Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs), due to complex geometries, interactions between physical variables, and the lack of large amounts of high-resolution training data. To address these issues, we propose Codomain Attention Neural Operator (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to the function space. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations and fluid-structure interactions, we found CoDA-NO to outperform existing methods on the few-shot learning task by over $36\%$. The code is available at https://github.com/ashiq24/CoDA-NO.

该研究提出了针对复杂几何结构、物理变量之间的相互作用和缺乏大量高分辨率训练数据等问题，通过在值域或通道空间中对函数进行分词，扩展位置编码、自注意力和标准化层，从而能够使用单一模型学习不同的偏微分方程系统，以实现学习多物理偏微分方程的目标。与现有方法相比，该方法在少量数据的复杂后续任务中，如流体流动模拟和流固相互作用中，通过少样本学习任务的表现超过36%。

基于预训练的共域注意力神经算子用于求解多物理偏微分方程