Operator Barron Regularity for Elliptic Equations and Application to Neural Operator Approximation

We develop a spectral Barron regularity framework for nonlinear operators between infinite-dimensional function spaces. Using Hilbert-valued Fourier representations of finite-dimensional input truncations, we establish bounds that are uniform in the input truncation dimension and prove arbitrary-order Barron regularity for a class of elliptic solution operators. Second, we use this operator Barron regularity to derive approximation error bounds for an FNO-type neural operator, with explicit dependence on the input truncation dimension, output reconstruction dimension, and network width. For a fixed input truncation dimension, optimizing the neural approximation bound over the Barron regularity order yields stretched-exponential decay of the approximation error in the network width. We also show some numerical experiments to support the predicted rate.