  This paper introduces a new neural-network-based approach, namely In-Context
Operator Networks (ICON), to simultaneously learn operators from the prompted
data and apply it to new questions during the inference stage, without any
weight update. Existing methods are limited to using a neural network to
approximate a specific equation solution or a specific operator, requiring
retraining when switching to a new problem with different equations. By
training a single neural network as an operator learner, we can not only get
rid of retraining (even fine-tuning) the neural network for new problems, but
also leverage the commonalities shared across operators so that only a few
demos in the prompt are needed when learning a new operator. Our numerical
results show the neural network's capability as a few-shot operator learner for
a diversified type of differential equation problems, including forward and
inverse problems of ordinary differential equations (ODEs), partial
differential equations (PDEs), and mean-field control (MFC) problems, and also
show that it can generalize its learning capability to operators beyond the
training distribution.
