Resolution invariant deep operator network for PDEs with complex geometries

Abstract

Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO has demonstrated the superiority of solving partial differential equations (PDEs) over other deep learning methods. However, for the widely used Fourier neural operator (FNO), the spatial domain of its input function needs to be identical to its output, i.e., FNO fails to approximate the map from boundary conditions to PDE solutions, which limits its applicability. 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 FNO fails. Various numerical experiments demonstrate the advantage of our method over DeepONet and FNO.