提出多输入神经算子在Sobolev空间中的泛化保证，揭示各输入源对误差的贡献。
Generalization Guarantees for Multi-Input Neural Operator Learning in Sobolev Spaces
- 构建多输入算子学习的理论框架，支持不同域、维数与正则性的输入函数
- 误差界明确量化各输入空间对总误差的影响，平衡情形下受输入维数、正则性与Sobolev阶交互制约
- 适用于偏微分方程与科学计算中的算子学习问题，适合理论研究者参考
我们为多输入神经算子建立了逼近与泛化误差估计，输出误差以Sobolev范数度量。与传统单输入算子学习不同，本框架允许多个输入函数定义在可能不同的域上，具有不同维度和Sobolev正则性。推导出的误差率显式量化了每个输入空间对最终误差界的影响。特别地，在平衡情况下，逼近与泛化率由输入维数、正则性与Sobolev阶之间的相互作用决定，而模型复杂度的依赖保持\(\log\log/\log\)型结构。该分析为多输入算子学习提供了通用理论框架，包括Sobolev训练，适用于来自偏微分方程和科学计算的算子学习问题。
原文摘要 · Abstract (English)
We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms. In contrast to standard operator-learning settings with a single input function, our framework allows multiple input functions defined on possibly different domains, with different dimensions and Sobolev regularities. The derived rates explicitly quantify the contribution of each input space to the final error bound. In particular, in the balanced regime, the approximation and generalization rates are governed by the interaction between the input dimensions, regularities, and Sobolev orders, while the dependence on the model complexity retains a \(\log\log/\log\)-type structure. Our analysis provides a general theoretical framework for multi-input operator learning, including Sobolev training, and is applicable to operator learning problems arising from partial differential equations and scientific computing.
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。