Kati8/ICLR_2023_2025 huggingface dataset:
authors: Yana Khassan Nibal
authors: Jiexing Gao
authors: Yimin Huang
authors: Fedor Buzaev
reviews: 
Comments: This paper proposes a resampling procedure for PINN training when the right-hand side is sharply peaked.  The idea is to use flow matching to generate new samples in regions where the residual is high; in each iteration, more quadrature points are added to the PINN objective function to penalize points where the PDE is harder to solve.  The method is tested on the Poisson equation and a simple version of elasticity.
Strengths: Resampling for PINNs makes sense and can help identify regions where the solver needs extra "help" to find the right solution.
Weaknesses: Overall, the method seems to be a fairly heuristic extension of existing works.  Although I see the authors are applying ideas from the flow matching work, they do not articulate important details of their algorithm, e.g. the distribution from which flow matching is attempting to draw samples.  The tests here show preliminary evidence that the method has some value but are far from conclusive (just 3 PDE examples total as far as I can tell).
Recommendation: 3
Reviewer confidence: 2
Soundness/correctness: 2
Clarity: 2
reviews: 
Comments: The authors propose a new strategy for re-sampling new training data points for training PINN networks. At each stage of the PINN training algorithm, they train a new flow-based generative model for sampling points in regions where the PDE residuals have large values. At each epoch, the training points are the union of the previous points and the new generated points.
Strengths: - Both the objective and the proposed algorithm are well explained. 
- Most of the related works and useful concepts are explained.
- The contribution makes sense. Indeed, using normalizing flows for resampling points seems to be an overkill, as it is not necessary to have explicit probability densities and invertibility.
- Different experimental conditions
Weaknesses: - You only compare with DAS PINN and no other methods. As you mention AAS-PINN and RAR in the related works, you should also compare with these methods. Especially, it would be interesting to have comparisons with methods that do not use generative models for re-sampling.
- Moreover I am questioning how fair is your comparison with DAS PINN. Indeed, you use the same number of training epochs / training points for both models, but both do not share the same architecture. It seems in Figure 4 b) and 10 that DAS-PINN has not fully converged yet. Can you provide the same curves with a larger number of epochs for both methods ? 
- The "weighted bootstrap resampling " algorithm should be clearly explained and detailed. It is a major block of the method, and it is mentioned without further explanations.
- There are several typos, missing commas, english mistakes that need to be corrected.
- Color inconsistency in the plots of Figure 16.
- Sections 3.2 and 3.3 seem to repeat / should be merged.
- $p_1(x)$ is used without being defined. Is it a pushforward measure ? 
- The  explanation of the AAS method is not clear enough for me.
Recommendation: 3
Reviewer confidence: 2
Soundness/correctness: 2
Clarity: 2
reviews: 
Comments: This paper presents a method for automatic and adaptive sampling of domain points during the training of a physics-informed neural network (PINN) model. The main idea is to use a flow-matching technique to learn the distribution of the ''correct'' density of the domain points from the current residuals. This is similar to the ''mesh refinement'' procedures common in PDE applications. At a typical ''refinement'' step in the current method, the residuals are used to train a flow-matching model to arrive at the unknown distribution of the domain points. Now, this model is used to generate point samples for the next loop of training. The authors present two examples with singular behaviour, and  compare their results with one other work, which is called ''DAS-PINN.'' In both these cases, "FMS-PINN" is seen be the superior method.

`Overall impression:` The major idea, i.e., using a generative model to generate PINN points for training is not new. The marginal contribution here is using flow-matching technique to do so. Nevertheless, it makes for an interesting read, and the results, though neither comprehensive nor conclusive, are interesting as well. Decent paper, but not ready for publication.
Strengths: * The paper is mostly well written, but some parts are confusing (see questions).
* Interesting application of flow-matching to generate sample points for PINN training.
Weaknesses: * The mathematical exposition (Sec 3.4) is not well written (see questions).

* The authors compare their results with one existing method (DAS-PINN, Tang et al., 2023). But only one of the examples from Tang et al. is recreated in this paper, and that too is relegated to the appendix. This is the only point of exact comparison, and it is inconclusive whether one method is better than the other.

* (minor) Line 325: (probably) incorrectly refers to Figure 12
Recommendation: 3
Reviewer confidence: 4
Soundness/correctness: 2
Clarity: 2