Title: RIGNO: A Graph-based Framework For Robust And Accurate Operator Learning For PDEs On Arbitrary Domains
Venue: NeurIPS 2025
Accepted: yes
Keywords: scientific machine learning

operator learning

partial differential equation

neural operator

graph neural network

AI for science

computational science
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. The approach includes novel elements aimed at ensuring spatio-temporal resolution invariance. 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 resolutions both in space and in time. Our code is publicly available at github.com/camlab-ethz/rigno.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents RIGNO, a new graph-based neural operator framework for solving PDEs on arbitrary domains. RIGNO proposes a new approach for encoding sets of points via a regional mesh approach, which enable to compress the physical information. It then uses a message passing architecture to predict future time-steps via an all2all training approach. This allows to generalize to different timestep intervals. To enable continuous timestep evaluation, they propose temporal fractional pairing to allow generalization to intermediate timesteps not accessible in the training set. Their approach is evaluated on a range of PDEs and exhibit competitive or sota performance.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nI think the paper is clearly written, in particular, there are very nice illustrations in appendix that clearly help for the comprehension of the paper. \n\nThe method has been evaluated on a large range of PDEs with substantial experiments. On all datasets, Rigno-18 outperforms the considered baselines. the proposed framework is a simple but effective encode-process-decode approach, which allows to process arbitrary geometries. \n\nI think the paper also tackles an important problem: generalization on arbitrary geometries. The authors could other important works that differ from graphs or transformers such as INRs, which is another direction for learning on arbitrary geometries [1, 2].\n\nI also appreciated the ablations studies and additional experiments provided, notably for the time processing component. \n\nWeaknesses:\n\nRegarding the novelty of the paper, the method is relatively simple and follows a similar direction than existing approaches. It follows the encode-process-decode approach that is particularly effective for graph-based neural operators and make use of the training strategy proposed in Poseidon. Their contributions include the edge masking and the Fractional pairing strategy which are relatively simple. However, they provide ablations showing the advantage of their incorporation.\n\nFor the ablations on the dataset and model size, it would have been nice to test until when model size or data size reaches a plateau. It has been on relatively small/medium setups only.\n\nThere are other important baselines that could have been used such as [1] and [3] for instance.\n\nAs you mentioned in the introduction, efficiency is a key property for learning efficient neural operators on arbitrary geometries. Graphs are known to be particularly costly. It is important to add additional experiments to show how RIGNO behaves in terms of computation with respect to considered baselines.\n\n[1] Serrano et al. Operator Learning with Neural Fields: Tackling PDEs on General Geometries, NeurIPS 2023.\n\n[2] Wang et al. GridMix: Exploring Spatial Modulation for Neural Fields in PDE Modeling, ICLR 2025.\n\n[3] Wu et al. Transolver: A Fast Transformer Solver for PDEs on General Geometries, ICML 2024.\n\nQUESTIONS:\nDid you try to test your model on out-of-domain data, e.g. on new PDE coefficients? As you give the pde coefficients as input, it could be interesting to check how the model behaves on unseen coefficients.\n\nIn the all2all training, you only provide future timesteps to predict right? Could the model actually learns to solve an inverse problem ? e.g. predicting the past frames?\n\nLIMITATIONS:\nLimitations have been mentioned. Negative societal impacts have not been mentioned in the work.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents the Region Interacting Graph Neural Operator (RIGNO), a multi-scale, resolution-invariant graph neural operator that builds upon established architectural principles while introducing several interesting enhancements. The work follows a solid foundation using latent space representations and adopts the encoder-processor-decoder structure familiar from MeshGraphNet, enhanced with a novel multiscale regional mapping component.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\nThe paper demonstrates several commendable qualities. The experimental evaluation is particularly thorough, encompassing a comprehensive suite of PDE test problems and comparing against a diverse set of baseline methods. The reported performance improvements are impressive and suggest the method's practical value. The incorporation of edge masking for regularization and temporal processing through adaptive layer normalization with gating mechanisms shows thoughtful engineering.