Kati8/ICLR_2023_2025 huggingface dataset:
authors: Nikolay Yavich
authors: Alexander Ryabov
authors: Evgeny Burnaev
authors: Vladimir Vanovskiy
reviews: 
Comments: This paper explores the use of graph neural networks for solving forward and inverse problems and particularly focuses on the incorporation of implicit solver. However, the writing is subpar and the procedures and advantages are not well explained. The experiments are also lack comparison with other methods.
Strengths: The question is interesting and combining GNN with finite element method seems natural.
Weaknesses: 1. The writing is subpar. There are many typos and grammatical errors. For example, "Compute $\nabla_bL$  with (12), whats is equivalent so the solution of a single linear system." should be "Compute $\nabla_bL$ with (12), which is equivalent to solving a single linear system."
2. One main focus of this paper is the incorporation of implicit solver. However, using an iterative solver and in a deep learning setting is well-studied in the Deep Equilibrium Models (DEQ) literature. The authors should compare their method with DEQ.
3. The experiments are not very convincing. The results in Section 3.4 is very poor and in no experiments the authors compare their method with other methods.
Recommendation: 1
Reviewer confidence: 3
Soundness/correctness: 2
Clarity: 1
reviews: 
Comments: This paper proposes an integrated approach for solving forward and inverse problems by creating a new pipeline that combines Graph Neural Networks (GNNs) with Finite Volume Methods (FVM) to enable automatic differentiation with implicit-time stepping.
Strengths: Figure 1 effectively illustrates the overall pipeline, demonstrating experimental results that apply the combination of GNN and FVM to both forward and inverse problems.
Weaknesses: First and foremost, the paper feels incomplete. The biggest concern is the lack of discussion about other approaches that use GNNs or integrate FVM with deep learning to solve PDEs. A “Related Work” section should be added to explain how the proposed model differs from recent studies and highlight its novelty. Although Section 2 on theory explains the problem setup to some extent, more detailed steps and methods for training the proposed approach should be included. Section 3, the experimental part, merely lists the results for forward and inverse problems without discussing how this method compares to existing GNN- and FVM-based approaches. For instance, the study "Learning to Solve PDE-constrained Inverse Problems with Graph Networks" solves inverse problems using GNNs—how does the proposed method differ from this approach, and what advantages does it offer? Experimentally, does it outperform in solving inverse problems?
Recommendation: 3
Reviewer confidence: 4
Soundness/correctness: 1
Clarity: 1
reviews: 
Comments: The work considers mesh coarsening, forward, and inverse problems and investigates the implicit solver. In the numerical experiments, the authors evaluated the performance of the method regarding each problem.
Strengths: * The work tries to build a framework that works with mesh coarsening, forward, and inverse problems.
Weaknesses: * The novelty of the work is limited. Incorporating FVM into GNN is not new and considered in, e.g., [Jessica et al. ICML 2024 https://arxiv.org/abs/2311.14464 ] and [Horie et al. ICML 2024 https://arxiv.org/abs/2405.16183v1 ]. The construction of gradients presented in Section 2.3 seems strongly related to the adjoint method, which is a standard way to deal with inverse problems. The implicit method for GNN is considered in the area of implicit GNNs, e.g., [Gu et al. NeurIPS 2020 https://arxiv.org/abs/2009.06211 ]. The authors state that these are their novelty, but there is existing work for each. The authors should cite these works and clarify the added novelty from the authors.
* The evaluation is weak. There is only one baseline for the experiment in Section 3.2 and nothing for the ones in Section 3.3 and 3.4. With the current form, the reviewer cannot asses the effectiveness and superiority of the model.
* The presentation is not clear. The figure may miss the labels (a), (b), and so on for Figures 2, 3, and 4. It is not clear what is "data 1", "fitting 1", "data 2", and "fitting 2" in Figures 2 and 3.
Recommendation: 1
Reviewer confidence: 3
Soundness/correctness: 2
Clarity: 1
reviews: 
Comments: This paper introduces a novel framework that combines graph neural networks with the finite volume method, to address implicit schemes. (There was a challenge that differential equation solvers typically avoid due to the additional computation complexity of handling implicit equations.)
Strengths: By employing an implicit scheme with optimized gradient computation, the proposed method reduces the required number of time steps. 
They present a differentiable framework for both forward and inverse methods, enabling a learnable numerical approach based on discrete time steps. 
Additionally, the paper explores applications in inverse problems, often employing irregular unstructured grids as used in practical scenarios.
Weaknesses: While the underlying idea is promising, the paper would benefit from stronger experimental or theoretical justification for the proposed methodology. Additional clarity and motivation for the approach would enhance the paper’s impact.
