reviews: 
Comments: In this paper the authors introduce an adaptive sampling technique that for sampling the input points (also referred to as collocation points) for training a PINN architecture on a domain. The main motivation behind it is that for highly irregular PDEs, uniformly sampling collocation points will not work well and result in high residual error for PINNs. 

The idea is similar to the adaptive sampling techniques that have been introduced in previous work by Tang et al (2023) and Gao et al (2023), however instead of directly minimizing a pushforward or adaptive sampling, the authors in this paper enforce a wasserstein loss to ensure that the residual error is uniformly distributed across the grid. 

That is, the training is done such that the error induced by PINNs is uniformly distributed across the grid, and hence the points are sampled accordingly. The authors operationalize this using a WGAN setup, wherein they train a pushforward map to learn p(x), such the wasserstein-1 loss between the the density of the residual and uniform distribution. This is similar in vein of the adaptive sampling techniques introduced in Gao et al (2023) but the weights are instead learned using a GAN type architecture.
Strengths: The idea to use a GAN type loss with to ensure that the density of the residual is uniform across the domain is very interesting. 

In their experiments, the authors are able to do well on very sparse data, i.e., PDEs that have two-peaks, something that PINNs sometimes don’t do well in, and are able to get better results than the previous baselines by Tang et al (2023), thus showing the benefits of using the wasserstein formulation for learning $p_\alpha$.
Weaknesses: In general the work is very similar to that by Tang et al (2023), there the authors are using a network defined using a normalizing flows type of an architecture whereas here the authors are using a GAN instead. 

The paper is very hard to read with sometimes the notation and the terms used by the authors for different quantities is not clear. For example

- The authors mention that this methodology achieves variance reduction, what is the proof for that? is it similar in flavor of Tang, et al (2023)
- In Assumption 1, the authors refer to $r$ as an operator. Is that an operator, or the residual between the loss. If the authors mean that $r$ is an operator, then from which function space to what other function space? (I presume it is U x F → U).
- In theorem 1, $\nu$ is used to define the density of the residuals, whereas in the derivation under equation 6, it is defined by $\mu_r$. I think that they are the same quantity, however if not, what is $\nu$ and what is $\mu_r$?
- It is unclear as to what is being approximated by a KRNet?, since the authors first use $p_\alpha$ to define the density that they are trying to train, however, there is no mention of it after section 3.2
- Also, KRNet is not cited in the last paragraph of page 3.
- In equation 9, what is p*?

Few other questions that I think would help with the understanding of the usability of the techniques are: 

- What are the implementation details (in terms of number of parameters) etc of the networks approximating the solution $u_\theta$ and approximating the density network, i.,e $p_\alpha$.
- Since the authors are doing a min-max optimization, that is usually hard to train and get right, esp for for PDEs that may have some advective terms.

While the methodology provided by the authors seems to do better than the baselines, given the presentation and the lack of clarity of the precise steps, I think the paper would benefit from a revision
Recommendation: 6: marginally above the acceptance threshold
Reviewer confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness/correctness: 2 fair
Clarity: 2 fair