Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: Learning Interface Conditions in Domain Decomposition Solvers

In this work, the authors use machine learning tools to enhance the domain decomposition solvers. It uses a graph convolutional network to learn the preconditioner. Specifically, the GNN takes the domain D and its decomposition as inputs, and outputs the boundary matrices L.

Numerically, the paper mainly considers a Helmholtz equation with constant coefficient and inhomogeneous boundary conditions. The experiments show the proposed method has a better convergence rate compared to the state-of-the-art solvers. 
strengths_and_weaknesses: Strengths: 
- The purposed method can be used to solve problems on larger domains (>10000 nodes). 
- The method is theoretical guarantees inherited from the Domain Decomposition solver.
- And it has a better convergence rate than the numerical solvers.
- Linear complexity

Weakness: 

- In the reviewer’s opinion, the testing problem is somehow too simple. The paper mentioned non-linear equations, but only a linear Helmholtz equation is considered. It will be also interesting to study the time-dependent equations. Can the iterative solver be used to solve hyperbolic PDEs?
- The new loss function appears somewhat increment to the reviewer.
questions: There could be more definitions or an example of the matrices L. What is its shape and what do the entries represent? 

limitations: The authors discuss that the study is limited to the Helmholtz equation. 
Besides, in the reviewer's opinion, the decomposition method as an iterative method is limited to elliptic PDEs. It will be interesting to see how the decomposition work for time-dependent equation and discontinuous problem. Do we need to change the decomposition over time?
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: Domain decomposition solvers are effective for large-scale PDE problems and optimized Schwarz domain decomposition methods are commonly used for structured-grid problems. In this work, the authors proposed to learn the parameters in Schwarz preconditioner that generalize it to unstructured grid and enable faster convergence. The main difference compared to the prior work, which works with a different preconditioner, is the proposed new loss function inspired by Gelfand's formula; and a practical approximation of the loss function via stochastic sampling.  Experiments on the 2D Helmholtz equation have been done to demonstrate the effectiveness of the approach.   
strengths_and_weaknesses: originality - the proposed loss function is different from the prior works and the stochastic sampling technique used to approximate the loss function is new to be applied to this problem, to the best of my knowledge. 

quality/clarity - the figures are clear and the article is well-written. More detailed captions could be added to facilitate reading. It would be nice to add a list of contributions by the end of the introduction to make the contributions clear. 

significance - the proposed approach demonstrates its effectiveness in the 2D Helmholtz equation problem. The approach scale linearly with the problem made is potentially be used for larger-scale problems. 
questions: 1. how are boundary nodes handled for the input graph to the GNN?
2. it's a bit unclear to be for the input/output of the GNN and how is that done. In line 139, it is noted that the input $i$ the $D$ (total domain) and its decomposition, and the sparsity constraint $L_i$, and predict the matrices. How exactly the "decomposition", and "sparsity constraints" are encoded in the input graph? According to the supplement, the output is the "learned interface edge features" and that will give us the predicted matrices, is this understanding correct? 
limitations: There is only one experiment setting with the Helmholtz equation. It would be more appealing if we have another experiment with a different PDE and boundary condition.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper presents a learning-based approach to find the optimal boundary values for domain-decomposed PDE solvers. The major contribution is the application of TAGConv to solving PDEs, and a new numerical method to compute the loss defined on the spectral radius of the difference matrix.
strengths_and_weaknesses: The method looks solid, and is well-validated. The results look promising, and the application of TAGConv to domain-decomposition is novel. 

However, the paper doesn't have much contribution to machine learning itself, and thus I'm not sure it fits the scope of NeurIPS. I would recommend the author resubmit this work to a journal for numerical PDE solvers.
questions: None. The paper is well-written and comprehended. 
limitations: The author only tested their method on Helmholtz equation, which is the major limitation. But the authors also have mentioned this limitation and further validation remains for future works.
review_title: null

Meta-review: metareview: The paper proposes a scheme to learn preconditioners for domain decomposition solvers. Graph neural network is used to learn interface matrices, and the training dataset consist of many unstructured grids for which optimal parameters are learning.
 The loss function is adapted from Daulbaev et.and is based on the Gelfand formula (with average replaced by maximum). Theoretical study is provided. The paper is well-presented, contains a useful and practical algorithm which potentially will be used by many researchers doing numerical simulations. 
confidence: Less certain
Decision: Accept