PDE generalization of in-context operator networks: A study on 1D scalar nonlinear conservation laws

Abstract

Can we build a single large model for a wide range of PDE-related scientific learning tasks? Can this model generalize to new PDEs 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 positive evidence for the second question above, through a study on 1D scalar nonlinear conservation laws, a family of PDEs with temporal evolution. In particular, we show that an ICON model trained on conservation laws with cubic flux functions can generalize well to some other flux functions of more general forms, without fine-tuning. 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.