Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

The paper builds on a powerful but less-known result: a neural network with a single hidden layer can accurately approximate not only continuous functions but any nonlinear continuous operator, a mapping from one function to another. The authors extend this universal approximation theorem of operators to deep neural networks and design a new architecture, the deep operator network (DeepONet), engineered for small generalization error. DeepONet consists of two subnetworks: a branch net that encodes the discrete input function space and a trunk net that encodes the domain of the output functions.

The authors demonstrate that DeepONet can learn a range of explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. They study different formulations of the input function space and their effect on generalization error across 16 diverse applications, showing that neural networks can efficiently learn mathematical operators, an important capability for modeling complex systems such as robotics control.