发表机构
Peking University; Chongqing Research Institute of Big Data, Peking University; China University of Mining and Technology（北京大学; 北京大学重庆大数据研究院; 中国矿业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文对随机神经网络求解椭圆界面问题进行了统一误差分析，推导了Barron函数积分表示和复合损失的O(n^{-1})泛化界，结合优化保证得到先验L^2误差估计，并通过数值实验验证。
AI 中文摘要
本文提出了随机神经网络（RaNNs）用于椭圆界面问题的统一误差分析。我们首先利用标准tanh激活函数推导出Barron函数的积分表示，从而得到一个C^2逼近界，该界对任何支撑在立方体上且密度严格为正的隐参数分布均成立。然后，我们建立了复合损失的定量O(n^{-1})泛化界，该损失同时强制执行偏微分方程、边界和界面条件，其中n表示训练样本数。将这些结果与RaNNs的优化保证相结合，我们得到了在存在界面不连续情况下神经网络解的先验L^2误差估计。最后，通过数值实验验证了理论结果。
英文摘要
This paper presents a unified error analysis of randomized neural networks (RaNNs) for elliptic interface problems. We first derive an integral representation of Barron functions using the standard $\tanh$ activation, which yields a $C^2$-approximation bound valid for any hidden-parameter distribution supported on a cube with strictly positive density. We then establish a quantitative $\mathcal{O}(n^{-1})$ generalization bound for the composite loss that enforces the partial differential equation, boundary, and interface conditions simultaneously, where $n$ denotes the number of training samples. Combining these results with the optimization guarantees for RaNNs, we obtain an a priori $L^2$ error estimate for the neural network solution in the presence of interface discontinuities. Numerical experiments are presented to validate the theoretical results.
