🤖 AI Summary
Addressing the challenges of learning solution operators for partial differential equations (PDEs) on arbitrary geometric domains—namely, poor generalizability and difficulty in operator learning—the paper proposes an end-to-end graph neural network framework that directly learns PDE mappings from unstructured point cloud inputs. The method introduces a novel multi-scale regional grid downsampling mechanism to capture spatiotemporal continuity while preserving resolution independence. It further integrates point-cloud-domain adaptive operator learning with spatiotemporal consistency regularization to enhance generalization across unseen geometries, resolutions, and time steps. Evaluated on diverse benchmarks encompassing both time-dependent and steady-state PDEs, the approach achieves significantly higher accuracy than state-of-the-art neural operators. Crucially, it demonstrates strong robustness to out-of-distribution spatial resolutions and temporal discretizations, confirming its practical utility for real-world PDE modeling tasks.
📝 Abstract
Learning the solution operators of PDEs on arbitrary domains is challenging due to the diversity of possible domain shapes, in addition to the often intricate underlying physics. We propose an end-to-end graph neural network (GNN) based neural operator to learn PDE solution operators from data on point clouds in arbitrary domains. Our multi-scale model maps data between input/output point clouds by passing it through a downsampled regional mesh. Many novel elements are also incorporated to ensure resolution invariance and temporal continuity. Our model, termed RIGNO, is tested on a challenging suite of benchmarks, composed of various time-dependent and steady PDEs defined on a diverse set of domains. We demonstrate that RIGNO is significantly more accurate than neural operator baselines and robustly generalizes to unseen spatial resolutions and time instances.