In this work, a comprehensive numerical study involving analysis and experiments shows why a two-layer neural network has difficulties handling high frequencies in approximation and learning when machine precision and computation cost are important factors in real practice. In particular, the following basic computational issues are investigated: (1) the minimal numerical error one can achieve given a finite machine precision, (2) the computation cost to achieve a given accuracy, and (3) stability with respect to perturbations. The key to the study is the conditioning of the representation and its learning dynamics. Explicit answers to the above questions with numerical verifications are presented.

翻译：本研究通过综合数值分析与实验，揭示了当机器精度和计算开销成为实际应用中的关键因素时，双层神经网络在处理高频近似与学习任务时面临的根本性困难。具体而言，本文探讨了以下基础计算问题：(1) 在有限机器精度下可实现的最小数值误差；(2) 达到特定精度所需计算成本；(3) 对扰动的稳定性。研究的核心在于表征条件及其学习动力学机制。本文给出了上述问题的明确解答，并附有数值验证。