在经验上取得了成功，支持这些架构的理论结果却很少，通常以通用逼近定理的形式呈现。特别地，尽管现有文献为用户提供了设计卷积自编码器的指导方针，但后续学习潜在特征这一挑战却鲜有研究。此外，许多实际问题仍未得到解答，例如实现收敛所需的快照数量或神经网络训练策略。在本工作中，我们利用稀疏高维函数逼近的最新技术，通过提供一个新的基于CNN的自编码器实用存在定理（当参数到解的映射是全纯时），填补了其中一些空白。这一正则性假设出现在许多相关类型的参数化偏微分方程中，例如参数化扩散方程，我们针对该方程讨论了通用理论的一个显式应用。