In recent times machine learning methods have made significant advances in becoming a useful tool for analyzing physical systems. A particularly active area in this theme has been "physics-informed machine learning" which focuses on using neural nets for numerically solving differential equations. In this work, we aim to advance the theory of measuring out-of-sample error while training DeepONets -- which is among the most versatile ways to solve PDE systems in one-shot. Firstly, for a class of DeepONets, we prove a bound on their Rademacher complexity which does not explicitly scale with the width of the nets involved. Secondly, we use this to show how the Huber loss can be chosen so that for these DeepONet classes generalization error bounds can be obtained that have no explicit dependence on the size of the nets. We note that our theoretical results apply to any PDE being targeted to be solved by DeepONets.

翻译：近期，机器学习方法在成为分析物理系统的有用工具方面取得了显著进展。这一领域中一个特别活跃的方向是“物理信息机器学习”，其专注于使用神经网络进行微分方程的数值求解。在本工作中，我们旨在推进训练DeepONet（一种用于一次性求解偏微分方程系统的最通用方法之一）时测量样本外误差的理论。首先，对于一类DeepONet，我们证明了其Rademacher复杂度的界，该界不显式地依赖于所涉及网络的宽度。其次，我们利用这一结果展示了如何选择Huber损失，使得对于这些DeepONet类别，可以获得与网络大小无显式依赖的泛化误差界。我们指出，我们的理论结果适用于DeepONet旨在求解的任何偏微分方程。