Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning
Provides theoretical guarantees for multi-task operator learning, establishing that it follows the same scaling laws as single-task learning, which is important for practitioners in scientific computing and engineering.
This paper proves that multi-task operator learning with Multiple Neural Operators (MNO) achieves near-optimal approximation and generalization rates, showing that shared representations across tasks do not increase the overall cost compared to single-task learning.
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.