Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids

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.