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