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Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured grid

Subhankar Sarkar, Souvik Chakraborty

TL;DR

This paper introduces a novel approach, referred to as Spatio-Spectral Graph Neural Operator (Sp$^2$GNO) that integrates spatial and spectral GNNs effectively and enables the learning of solution operators across arbitrary geometries, thus catering to a wide range of real-world problems.

Abstract

Scientific machine learning has seen significant progress with the emergence of operator learning. However, existing methods encounter difficulties when applied to problems on unstructured grids and irregular domains. Spatial graph neural networks utilize local convolution in a neighborhood to potentially address these challenges, yet they often suffer from issues such as over-smoothing and over-squashing in deep architectures. Conversely, spectral graph neural networks leverage global convolution to capture extensive features and long-range dependencies in domain graphs, albeit at a high computational cost due to Eigenvalue decomposition. In this paper, we introduce a novel approach, referred to as Spatio-Spectral Graph Neural Operator (Sp$^2$GNO) that integrates spatial and spectral GNNs effectively. This framework mitigates the limitations of individual methods and enables the learning of solution operators across arbitrary geometries, thus catering to a wide range of real-world problems. Sp$^2$GNO demonstrates exceptional performance in solving both time-dependent and time-independent partial differential equations on regular and irregular domains. Our approach is validated through comprehensive benchmarks and practical applications drawn from computational mechanics and scientific computing literature.

Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured grid

TL;DR

This paper introduces a novel approach, referred to as Spatio-Spectral Graph Neural Operator (SpGNO) that integrates spatial and spectral GNNs effectively and enables the learning of solution operators across arbitrary geometries, thus catering to a wide range of real-world problems.

Abstract

Scientific machine learning has seen significant progress with the emergence of operator learning. However, existing methods encounter difficulties when applied to problems on unstructured grids and irregular domains. Spatial graph neural networks utilize local convolution in a neighborhood to potentially address these challenges, yet they often suffer from issues such as over-smoothing and over-squashing in deep architectures. Conversely, spectral graph neural networks leverage global convolution to capture extensive features and long-range dependencies in domain graphs, albeit at a high computational cost due to Eigenvalue decomposition. In this paper, we introduce a novel approach, referred to as Spatio-Spectral Graph Neural Operator (SpGNO) that integrates spatial and spectral GNNs effectively. This framework mitigates the limitations of individual methods and enables the learning of solution operators across arbitrary geometries, thus catering to a wide range of real-world problems. SpGNO demonstrates exceptional performance in solving both time-dependent and time-independent partial differential equations on regular and irregular domains. Our approach is validated through comprehensive benchmarks and practical applications drawn from computational mechanics and scientific computing literature.
Paper Structure (18 sections, 2 theorems, 46 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 2 theorems, 46 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup and Background
  3. Problem Setup
  4. Neural Operator
  5. Spectral Graph theory and Spectral Graph Convolution
  6. Spatial Graph Convolution
  7. Spatial-Spectral Graph Neural Operator (Sp$^2$GNO)
  8. Overall architecture of Sp$^2$GNO
  9. Sp$^2$GNO Block
  10. Global Spectral Graph Convolution based on truncated GFT
  11. Local Convolution through Spatial GNN with Gating Mechanism
  12. Collaboration between Spatial and Spectral GNN
  13. Numerical experiments
  14. Darcy Flow Problem
  15. Flow Around Airfoils
  16. ...and 3 more sections

Key Result

Lemma 1

Consider the spectral graph convolution $\bm y = \mathbf Q g_{\bm \theta} \mathbf Q^T \mathbb X$, where $\mathbf Q$ is the $N \times N$ matrix of eigenvectors of the graph Laplacian, and $g_\theta$ is a diagonal matrix representing the spectral filter. Let $\bm{y_m} = \mathbf{Q_m} g_{\bm \theta}^{(m

Figures (7)

  • Figure 1: The overall working principal of the proposed Spatio-Spectral Graph Neural Operator. It involves graph structure generation from point clouds, uplifting the feature, iterating through Sp$^2$GNO block and then downlifting.
  • Figure 2: Neural architecture of a Sp$^2$GNO block. It exploits spectral and spatial graph neural network to formulate the kernel integration operator. The overall architecture involves multiple such blocks.
  • Figure 3: Predictions in Darcy dataset: We have plotted input coefficients, ground truth, predictions, and errors of four different test examples in four columns. The first row represents the permeability coefficients for each of the four examples. The second and third rows represent the ground truth and predictions for the same. The fourth row shows the MSE error.
  • Figure 4: Super resolution Predictions in Darcy dataset: The leftmost figure is the prediction in a $141 \times 141$ resolution. The figure in the middle is the ground truth in $85 \times 85$ resolution. The figure on the right-hand side is the MSE error.
  • Figure 5: Predictions in Airfoil dataset: We have plotted input mesh coordinates, ground truth, predictions, and errors of four different test examples in four columns. The first row represents the mesh coordinates for each of the four examples, which serves as the input for the model. The second and third rows represent the ground truth and predictions for the same. The fourth row shows the MSE error.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Lemma 1
  • proof
  • Theorem 1: Bourgain Embedding Theorem