GraphDeepONet: Learning to simulate time-dependent partial differential equations using graph neural network and deep operator network

Sung Woong Cho, Jae Yong Lee, Hyung Ju Hwang

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Keywords: Physical simulations, Graph neural network, Message passing, neural PDE solvers, Deep operator network, DeepONet

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TL;DR: We propose an autoregressive model GraphDeepONet based on graph neural networks, to effectively adapt deep operator network.

Abstract:

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 outperforms existing GNN-based PDE solver models by accurately predicting solutions, even on irregular grids, while inheriting the advantages of DeepONet, allowing 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.

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Submission Number: 7592