Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning

2026-05-21

AI Summary

This work addresses the approximation and statistical complexity challenges in multi-task operator learning by proposing a Multiple Neural Operators (MNO) architecture that enables joint learning across tasks under a shared representation. By integrating function space approximation theory with statistical learning analysis, the study demonstrates that multi-task learning exhibits the same scaling laws as single-task learning, indicating that shared representations do not incur additional overall learning costs. The authors establish minimax lower bounds on parametric complexity and derive near-optimal upper bounds for both approximation and generalization errors. Theoretical analysis further reveals that, in the worst case, MNO achieves convergence rates comparable to those of input-concatenated DeepONet extensions, thereby confirming its statistical efficiency and expressive power.

Abstract

We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling laws as single operator learning. We also compare MNO with a multi-task extension of DeepONet based on concatenated task inputs and show that, from a worst-case approximation-complexity perspective, both architectures satisfy essentially the same asymptotic rates.

Problem

• multi-task learning
• neural operators
• approximation complexity
• statistical generalization
• minimax rates

Innovation

• Multiple Neural Operators
• multi-task operator learning
• approximation complexity
• statistical generalization
• minimax rates