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.

翻译：我们能否构建一个统一的大模型来处理各类偏微分方程（PDE）相关的科学学习任务？该模型能否在不经微调的情况下泛化至新形式甚至全新类型的PDE问题？上下文算子学习及其对应模型In-Context Operator Networks（ICON）是对上述问题的初步探索。已有研究证明了ICON在第一个问题上的可行性。本文详细阐述了利用ICON求解PDE问题的方法论，并展示了单一ICON模型如何在适当设计的数据提示下，对不同方程实现不同步长的正向预测与反向推理。针对第二个问题，我们提供了积极证据，即ICON无需任何微调即可良好泛化至某些新形式PDE。我们以一维标量非线性守恒律（具有时间演化特性的PDE族）为例验证了该能力。此外，我们还展示了如何通过将函数与方程变换至ICON能力范畴来扩展模型可处理的问题范围。我们相信，本文取得的进展标志着在上下文算子学习框架下训练PDE基础模型这一目标迈出了重要一步。