I believe that the exposition of the intuition behind the main results needs to be improved. In particular, I found the discussion after Thm. 2.1 to be hard to follow.

Also, some polishing is needed. I have found several typos and sentences with references to equations that do not flow correctly.

The paper is two essentially disjoint contributions. The first is to propose the eikonal equation as a model of deep network dynamics, and the second is to consider deep learning as a solution technique for this class of PDEs. Each contribution is thus allocated one half of the space, and this compression means that neither idea is explored very deeply.

Unquestionably, the eikonal equation is a PDE of vast importance, in optics and far beyond. Its connection to optimal control and to Hamilton-Jacobi theory suggests, but only very roughly, a possible connection to neural networks (ResNets), in particular as a way to study the effect of depth. The authors do make clear (in Section 2) that this is not an entirely compelling connection (in the paragraph starting on line 189). The problem, though, is that one is left without much conviction about the eikonal equation as a model. The rest of the section is devoted to criticizing (rightly) smoothness-based approximations, but the criticism doesn't imply that the eikonal equation is the right way to proceed. Further work is needed on this.

In section 3, the paper turns to a rather different class of problem, which is the numerical solution of the eikonal PDE via deep learning. This material is better developed, but again one is left with a feeling that more work is needed. The numerical solution of the eikonal equation is the subject of a vast literature, and the putative advantages of neural network solutions needs a more sustained investigation. Also, the restriction to strong solutions is quite restrictive (as the authors are clear in pointing out).

• The paper is very hard to read and the writing seems rushed. The paper contains several typos or mathematical inconsistencies, such as:
  • Line 155: Pointing to equation 23 instead of (I think) equation 5.
  • Line 164: Should it not be $\chi_{\epsilon}(\varphi_{V_u}(x,t))$ instead of $\varphi_{\chi_{\epsilon} V_u}(x,t))$?
• There are several definitions missing, such as the definition of flow map in line 155.
• In the approximation Theorem provided by the authors (Theorem 2.1), the definition of the error term is a bit odd, and makes me wonder if a better result could not be achieved. The error term that I am referring to is $E_{\mathcal W}(V_u)=\inf \|W - \tilde V_u\|$. I wonder if it did not make more sense to have $E_{\mathcal{W}}(V_u)=\inf\||x-x_s|W/|W| - \tilde V_u\|$, per the definition of $\tilde V_u$.
• The loss is not explained at all. $\bar u_\theta$ is not defined.
• The experiments are also insufficient, and there are no comparisons with other PINN methods for solving the eikonal equation.

• A pressing concern is the unreadability of the paper to an ML audience. I have some background in dynamical systems and ODEs but still found it hard to understand the ideas and the contributions. I list some examples below, that I hope help the authors. This is not to say that the paper is not suitable for ICLR, but rather the way it is presented is unreadable for the ICLR audience. I used some of the cited papers to understand better the approach. I believe it should be presented more suitably for ML readers, in a self-contained way.
• The theoretical contributions largely build on previous works and are not very involved. Moreover, the associated error if the flow does not follow the Eikonal equations is not characterized or discussed. The Eikonal equations are a special case of the Hamilton-Jacobi Equation, so focusing on the latter will give more general results.
• The experiments focus on flows that follow the eikonal equation, and it is unclear to me how the approach works otherwise.

Structure

• Some theorems are given in the related works section

Dynamical systems terms and difficult to follow

• flow idealization -- appears first in the introduction, making it hard to understand the idea for an ML reader
• robustness to spacial resolution is unclear in the abstract for people unfamiliar with the Eikonal Equation; perhaps it is better to explain in simple terms that is beneficial
• It is hard to understand how to implement the method proposed in Sec. 3. Provide steps or pseudo code that are self-contained. Explain the symbols in Figure 2 and repeat the most important notation in the caption.

Writing

Abstract. After reading solely the abstract I could not understand the contributions. As a person not familiar with the Eikonal equation, I was confused. I hope this feedback helps improve it.

• The motivation posed in the abstract is the lack of approximation rates in terms of depth. The contributions as listed in the abstract are unclear on how they relate to this motivation--what is the rate shown in terms of depth? Please list the precise rate.
• The text in the brackets "(corresponding to time)" is also not very clear. I suggest you avoid writing in brackets here and improve this part.
• The last two sentences are unclear: I did not understand by this point that the goal is to solve the Eikonal equation. I also did not know that the spatial resolution needs to be specified and why we want to be robust to it.

Minor

• typo line 203: will determined
• use citing with brackets where appropriate