reviews: 
Comments: The paper proposes a new objective to sample collocation points for PINNs adaptively. Specifically, the collocation points for training the PINN are sampled from a normalizing flow with soft Lipschitz constraint, enforced by Sobolev regularization. The PINNs and normalizing flows are optimized in an alternating fashion, where the normalizing flow is trained to maximize the expected PINN residual (w.r.t. to the distribution given by the pushforward of the normalizing flow). This formulation is then connected to the dual form of an optimal transport problem. In this context, it is shown that there exists an optimal solution to the proposed min-max problem where the (normalized) squared residual converges to a uniform distribution in the Wasserstein distance. The approach's effectiveness is further demonstrated on two toy problems and one high-dimensional nonlinear PDE and compared to three related methods.
Strengths: 1) Tackling adaptive sampling for PINNs using optimal transport seems to be a promising direction.
2) The proposed objective seems to be novel and can bring numerical benefits on the considered examples.
Weaknesses: The theoretical as well as numerical contributions need to be significantly improved:

1) Further challenging problems (as in Subsection 5.3), as well as baselines (as enumerated in the section on related works), are needed to judge the performance of the proposed algorithm. 
2) "[...] which is the first time to minimize the residual and seek the optimal training set simultaneously for PINN.": Analogous to DAS-PINNs, it seems that the final algorithm is still alternating between optimizing the two networks, see also Question 2) below. In general, it should be made more explicit what the novelty of the present work is and how it theoretically compares to related work.
3) For the numerical results, training times and standard deviations (w.r.t. different seeds) are missing. Especially given that the adversarial training slows down training.
4) The loss for learning the normalizing flow seems to use the REINFORCE trick, which, however, is known to suffer from high variance.
5) Since the residual continuously changes over the course of optimization, it should be better motivated what the advantage of learning a generative modeling is (as compared to just using a method to sample, e.g., according to squared residual)?
6) It seems that the precise connection to optimal transport remains a bit unclear:
	* "So the minimization step will reduce not only the residual but also the Wasserstein distance between $\mu_r$ and the uniform distribution". Since there is only an upper bound shown on page 6, it is unclear why the minimization step is guaranteed to decrease the Wasserstein distance.
	* "The evolution of the residual-induced distribution has a clear path": According to the theorem, there only *exists* such a path, and it is not clear whether this path is taken by GD. It would be good to have at least plots of the residual evolution for all experiments.

**Minor issues:** 
1) use `\citep`.
2) In Eq. (4), min-max should be written on both sides of the equation.
3) What is a 'proper' constraint?
4) The last paragraph in Section 3 seems a bit convoluted. The choice for minimal variance could just be motivated by the optimal choice for importance sampling. 
5) It is unclear why we would consider $I^D$ instead of $\Omega$ when the KRnet is first introduced.
6) The text for the figures of the training set is barely readable.
7) typo: resultd
Recommendation: 5: marginally below the acceptance threshold
Reviewer confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Soundness/correctness: 2 fair
Clarity: 1 poor