Kati8/ICLR_2023_2025 huggingface dataset:
authors: Kejun Tang
authors: Jiayu Zhai
authors: Xiaoliang Wan
reviews: 
Comments: In this article, authors raise a significant problem that linked with numerical difficulty is that random samples in the training dataset introduce statistical errors into the functional loss which may become the dominant error in the approximation, and therefore overshadow the modeling capability of the neural network. A new approach was proposed to optimize both the approximate solution and random samples in the training set by using a min max formulation. This approach is called adversarial adaptive sampling (AAS). The main idea of AAS is minimizing the residuals and meanwhile push the residual-induced distribution to a uniform one.  AAS can be divided into two parts. In the maximization part, the deep generative model helps define the difference between the residual-induced distribution and a uniform one using Wasserstein distance. In the minimization one, this difference is minimized together with the residuals.   Also, they used some benchmark test problems of comparison AAS algorithm with another state-of-the-art algorithms as DAS and RAR.
Strengths: The motivation is clear, and the method is novel and interesting. Authors clearly described the theory of proposed method and the algorithm.
Weaknesses: A numerical result is given for one type of PDEs. It would be interesting to see how AAS algorithm performs for other types of PDEs. There is no discussion of further development or limitations of the proposed method.
Recommendation: 8: accept, good paper
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: 3 good
Clarity: 4 excellent
reviews: 
Comments: A uniform approximation error is crucial in representing functions in a given vector base.  The same applies to compound function representations like neural networks. It also crucial where the errors are measured. The Authors combine both aspects, a  Wasserstein distance between a uniform and error distributions,  and an adversarial adaptive sampling of points for the PINN collocation.

The loss is divided to an analytical part that reflects the difference of the neural network from the solution and a statistical part that comes from using a finite sample.  By tuning the residual to become uniform, the density of samples becomes optimally non-uniform.

The method is used for three different PDEs with encouraging results compared to other adaptive learning techniques.
Strengths: An excellent way to bring confidence to the inference with good proofs of convergence for the adaptive learning distributions.

Presentation is clear and not hard to follow.
Weaknesses: Looking at the point clouds one visually can guess that the Delaunay triangulation link length distribution of  the point cloud contains very short distances i.e the point cloud is not locally smooth. Usually building meshes for FEM one prefers same size of elements locally and smaller elements in those areas do not provide more accuracy. 

I suppose this is the same for collocation points in PINNs. Currently the point clouds do not look like node distributions for FEM calculus. This is increasingly so, when creating points in the higher dimensions. The amount of wasted calculations increases and may drive this to a curse of high dimensionality.
Recommendation: 10: strong accept, should be highlighted at the conference
Reviewer confidence: 5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.
Soundness/correctness: 4 excellent
Clarity: 4 excellent