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.

神经操作符（NO）是一种具有功能输出的离散不变深度学习方法，可以近似任意连续算子。然而，其输入函数的空间域与输出必须相同，这限制了其适用性。为了解决这个问题，我们提出了一种名为分辨率不变深度操作符（RDO）的创新框架，它解耦了输入和输出的空间域。RDO能够保持NO的分辨率不变性，解决了具有复杂几何结构的PDE问题。通过各种数值实验证明了我们的方法相比DeepONet和FNO的优势。

具有复杂几何结构的PDE的分辨率不变深度操作网络