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Multipole Graph Neural Operator for Parametric Partial Differential Equations

Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar

TL;DR

<3-5 sentence high-level summary> MGKN addresses the challenge of learning discretization-invariant solution operators for parametric PDEs by introducing a multipole-inspired multi-scale graph kernel network. It unifies GNNs with multi-resolution matrix factorization via a V-cycle algorithm and Nyström-induced points to achieve linear time complexity while capturing long-range interactions. Empirical results on Darcy flow and Burgers' equation show discretization invariance, scalability to high-resolution meshes, and competitive accuracy against standard baselines. This approach offers a scalable, mesh-agnostic framework for neural operators applicable to complex multi-scale PDEs.

Abstract

One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.

Multipole Graph Neural Operator for Parametric Partial Differential Equations

TL;DR

<3-5 sentence high-level summary> MGKN addresses the challenge of learning discretization-invariant solution operators for parametric PDEs by introducing a multipole-inspired multi-scale graph kernel network. It unifies GNNs with multi-resolution matrix factorization via a V-cycle algorithm and Nyström-induced points to achieve linear time complexity while capturing long-range interactions. Empirical results on Darcy flow and Burgers' equation show discretization invariance, scalability to high-resolution meshes, and competitive accuracy against standard baselines. This approach offers a scalable, mesh-agnostic framework for neural operators applicable to complex multi-scale PDEs.

Abstract

One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.

Paper Structure

This paper contains 34 sections, 17 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Main contributions.
  3. Operator Learning
  4. Parametric PDEs.
  5. Learning the Operator.
  6. Graph Kernel Network (GKN)
  7. Kernel operator.
  8. Kernel convolution on graphs.
  9. Domain of Integration.
  10. Nyström approximation.
  11. Multipole Graph Kernel Network (MGKN)
  12. Decomposition of the kernel
  13. Multi-scale graphs.
  14. Linear complexity.
  15. V-cycle Algorithm
  16. ...and 19 more sections

Figures (4)

  • Figure 1: V-cycle
  • Figure 2: Hierarchical matrix decomposition
  • Figure 3: Properties of multipole graph kernel network (MGKN) on Darcy flow
  • Figure 4: Comparsion with benchmarks on Burgers equation, with examples of inputs and outputs.