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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

TL;DR

This work introduces GraphDeepONet, a graph-based extension of DeepONet for learning time-dependent PDE solution operators on irregular grids. By embedding the initial state through a GNN-based branch, evolving time via autoregressive latent-space updates, and reconstructing solutions with a global basis (trunk net), the model achieves time extrapolation and grid-independent predictions with strong accuracy. The authors provide universal approximation theory showing convergence across arbitrary time intervals and demonstrate robust performance on Burgers', shallow water, and Navier–Stokes datasets, outperforming several GNN-based solvers on irregular grids. The approach enables accurate, continuous spatial predictions and hard boundary enforcement, offering practical impact for real-time surrogate modeling in complex geometries, while identifying opportunities to improve regular-grid performance and extend to more complex 2D PDEs.

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 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.

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

TL;DR

This work introduces GraphDeepONet, a graph-based extension of DeepONet for learning time-dependent PDE solution operators on irregular grids. By embedding the initial state through a GNN-based branch, evolving time via autoregressive latent-space updates, and reconstructing solutions with a global basis (trunk net), the model achieves time extrapolation and grid-independent predictions with strong accuracy. The authors provide universal approximation theory showing convergence across arbitrary time intervals and demonstrate robust performance on Burgers', shallow water, and Navier–Stokes datasets, outperforming several GNN-based solvers on irregular grids. The approach enables accurate, continuous spatial predictions and hard boundary enforcement, offering practical impact for real-time surrogate modeling in complex geometries, while identifying opportunities to improve regular-grid performance and extend to more complex 2D PDEs.

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 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.
Paper Structure (31 sections, 7 theorems, 55 equations, 11 figures, 4 tables)

This paper contains 31 sections, 7 theorems, 55 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. GraphDeepONet for time-dependent PDEs
  4. Problem statement
  5. DeepONet for time-dependent PDEs
  6. Proposed model: GraphDeepONet
  7. Encoder-Processor-Decoder framework
  8. Encoder $\epsilon$.
  9. Processor $\phi,\psi$.
  10. Decoder1 - Soft attention aggregation $\omega$.
  11. Decoder2 - Inner product of coefficients and basis $\boldsymbol{\tau}$.
  12. Recursive time prediction in latent space
  13. Distinctions between GraphDeepONet and other models
  14. Theoretical analysis of GraphDeepONet for time-dependent PDEs
  15. Experiments
  16. ...and 16 more sections

Key Result

Theorem 3.1

(Universality of GraphDeepONet) Let $\mathcal{G}^{(k)}: H^{s}(\mathbb{T}^{d}) \rightarrow H^{s}(\mathbb{T}^{d})$ be a Lipschitz continuous operator for each $k=1,...,K_{\text{frame}}$, and let $\mu$ be a probability measure on $L^2(\mathbb{T}^{d})$ characterized by a covariance operator with a bound

Figures (11)

  • Figure 1: The original DeepONet structure lu2021learning for simulating time-dependent PDE.
  • Figure 2: Framework of the proposed GraphDeepONet
  • Figure 3: Solution profile in Burgers' equation for time extrapolation simulation using DeepONet, VIDON, and GraphDeepONet.
  • Figure 4: Prediction of 2D shallow water equations on irregular sensor points with distinct training sensor points using graph-based models and GraphDeepONet. The Truth (irregular), MP-PDE, and MAgNet plot the solutions through interpolation using values from the irregular sensor points used during training, whereas GraphDeepONet predicts solutions for all grids directly.
  • Figure 5: Diagram of the GraphDeepONet
  • ...and 6 more figures

Theorems & Definitions (15)

  • Theorem 3.1
  • Theorem 3.2
  • Definition 2.1
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 5 more