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Graph-Based Operator Learning from Limited Data on Irregular Domains

Yile Li, Shandian Zhe

TL;DR

This work tackles operator learning for PDE solution operators on irregular domains by introducing GOLA, a framework that combines an learnable Fourier encoder with attention-enhanced graph neural networks to model global and local dependencies from irregular samples. By constructing proximity graphs and projecting inputs into a spectral space, GOLA achieves strong data efficiency and robust generalization across multiple PDE types and sampling regimes. Theoretical analysis establishes a universal approximation guarantee under sufficient capacity, and empirical results demonstrate state-of-the-art performance versus baselines like DeepONet and FNO, especially in data-scarce scenarios. Overall, GOLA provides a flexible, mesh-free approach to learning PDE operators on non-Euclidean domains with meaningful practical impact for irregular geometries and sparse data.

Abstract

Operator learning seeks to approximate mappings from input functions to output solutions, particularly in the context of partial differential equations (PDEs). While recent advances such as DeepONet and Fourier Neural Operator (FNO) have demonstrated strong performance, they often rely on regular grid discretizations, limiting their applicability to complex or irregular domains. In this work, we propose a Graph-based Operator Learning with Attention (GOLA) framework that addresses this limitation by constructing graphs from irregularly sampled spatial points and leveraging attention-enhanced Graph Neural Netwoks (GNNs) to model spatial dependencies with global information. To improve the expressive capacity, we introduce a Fourier-based encoder that projects input functions into a frequency space using learnable complex coefficients, allowing for flexible embeddings even with sparse or nonuniform samples. We evaluated our approach across a range of 2D PDEs, including Darcy Flow, Advection, Eikonal, and Nonlinear Diffusion, under varying sampling densities. Our method consistently outperforms baselines, particularly in data-scarce regimes, demonstrating strong generalization and efficiency on irregular domains.

Graph-Based Operator Learning from Limited Data on Irregular Domains

TL;DR

This work tackles operator learning for PDE solution operators on irregular domains by introducing GOLA, a framework that combines an learnable Fourier encoder with attention-enhanced graph neural networks to model global and local dependencies from irregular samples. By constructing proximity graphs and projecting inputs into a spectral space, GOLA achieves strong data efficiency and robust generalization across multiple PDE types and sampling regimes. Theoretical analysis establishes a universal approximation guarantee under sufficient capacity, and empirical results demonstrate state-of-the-art performance versus baselines like DeepONet and FNO, especially in data-scarce scenarios. Overall, GOLA provides a flexible, mesh-free approach to learning PDE operators on non-Euclidean domains with meaningful practical impact for irregular geometries and sparse data.

Abstract

Operator learning seeks to approximate mappings from input functions to output solutions, particularly in the context of partial differential equations (PDEs). While recent advances such as DeepONet and Fourier Neural Operator (FNO) have demonstrated strong performance, they often rely on regular grid discretizations, limiting their applicability to complex or irregular domains. In this work, we propose a Graph-based Operator Learning with Attention (GOLA) framework that addresses this limitation by constructing graphs from irregularly sampled spatial points and leveraging attention-enhanced Graph Neural Netwoks (GNNs) to model spatial dependencies with global information. To improve the expressive capacity, we introduce a Fourier-based encoder that projects input functions into a frequency space using learnable complex coefficients, allowing for flexible embeddings even with sparse or nonuniform samples. We evaluated our approach across a range of 2D PDEs, including Darcy Flow, Advection, Eikonal, and Nonlinear Diffusion, under varying sampling densities. Our method consistently outperforms baselines, particularly in data-scarce regimes, demonstrating strong generalization and efficiency on irregular domains.

Paper Structure

This paper contains 21 sections, 20 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology
  4. Problem Formulation
  5. Graph Construction
  6. Fourier Encoder
  7. Message Passing
  8. Multi-Head Self-Attention
  9. Message Passing with Attention
  10. Training
  11. Theoretical Analysis
  12. Proposition.
  13. Experiments
  14. Conclusion
  15. Additional Results
  16. ...and 6 more sections

Figures (8)

  • Figure 1: GOLA Model
  • Figure 2: Test errors across PDE benchmarks with varying training data sizes
  • Figure 3: Error reduction heatmaps across training data sizes and sample densities for PDE Benchmarks
  • Figure 4: Test error trend with test sample density
  • Figure 5: Test error trend with train data size
  • ...and 3 more figures