\n\n## Weaknesses and Questions\nWhile the work shows strong empirical results, I believe it could benefit from deeper analysis in a few key areas:\n- Novelty Clarification: The paper skillfully combines several existing techniques, but it would strengthen the contribution to more clearly articulate which specific innovations drive the state-of-the-art performance. A more explicit discussion of the novel aspects beyond the multiscale regional mapping would help readers understand the core contributions.\n- Ablation study: The work would significantly benefit from ablation studies examining each major component's contribution to overall performance. This would help identify which design choices are most critical and provide valuable insights for future work in this area.\n- Computational Efficiency: Including runtime and memory usage comparisons would provide important practical insights, especially given the multi-scale nature of the approach, which readers would find valuable for implementation considerations.\n\nQUESTIONS:\nBased on my understanding, the paper proposes to use graph neural operators for the flexibility of graph. Which component from RIGNO makes it signficiantly outperform previous state-of-the-art model such as CNO and Poseidon/SCOT on regular grids? Are these components applicable to other models?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose RIGNO (Region Interacting Graph Neural Operator), an end-to-end graph neural network for training the solution operators of PDEs on arbitrary domains. The main motivation for the article is that many existing Neural Operators (NOs) assume that the underlying domain of the PDEs is discretized on a regular grid. The results demonstrate that RIGNO outperforms the baselines on both unstructured and structured grids.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n* RIGNO demonstrate better performance compare with state-of-art baselines across a wide and challenging range of benchmarks.\n* The paper introduces several novel techniques: \n\t* **edge masking**: the core idea is to randomly disable, or \"mask,\" a fraction of the edges in the graph before each message-passing step. This process is dynamic, with a different random set of edges being masked at every step and epoch.\n\t* **temporal fractional pairing** is a novel fine-tuning strategy designed to improve the generalization of time-continuous neural operators to time steps smaller than those seen during training.\n* The authors are also write about the limitations of their approach.\n* The introduction and related work sections clearly formulated the gap in the literature (the need for a multi-scale, resolution-invariant GNN operator) and position RIGNO as the solution.\n\n\n**Weaknesses**\n* The paper is purely empirical. It lacks theoretical justification for its key components:\n\t* Are there theoretical guarantees on why edge masking improve resolution invariance? It is presented only as an empirical result.\n* RIGNO contains a lot of hyperparameters. The article does not provide a sensitivity analysis for these hyperparameters. This complexity could make the model difficult to tune and apply for new problems, potentially hindering its practical application.\n* For a method to be a practical surrogate solver, its speed is as important as its accuracy. Therefore, it would be beneficial to include information in the article about the training and inference times for both RIGNO and the baselines.\n\nQUESTIONS:\nSee the weaknesses\n\nLIMITATIONS:\nThe authors discussed the method limitations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper proposes  a novel graph-based framework for learning solution operators of PDEs on arbitrary, non-Cartesian domains. RIGNO uses an end-to-endGNN with a hierarchical "regional mesh" to achieve space-time resolution invariance and handle irregular point clouds. The framework introduces key innovations including regional mesh construction, edge masking, and temporal fractional pairing for robust and generalizable operator learning. Experiments on 13 unstructured and 8 Cartesian-grid PDE datasets show that RIGNO significantly outperforms prior methods (e.g., MeshGraphNet, FNO, CNO) in both accuracy and generalization.

STRENGTHS AND WEAKNESSES:
**Strengths**
1. The paper is novel to the best of my knowledge. It introduces a unique combination of graph neural operators and multi-scale regional meshes to handle non-Cartesian geometries with space-time resolution invariance, which is not sufficiently addressed in prior work.

2. Writing is clear and well-structured.

**Weakness**
1. The current implementation focuses only on 2D domain. No empirical evidence or 3D experiments are shown.
2. Fig 2 is small and not clear.
3. Table1 looks strange. I believe the authors could find more baselines to fill the table.

QUESTIONS:
None.

LIMITATIONS:
None.
TITLE: Official Review
CLARITY: 4
IMPACT: 4
ORIGINALITY: 4
DATE: 2025-07-16