论文标题
朝着深度运算符网的尺寸独立的概括范围
Towards Size-Independent Generalization Bounds for Deep Operator Nets
论文作者
论文摘要
最近，机器学习方法在成为分析物理系统的有用工具方面已取得了重大进步。该主题中一个特别活跃的领域是“物理知识的机器学习”，其重点是使用神经网以数值求解微分方程。在这项工作中，我们旨在推进训练DeepOnets时测量样本外误差的理论 - 这是用来搜索P.D.E系统的最通用的方法之一。首先，对于一类DeepOnets，我们证明了它们的Rademacher复杂性的束缚，该复杂性并未与所涉及的网的宽度明确扩展。其次，我们使用它来说明如何选择Huber损失，以便对于这些DeepOnet类别，可以获得对网络大小没有明确依赖的概括误差界限。我们得出的deponets的有效能力度量也证明与实验中的概括误差的行为相关。
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 P.D.E 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. The effective capacity measure for DeepONets that we thus derive is also shown to correlate with the behavior of generalization error in experiments.
