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.

翻译：我们研究共享多任务设置中学习算子集合的逼近和统计复杂度，重点关注多种经算子（MNO）架构。对于广泛的Lipschitz多算子映射类，我们推导出逼近和统计泛化能力的近最优上界。在下界方面，我们建立了参数复杂度的"维数灾难"并证明了相应的极小极大速率。这些结果共同表明任务间共享表示不会增加总体成本：多任务算子学习遵循与单算子学习相同的标度律。我们还将MNO与基于拼接任务输入的DeepONet多任务扩展进行对比，证明从最坏情况逼近复杂度角度，两种架构满足本质相同的渐近速率。