Kati8/ICLR_2023_2025 huggingface dataset:
authors: Bocheng Zeng
authors: Mengtao Yan
authors: Ruizhi Chengze
authors: Yi Zhang
authors: Hongsheng Liu
authors: Zidong Wang
authors: Hao Sun
reviews: 
Comments: The paper introduces a graph learning approach called Physics-encoded Message Passing Graph Network (PhyMPGN) designed to model spatiotemporal PDE systems on coarse unstructured meshes using small training datasets. Authors incorporate a learnable Laplace block that encodes the discrete Laplace-Beltrami operator to constrain the GNN learning within a physically feasible solution space. Additionally, a boundary condition padding strategy is proposed to handle different types of boundary conditions, enhancing model convergence and accuracy. Extensive experiments demonstrate that PhyMPGN accurately predicts various spatiotemporal dynamics on coarse unstructured meshes with complex BCs, outperforming other baseline models with considerable gains.
Strengths: 1.	The integration of physics-encoded components within a message-passing GNN addresses the challenges of modeling spatiotemporal dynamics on unstructured meshes.
2.	The learnable Laplace block that encodes the discrete Laplace-Beltrami operator ensures solutions remain within a physically feasible space.
3.	The experiments are thorough, showcasing the model's ability to generalize across various spatiotemporal dynamics. The ablation studies further strengthen the validity of the proposed methods.
4.	The model demonstrates strong generalization capabilities despite being trained on small datasets.
Weaknesses: 1.	The authors state that traditional numerical methods require fine meshes and small time stepping. This overlooks implicit schemes, which can handle larger time steps while maintaining stability.
2.	The paper does not provide a comparative analysis of the computational speed or efficiency of PhyMPGN.
3.	The rationale behind focusing solely on coarse meshes is not explained.
Recommendation: 8
Reviewer confidence: 4
Soundness/correctness: 3
Clarity: 4
reviews: 
Comments: I love the paper for its outperforming MGN; it has a solid baseline by DeepMind! It introduces PhyMPGN, a Physics-encoded Message Passing Graph Network that effectively models spatiotemporal PDE systems on irregular meshes using limited data. By integrating physics through a learnable Laplace-Beltrami operator and a novel boundary condition padding strategy, the approach ensures physically accurate predictions. PhyMPGN significantly outperforms existing methods, achieving over 50% performance gains and demonstrating strong generalization across various PDEs and conditions. The model's efficiency and robustness make it a valuable advancement for scientific simulations where data is sparse or complex geometries are involved.
Strengths: Outperform a solid baseline; writing is good, presentation is good. The method is clear and new. The strength of this paper lies in its innovative integration of physics-based knowledge into graph neural networks, which enables accurate and efficient modeling of complex spatiotemporal PDE systems on irregular meshes with limited data. By employing a learnable Laplace block and a novel boundary condition padding strategy, PhyMPGN ensures solutions remain physically consistent and precise, overcoming limitations of traditional and purely data-driven methods. Additionally, the model's demonstrated ability to generalize well to different PDEs, geometries, and conditions, along with its significant performance gains over existing techniques, highlights its robustness and versatility for real-world scientific applications
Weaknesses: There is some need for some clarification when it is for convection-dominant problem.
Recommendation: 10
Reviewer confidence: 5
Soundness/correctness: 4
Clarity: 4
reviews: 
Comments: The manuscript introduces a way to approximate a non-linear PDE operator with a GNN block completed with a Laplace operator block.
The examples include a varity of linear and linear PDEs.
Strengths: The way to approximate a non-linear PDE operator with a GNN block completed with a Laplace operator block looks fairly original.
Weaknesses: 1.	The ultimate goal of the manuscript is to learn the nonlinear operator $F$.  However, it is not explained why it is needed while $F$ is known and given.
2.	The way how the authors approximate the nonlinear operator F, Eq. (6), evidently has some limitations. For example, it won’t work well when $F = (\Delta u)^2 $ and so on.  Consider being more specific in defing the operator F in Eq. (1).
3.	Section 4 is difficult to follow since the actual PDEs and their coefficients are not specified.
4.	It is not really explained why paddling is needed to incorporate boundary conditions.
Recommendation: 6
Reviewer confidence: 4
Soundness/correctness: 3
Clarity: 3
reviews: 
Comments: The paper is related to PDE systems solution in two-dimensional space on irregular domains. The problem is addressed with the mix of GNNs and classical numerical schemes. The authors propose several novel elements, such as trainable Laplace block, second order Runge-Kutta scheme for simulating temporal dynamics and a padding strategy to incorporate boundary conditions of different types. These improvements allowed the authors to achieve new quality of simulation. The obtained model has good generalization capabilities over initial conditions and over the unseen types of flow (such as another Reynolds number)
Strengths: * The authors have shown that spatiotemporal dynamics can be modeled by GNN in spatial domain together with 2nd order numerical scheme in temporal domain
* The authors proposed and tested a combination of reasonable improvements to the previous works with similar techniques (Pfaff et al. 2021). There key additions are Laplace block and padding strategy for different types of boundary conditions
* The proposed improvements allowed authors to outperform the competitors in terms of quality for chosen test problems
* Generalization and ablation studies substantiate the authors contributions
Weaknesses: The limitations of the work were not addressed properly. In particular:
* applicability to the PDE systems without the Laplace operator
* simulation for the big number of time steps (for example this occurs for weather simulations)
* boundary conditions padding for the domain with non-straight boundary – it’s not clear how to work with them

Also the experiments are insufficient:
* It’s important to investigate the computational requirements of the proposed approach for training and inference in dependence on the important parameters (number of time steps, size of the training dataset, etc.) in order to assess its applicability in practice
* Also it would be useful to understand how the segment size M in 3.4 influences the results and computational requirements

Also the paper is hard to read and it needs some restructurization. Probably  the introduction can be shortened a little bit together with the beginning of the abstract and authors could explain in more details how the model is trained and how different types of boundary conditions are introduced for curved boundary
Recommendation: 8
Reviewer confidence: 5
Soundness/correctness: 3
Clarity: 3