Research map / Point clouds / Point cloud classification
RIGNO: A Graph-based Framework For Robust And Accurate Operator Learning For PDEs On Arbitrary Domains
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes RIGNO, a multi-scale graph neural operator that maps between point clouds via a downsampled regional mesh, with edge masking, lead-time conditioning, and fractional pairing for resolution- and time-continuous operator learning.
Problem
Learning PDE solution operators on arbitrary domains and point clouds while achieving spatial and temporal resolution invariance and capturing long-range dependencies.
Approach
An encode-process-decode GNN that aggregates physical nodes into randomly subsampled regional nodes with radius-adaptive support regions, runs multi-scale message passing on the regional mesh, and uses random edge masking, lead-time-conditioned normalization, and a fractional-pairing fine-tuning to achieve resolution and time invariance.
Key results
- RIGNO-18 achieves the lowest median relative L1 error on all 13 unstructured/point-cloud datasets, e.g., 0.04% on Heat-L-Sines vs 0.15% for the best baseline Geo-FNO and 0.56% vs 0.73% for RIGNO-12 on NS-SVS.
- On 9 Cartesian-grid PDEGym datasets, RIGNO-18 is best on 8 of 9, e.g., NS-Gauss 2.74% vs 2.92% for scOT and 10.9% for CNO.
- Fractional pairing fine-tuning reduces errors for unseen small lead times, yielding monotone error accumulation for t >= t4 in NS-PwC/NS-SL experiments.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use RIGNO-18 instead of the strongest listed baseline | PDE operator learning on unstructured point clouds · 13 unstructured datasets | median relative L1 error (%) | Helped |
| Add edge masking during training (ablation) | RIGNO · NS-PwC resolution generalization | relative test error | Helped |
| Add edge masking during training (ablation) | RIGNO · NS-PwC training and inference | computational cost: by a factor of 2 when 50% of the edges are masked | Helped |
| Fine-tune with fractional pairing (ablation) | RIGNO · NS-PwC temporal generalization to smaller unseen time steps | test error | Helped |
| Use derivative time marching (alpha=1, beta=tau) instead of output or residual marching (ablation) | RIGNO · NS-SL temporal rollout | autoregressive test error | Helped |
Surprising findings
- Authors report that standard UPT yields very large errors on several datasets (e.g., 100% on NS-PwC and ACE), indicating strong baseline sensitivity.
- Poisson-Gauss on Cartesian grids is best handled by scOT (0.68%), with RIGNO not winning there, consistent with the claim that global interactions are less favored by local message passing.
Limitations
- Focuses on 2D PDEs; 3D extension not demonstrated due to lack of public 3D benchmarks.
- Gains are less pronounced for elliptic problems where global interactions matter.
- Standalone neural operators require many training samples for small error; fractional pairing can cause instability if intermediate approximations are inaccurate.
Experimental setup
Datasets: Heat-L-Sines Wave-C-Sines AF Elasticity PDEGym NS-Gauss PDEGym NS-PwC PDEGym NS-SL PDEGym NS-SVS PDEGym CE-Gauss PDEGym CE-RP ACE Wave-Layer
Benchmarks: Heat-L-Sines Wave-C-Sines NS-Gauss NS-PwC NS-SL NS-SVS CE-Gauss CE-RP ACE Wave-Layer Poisson-Gauss AF
Metrics: median relative L1 error
Baselines: MeshGraphNet Geo-FNO FNO DSE GINO UPT CNO scOT FNO
Builds on: MeshGraphNet all2all training Fourier Neural Operator GINO
Closest papers by meaning
- Geometry Aware Operator Transformer as an efficient and accurate neural surrogate for PDEs on arbitrary domains (NeurIPS 2025 · similarity 0.89)
- Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations (ICML 2024 · similarity 0.88)
- Latent Neural Operator for Solving Forward and Inverse PDE Problems (NeurIPS 2024 · similarity 0.88)
- Adaptive Coordinate Transforms for Neural Operators (NeurIPS 2026 · similarity 0.88)
- HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions (NeurIPS 2025 · similarity 0.88)
- Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems (ICLR 2025 · similarity 0.88)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.