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.

翻译：我们针对多输入神经算子发展了近似误差与泛化误差估计，其中输出误差在Sobolev范数下度量。与仅含单一输入函数的标准算子学习设定不同，我们的框架允许定义在可能不同域上的多个输入函数，这些函数具有不同的维度和Sobolev正则性。推导出的收敛速率明确量化了每个输入空间对最终误差边界的贡献。特别地，在平衡机制下，近似误差与泛化误差速率由输入维度、正则性及Sobolev阶数之间的相互作用主导，而对模型复杂度的依赖仍保持 \(\log\log/\log\) 型结构。我们的分析为多输入算子学习（包括Sobolev训练）提供了通用理论框架，适用于偏微分方程和科学计算中出现的算子学习问题。