reviews: 
Comments: This paper proposes a method using flow matching for adaptive sampling of residual points in physics-informed neural networks (PINNs), aimed at improving stability and accuracy, particularly for PDEs with singularities. The approach is compared against normalizing flow-based sampling, demonstrating advantages in certain scenarios.
Strengths: The paper demonstrates improvements of the flow matching-based sampling method over the previous normalizing flow approach. The proposed methodology might provide insights for the community.
Weaknesses: - The writing could be improved. Many citations should use `\citep` instead of `\citet`. Some symbols are not properly defined, e.g. $\mathbb{S_k}$ appears in Equations (2) and (3), long before it is introduced in Algorithm 1. The definition of the optimization problem in Equation (7) is nonstandard. Additionally, some acronyms are not properly introduced, such as DAS and AAS.
-  The structure of the paper is confusing. Section 3.4 appears to describe the author's original work but falls under "3. Related Work." Furthermore, there is a "Section 3.1" comes after Section 3.4, which disrupts the logical flow.
- Some figures have random gray lines surrounding them (e.g., Figures 2(b,c) and 4(b), among others). 
- The novelty of the proposed method seems limited. The work replaces GAN or normalizing flow in previous approaches with flow matching, which is known to have some advantages in certain scenarios. This incremental improvement may not be sufficient to justify the claims of significant advancement.
- While the work might be useful, its practical impact on solving PDEs appears limited. Practitioners typically care about whether they should use this method for solving their PDE problem, including aspects like accuracy and computational speed. The paper primarily focuses on showing that this ML method is more accurate than a previous ML method, without adequately addressing the broader context of whether PINNs are appropriate for the given problem compared to traditional numerical methods.
- The use of the term "source singularities" might be inappropriate. The source terms in the Poisson equations presented are sharp Gaussians at best, not truely singularities (delta function). These singularities can lead to discontinuous gradient or solutions. See methods for elliptic interface problems. Even in PINN, there are more exact method to handle the singularities, e.g, Tseng, Y.-H., Lin, T.-S., Hu, W.-F., Lai, M.-C., 2023. A cusp-capturing PINN for elliptic interface problems. Journal of Computational Physics.
- The two inset figures in Figure 1 ("train flow matching generative model" and "construct sample from flow matching generative model") do not appear to be original work. Similar images can be found in Figures 7 and 21 on [this website](https://mlg.eng.cam.ac.uk/blog/2024/01/20/flow-matching.html).
- The authors should provide a discussion on computational time. For instance, in Section 4.1, 28,000 points are introduced at each resampling stage, leading to a substantial increase in point count during the whole training process. Since"high computational costs" was mentioned as a drawback of previous approaches, it is unclear that the proposed method can alleviate this issue.
- The authors should report the mean and standard deviation of different trials, as sampling is required in this method. This would provide a better understanding of the variability and reliability of the results.
Recommendation: 3
Reviewer confidence: 3
Soundness/correctness: 1
Clarity: 1