  Can we build a single large model for a wide range of PDE-related scientific
learning tasks? Can this model generalize to new PDEs, even of new forms,
without any fine-tuning? In-context operator learning and the corresponding
model In-Context Operator Networks (ICON) represent an initial exploration of
these questions. The capability of ICON regarding the first question has been
demonstrated previously. In this paper, we present a detailed methodology for
solving PDE problems with ICON, and show how a single ICON model can make
forward and reverse predictions for different equations with different strides,
provided with appropriately designed data prompts. We show the positive
evidence to the second question, i.e., ICON can generalize well to some PDEs
with new forms without any fine-tuning. This is exemplified through a study on
1D scalar nonlinear conservation laws, a family of PDEs with temporal
evolution. We also show how to broaden the range of problems that an ICON model
can address, by transforming functions and equations to ICON's capability
scope. We believe that the progress in this paper is a significant step towards
the goal of training a foundation model for PDE-related tasks under the
in-context operator learning framework.
