Error estimates for DeepONets: a deep learning framework in infinite dimensions

Abstract DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite-dimensional Banach spaces. We analyze and prove estimates on the resulting approximation generalization errors. In particular, we extend universal property of to include measurable mappings in non-compact By decomposition error into encoding, reconstruction errors, both lower upper bounds total error, relating it spectral decay properties covariance operators, associated with underlying measures. derive almost optimal very general affine reconstructors random sensor locations well using covering number arguments. illustrate our four prototypical examples namely those arising forced ordinary differential equation, an elliptic partial equation (PDE) variable coefficients parabolic hyperbolic PDEs. While arbitrary Lipschitz by accuracy $\epsilon $ is argued suffer from ‘curse dimensionality’ (requiring neural networks exponential size $1/\epsilon $), contrast, all above concrete interest, rigorously that can break this curse dimensionality (achieving grow algebraically $).Thus, demonstrate efficient potentially large class machine framework.