  We present a novel training method for deep operator networks (DeepONets),
one of the most popular neural network models for operators. DeepONets are
constructed by two sub-networks, namely the branch and trunk networks.
Typically, the two sub-networks are trained simultaneously, which amounts to
solving a complex optimization problem in a high dimensional space. In
addition, the nonconvex and nonlinear nature makes training very challenging.
To tackle such a challenge, we propose a two-step training method that trains
the trunk network first and then sequentially trains the branch network. The
core mechanism is motivated by the divide-and-conquer paradigm and is the
decomposition of the entire complex training task into two subtasks with
reduced complexity. Therein the Gram-Schmidt orthonormalization process is
introduced which significantly improves stability and generalization ability.
On the theoretical side, we establish a generalization error estimate in terms
of the number of training data, the width of DeepONets, and the number of input
and output sensors. Numerical examples are presented to demonstrate the
effectiveness of the two-step training method, including Darcy flow in
heterogeneous porous media.
