  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.
