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挑战性探索，而选择更深还是更宽的困境始终是一个持久问题。本文对具有灵活层数的深度神经网络与具有受限隐藏层的宽度神经网络进行了比较，重点研究了它们在Sobolev损失下的最优泛化误差。分析研究表明，神经网络的架构会受到多种因素的显著影响，包括样本点数量、神经网络中的参数以及损失函数的正则性。具体而言，参数数量较多时倾向于采用宽度神经网络，而样本点数量增加和损失函数正则性更强时则倾向于采用深度神经网络。最终，我们将该理论应用于深度Ritz方法和物理信息神经网络方法处理偏微分方程，为神经网络的设计提供指导。