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.

使用深度学习中的卷积神经网络 (CNNs) 构建的自编码器在处理复杂的非线性问题时表现出色，并提供了新的数据驱动技术，如物理信息神经网络 (PINNs)、神经算子、深度算子网络 (DeepONets) 和基于深度学习的 ROMs (DL-ROMs)。然而，尽管基于 CNN 的自编码器在实践中取得了成功，但对于这些架构的理论结果仍然很少，只有一些万能逼近定理的陈述。本研究通过利用稀疏高维函数逼近的最新技术，填补了其中一些空白，并在参数 - 解映射为全纯函数时，提供了基于 CNN 的自编码器的一个新的实用存在定理。该规则性假设适用于许多相关的参数化偏微分方程类，例如参数化扩散方程，我们在其中讨论了我们通用理论的明确应用。