Constructing the architecture of a neural network is a challenging pursuit for the machine learning community, and the dilemma of whether to go deeper or wider remains a persistent question. This paper explores a comparison between deeper neural networks (DeNNs) with a flexible number of layers and wider neural networks (WeNNs) with limited hidden layers, focusing on their optimal generalization error in Sobolev losses. Analytical investigations reveal that the architecture of a neural network can be significantly influenced by various factors, including the number of sample points, parameters within the neural networks, and the regularity of the loss function. Specifically, a higher number of parameters tends to favor WeNNs, while an increased number of sample points and greater regularity in the loss function lean towards the adoption of DeNNs. We ultimately apply this theory to address partial differential equations using deep Ritz and physics-informed neural network (PINN) methods, guiding the design of neural networks.

翻译：构建神经网络架构是机器学习领域的一项具有挑战性的追求，而关于网络应选择更深还是更宽的两难问题始终悬而未决。本文深入探讨了具有灵活层数的深度神经网络与具有有限隐藏层的宽度神经网络在Sobolev损失下的最优泛化误差对比。理论分析表明，神经网络架构会受到多种因素的显著影响，包括样本点数量、网络内部参数数量以及损失函数的正则性。具体而言，较多的参数数量倾向于偏好宽度神经网络，而样本点数量的增加以及损失函数正则性的增强则有利于采用深度神经网络。最终，我们通过深度Ritz方法和物理信息神经网络方法将该理论应用于求解偏微分方程，从而指导神经网络的设计。