Graph-Based Operator Learning from Limited Data on Irregular Domains

TL;DR

This paper introduces GOLA, a graph-based operator learning framework with attention and Fourier encoding, enabling effective PDE modeling on irregular domains with limited data.

Contribution

GOLA combines graph neural networks with attention mechanisms and Fourier encoding to handle irregular domains and sparse data in operator learning for PDEs.

Findings

Outperforms baselines on 2D PDEs with irregular sampling.

Shows strong generalization in data-scarce regimes.

Efficiently models complex spatial dependencies.

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.