Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO have demonstrated the superiority of solving partial differential equations (PDEs) over other deep learning methods. However, the spatial domain of its input function needs to be identical to its output, which limits its applicability. For instance, the widely used Fourier neural operator (FNO) fails to approximate the operator that maps the boundary condition to the PDE solution. To address this issue, we propose a novel framework called resolution-invariant deep operator (RDO) that decouples the spatial domain of the input and output. RDO is motivated by the Deep operator network (DeepONet) and it does not require retraining the network when the input/output is changed compared with DeepONet. RDO takes functional input and its output is also functional so that it keeps the resolution invariant property of NO. It can also resolve PDEs with complex geometries whereas NO fail. Various numerical experiments demonstrate the advantage of our method over DeepONet and FNO.

翻译：神经算子是一种具有函数输出的离散化不变深度学习方法，能够逼近任意连续算子。与其他深度学习方法相比，神经算子在求解偏微分方程方面展现出优越性。然而，其输入函数的空间域必须与输出相同，这限制了其适用性。例如，广泛使用的傅里叶神经算子无法逼近将边界条件映射到偏微分方程解的算子。为解决此问题，我们提出一种称为"分辨率不变深度算子"的新框架，该框架解耦了输入与输出的空间域。RDO受深度算子网络启发，与DeepONet相比，当输入/输出发生变化时无需重新训练网络。RDO接受函数输入并产生函数输出，从而保持神经算子的分辨率不变特性。它还能求解神经算子无法处理的复杂几何PDE问题。多种数值实验证明了我们的方法相较于DeepONet和FNO的优势。