深度神经网络在偏微分方程逼近中展现超收敛，优于传统数值方法。
Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms
- 使用通用激活函数的深度网络，在Sobolev空间中逼近解
- 误差率超越有限元与谱方法，实现超收敛现象
- 为科学计算中的神经网络提供统一理论基础
本文建立了具有常见通用激活函数的深度全连接神经网络在Sobolev空间 $W^{n, ext{∞}}$ 中的完整逼近结果，误差以 $W^{m,p}$-范数衡量（$m < n$，$1\le p \le \infty$）。所得逼近率优于经典数值方法（如有限元法和谱方法），展现出我们称之为“超收敛”的现象。分析表明，采用通用激活函数的深度网络在逼近偏微分方程（PDE）弱解时，精度高于传统数值方法。此外，该工作填补了基于神经网络求解PDE的误差估计理论的重要空白，为科学计算中应用神经网络提供了统一的理论基础。
原文摘要 · Abstract (English)
This paper establishes a comprehensive approximation result for deep fully-connected neural networks with commonly-used and general activation functions in Sobolev spaces $W^{n,\infty}$, with errors measured in the $W^{m,p}$-norm for $m < n$ and $1\le p \le \infty$. The derived rates surpass those of classical numerical approximation techniques, such as finite element and spectral methods, exhibiting a phenomenon we refer to as \emph{super-convergence}. Our analysis shows that deep networks with general activations can approximate weak solutions of partial differential equations (PDEs) with superior accuracy compared to traditional numerical methods at the approximation level. Furthermore, this work closes a significant gap in the error-estimation theory for neural-network-based approaches to PDEs, offering a unified theoretical foundation for their use in scientific computing.
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