My general comment is that it is hard to place this work both wrt to existing numerical methods, but also with respect to neural solvers, neural surrogates. That makes it really tough to properly weigh the pros and cons wrt to e.g. Proper Orthogonal Decomposition, Dynamic Model Decomposition, but also operator learning methods or PINNs. Furthermore, I would like to see a test against a model which does not preserve time reversibility. - The only comparison is against Proper Orthogonal Decomposition (POD) which as stated in the introduction “is flawed in that it ignores the temporal dependence of state variables”. Naturally the question is for example how Dynamic Model Decomposition (DMD) is doing for the problems. - Also from a Deep Learning perspective, e.g. Figure 4 could be learned by an operator learning that takes the initial state as input and outputs the state of the system after e.g. every 100th step. On the other hand, if done with e.g. PINNs this could be performed completely without training data. Just using the boundary conditions for the loss and the equation for the residual loss. It would be really interesting to have a comparison wrt to speed and accuracy. - PyTorch and more specifically the automatic differentiation capability of Pytorch is used as an optimization tool. It is however hard to judge from the paper what exactly the parameters are that are optimized. How many parameters are optimized for the different problems? Algorithms 1 and 2 in the appendix help a great deal but it took me a great deal to scroll back and forth. Is it possible to place the algorithms in the main paper and write refer to the important parts in Section 3,4? - The loss should be a bit more central to the writing of the paper. In Fig 1 the most important components are the reduced-order fluid model and the trajectory-wise discrepancy loss. However, Sections 4.3 takes a huge part of the main paper and is actually hard to follow. In my opinion the readability of Section 4 can be improved by focusing more on the relevant parts and not let the reader figure out what they are. - Why is e.g. JAX not used for optimization? Did the authors consider that? - How do optimization time vs inference time relate between the different models? - How do the individual components of the algorithm relate to the performance, e.g. how does the performance change for other loss terms? - Is it correct that for the first benchmark a single trajectory is used, whereas in the second benchmark more trajectories are used for optimization? On which trajectories is the testing done afterwards? Shouldn’t there be more than just one trajectories to get a better performance estimate?
- Why is e.g. JAX not used for optimization? Did the authors consider that?