  Deep operator network (DeepONet) has demonstrated great success in various
learning tasks, including learning solution operators of partial differential
equations. In particular, it provides an efficient approach to predict the
evolution equations in a finite time horizon. Nevertheless, the vanilla
DeepONet suffers from the issue of stability degradation in the long-time
prediction. This paper proposes a {\em transfer-learning} aided DeepONet to
enhance the stability. Our idea is to use transfer learning to sequentially
update the DeepONets as the surrogates for propagators learned in different
time frames. The evolving DeepONets can better track the varying complexities
of the evolution equations, while only need to be updated by efficient training
of a tiny fraction of the operator networks. Through systematic experiments, we
show that the proposed method not only improves the long-time accuracy of
DeepONet while maintaining similar computational cost but also substantially
reduces the sample size of the training set.