Recommendation: 3
Reviewer confidence: 3
Soundness/correctness: 1
Clarity: 1
reviews: 
Comments: This paper addresses mesh coarsening alongside forward and inverse problems, with a focus on examining the behavior of an implicit solver. The authors present numerical experiments that assess the methodology's effectiveness across these different problem types.
Strengths: The work proposes an integrated framework capable of handling mesh coarsening, forward, and inverse problems in a unified manner.
Weaknesses: * Limited originality is evident in the contribution. The integration of FVM with GNN builds upon existing literature, including work by Jessica et al. (ICML 2024, https://arxiv.org/abs/2311.14464) and Horie et al. (ICML 2024, https://arxiv.org/abs/2405.16183v1). The gradient computation approach outlined in Section 2.3 bears significant similarity to the adjoint method, a well-established technique for addressing inverse problems. Implicit GNN architectures are already explored in prior work such as Gu et al. (NeurIPS 2020, https://arxiv.org/abs/2009.06211). While the authors claim these as novel contributions, related approaches exist in the literature. The authors should properly acknowledge these references and articulate what distinguishes their contributions from this existing body of work.
* The experimental validation is insufficient. Section 3.2 includes only a single comparison baseline, while Sections 3.3 and 3.4 lack any comparative analysis. In its current state, the evaluation does not provide adequate evidence for assessing the model's performance gains or comparative advantages.
* The clarity of presentation requires improvement. Subfigure labels (a), (b), etc. appear to be absent from Figures 2, 3, and 4. Additionally, the meaning of terms such as "data 1", "fitting 1", "data 2", and "fitting 2" in Figures 2 and 3 remains ambiguous and unexplained.
Recommendation: 1
Reviewer confidence: 3
Soundness/correctness: 2
Clarity: 1
reviews: 
Comments: This work investigates the application of graph neural networks to address both forward and inverse problems, with particular emphasis on the integration of implicit solvers. Nevertheless, the presentation quality is inadequate and the methodology along with its benefits lack clear exposition. Furthermore, the experimental evaluation is insufficient, as it does not include comparative analysis against alternative approaches.
Strengths: The research topic is compelling, and the integration of GNNs with finite element methods appears to be a logical and well-motivated approach.
Weaknesses: 1. The presentation contains numerous issues including typos and grammatical mistakes. For instance, the sentence "Compute $\nabla_bL$ with (12), whats is equivalent so the solution of a single linear system." should read "Compute $\nabla_bL$ with (12), which is equivalent to solving a single linear system." Such errors undermine the clarity of the paper.
2. A central theme of this work is the use of implicit solvers. However, the integration of iterative solvers within deep learning frameworks has already been extensively investigated in the Deep Equilibrium Models (DEQ) literature. The authors must provide comparisons with DEQ-based approaches to establish the novelty and advantages of their method.
3. The empirical validation is unconvincing. The outcomes presented in Section 3.4 demonstrate poor performance, and critically, no baseline comparisons are included in any of the experiments, making it impossible to assess the relative merit of the proposed approach.
Recommendation: 1
Reviewer confidence: 3
Soundness/correctness: 2
Clarity: 1
reviews: 
Comments: This paper addresses mesh coarsening alongside forward and inverse problem solving, with a focus on examining the behavior of an implicit solver. The authors conduct numerical experiments to assess how their proposed approach performs across these different problem types.
Strengths: The paper attempts to develop an integrated framework that simultaneously handles mesh coarsening, forward problems, and inverse problems.
Weaknesses: * The originality is questionable. The integration of finite volume methods with graph neural networks has already been explored in prior work, such as [Jessica et al. ICML 2024 https://arxiv.org/abs/2311.14464] and [Horie et al. ICML 2024 https://arxiv.org/abs/2405.16183v1]. The gradient construction methodology in Section 2.3 bears significant resemblance to standard adjoint-based approaches, which are well-established for inverse problem formulations. Implicit solvers in GNN architectures have also been addressed in the implicit GNN literature, e.g., [Gu et al. NeurIPS 2020 https://arxiv.org/abs/2009.06211]. While the authors claim these as novel contributions, comparable work exists for each component. The authors must properly acknowledge this prior work and distinctly articulate what their specific contributions add beyond existing literature.
* The experimental evaluation is insufficient. Section 3.2 presents results against only a single competing approach, while Sections 3.3 and 3.4 lack any comparative baselines whatsoever. In its present state, this severely limits the ability to evaluate the practical merit and competitive advantage of the proposed model.
* The clarity of presentation needs improvement. Figures 2, 3, and 4 appear to lack proper subfigure labels (a), (b), etc. Additionally, the meanings of "data 1", "fitting 1", "data 2", and "fitting 2" in Figures 2 and 3 are ambiguous and require clarification.
Recommendation: 1
Reviewer confidence: 3
Soundness/correctness: 2
Clarity: 1