Learning time-dependent PDE via graph neural networks and deep operator
  network for robust accuracy on irregular grids
Sung Woong Cho Jae Yong Lee Hyung Ju Hwang
Scientific computing using deep learning has seen significant advancements in
recent years. There has been growing interest in models that learn the operator
from the parameters of a partial differential equation (PDE) to the
corresponding solutions. Deep Operator Network (DeepONet) and Fourier Neural
operator, among other models, have been designed with structures suitable for
handling functions as inputs and outputs, enabling real-time predictions as
surrogate models for solution operators. There has also been significant
progress in the research on surrogate models based on graph neural networks
(GNNs), specifically targeting the dynamics in time-dependent PDEs. In this
paper, we propose GraphDeepONet, an autoregressive model based on GNNs, to
effectively adapt DeepONet, which is well-known for successful operator
learning. GraphDeepONet exhibits robust accuracy in predicting solutions
compared to existing GNN-based PDE solver models. It maintains consistent
performance even on irregular grids, leveraging the advantages inherited from
DeepONet and enabling predictions on arbitrary grids. Additionally, unlike
traditional DeepONet and its variants, GraphDeepONet enables time extrapolation
for time-dependent PDE solutions. We also provide theoretical analysis of the
universal approximation capability of GraphDeepONet in approximating continuous
operators across arbitrary time intervals.