Generalization Guarantees for Multi-Input Neural Operator Learning in Sobolev Spaces

2026-06-15

AI Summary

This work addresses the absence of generalization error theory for multi-input neural operators in Sobolev spaces, particularly when input functions are defined on heterogeneous domains with differing dimensions and regularity. The paper establishes the first unified Sobolev generalization framework by integrating function approximation theory, Sobolev space analysis, and statistical learning theory. It derives complexity-dependent approximation and generalization error bounds of logarithmic-logarithmic over logarithmic type, quantifies the contribution of each input space to the overall error, and reveals the coupling mechanism among input dimensionality, regularity, and Sobolev smoothness order. The resulting theory is applicable to operator learning tasks in PDE solving and scientific computing, accurately characterizing the impact of multi-input interactions on learning performance under balanced settings.

Abstract

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.

Problem

• multi-input neural operators
• generalization guarantees
• Sobolev spaces
• operator learning
• approximation error

Innovation

• multi-input neural operators
• Sobolev spaces
• generalization error
• approximation theory
• operator learning