提出对称神经网络，突破高维函数逼近的维度诅咒。
On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functions
- 设计对称结构的深度网络，适配具有置换对称性的函数。
- 逼近误差与常数因子仅随维度多项式增长，无指数爆炸。
- 适合学习高维对称物理函数，如分子能量预测等场景。
深度神经网络被广泛用作具有内在物理结构（包括置换对称性）函数的通用逼近器。本文构建了用于逼近对称Korobov函数的对称深度神经网络，并证明其收敛速率和常数前因子最多随环境维度多项式增长。这相较于以往受维度诅咒影响的逼近保证有显著提升。基于这些逼近界，进一步推导出学习对称Korobov函数时的泛化误差率，其主导因子同样避免了维度诅咒。
原文摘要 · Abstract (English)
Deep neural networks have been widely used as universal approximators for functions with inherent physical structures, including permutation symmetry. In this paper, we construct symmetric deep neural networks to approximate symmetric Korobov functions and prove that both the convergence rate and the constant prefactor scale at most polynomially with respect to the ambient dimension. This represents a substantial improvement over prior approximation guarantees that suffer from the curse of dimensionality. Building on these approximation bounds, we further derive a generalization-error rate for learning symmetric Korobov functions whose leading factors likewise avoid the curse of dimensionality.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。