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This paper proves the first universal approximation theorems for nonlinear operators and their derivatives in infinite-dimensional settings, extending classical results to operator learning architectures like DeepONet and PCA-Net.
This paper establishes quantitative Sobolev approximation bounds for neural operators, proving that operators can be uniformly approximated with explicit complexity-error relations. It validates these theoretical bounds using Fourier Neural Operators on the Burgers' equation, demonstrating that Sobolev-space approximation theory accurately predicts scaling behavior.