Mixtures of Neural Operators Reduce Active Complexity in Operator Learning

Operator-learning systems are not governed solely by total parameter count; for one query, the relevant bottleneck can be the model that must be loaded and evaluated. We study this distinction for classical neural operators on compact Sobolev subsets through a constructive comparison between routed mixtures of neural operators (MoNOs) and a fixed single-neural-operator construction. A MoNO routes each input function through a tree to one expert. Our main theorem shows that every scalar uniformly continuous nonlinear operator with bounded output Sobolev radius on the approximation set admits a MoNO approximation whose active expert has smaller depth and width scaling, and no larger rank scaling, than the analyzed single-neural-operator construction; for Lipschitz targets these expert quantities are bounded by $\mathcal{O}({\epsilon }^{-1})$. A coefficient-quantization tree also gives polynomial routing work for Lipschitz targets under an explicit coefficient-access model, while storage remains exponential. For bounded $C^{1,1}$ functionals, affine experts improve the exponential storage power over constant lookup under target-independent routing. We also prove a quantitative universal approximation theorem for the underlying neural-operator architecture, with explicit dependence on compact-set diameter and modulus of continuity.