In this paper we propose a physics-aware neural network for inpainting fluid flow data. We consider that flow field data inherently follows the solution of the NavierStokes equations and hence our network is designed to capture physical laws. We use a DenseBlock U-Net architecture combined with a stream function formulation to inpaint missing velocity data. Our loss functions represent the relevant physical quantities velocity, velocity Jacobian, vorticity and divergence. Obstacles are treated as known priors, and each layer of the network receives the relevant information through concatenation with the previous layer's output. Our results demonstrate the network's capability for physics-aware completion tasks, and the presented ablation studies show the effectiveness of each proposed component.
In this paper the authors adopt prior work in image inpainting to the problem of 2d fluid velocity field inpainting by extending the network architecture and using additional loss functions. Specifically, the U-net network is extended with a DenseBlock in the middle, and a separate branch of the network is added which predicts the stream function (a different representation of the velocity field which guarantees incompressibility). The additional losses are L1 for various derivatives of the flow field (Jacobian, divergence, vorticity). Experiments presented in the paper show that these new elements improve the flow field error compared to a baseline model originally developed for image inpainting. The suggested application for this model is filling gaps in experimental measurements that are missing or impossible to obtain, and where such a model could be computationally cheaper than an actual fluid solver.
The paper discusses prior work at the appropriate level of depth, and specifies hyperparameters that were used for training.
As is, I do not believe the paper fully meets the bar for novelty and potential impact for acceptance at ICLR (but it could perhaps be a good fit for a more specialized venue). I am specifically not convinced about the practical applicability of the results. The novelty of the approach also seems quite limited, and the findings do not seem particularly surprising or insightful (i.e. that extending the vanilla U-net architecture and adding losses that directly bias the network towards physically meaningful solutions is better than the baseline).
Since all training and testing data was generated from simulation, it is unclear how well the networks would cope with real world measurement noise. Furthermore, the authors note that the flow field from the simulation is sometimes not divergence-free. It is surprising that this could be a problem to a level where it could impact evaluations. If it indeed is, then perhaps it would make sense to either simulate the flows with a denser grid (inpainting could still be done on sparsified results). I found it surprising that the authors chose instead to use Eq. 8 and "force" the network to learn a flow field from the solver which is known to not be strictly physically valid.
It is also unclear to me that the masking schemes used in the experiments are relevant to actual measurements -- for instance, 2nd row of Fig. 3 or 1st row of Fig. 4 seem completely artificial. I appreciated the ablation study variants (but see also comments below for at least one more configuration that I believe should be discussed in addition to the existing ones). It would however be more informative (and important for potential practical applications) to include some sort of breakdown by mask type and flow field configuration/structures (steady, unsteady, wakes, jets, vortices, etc). The included images show the model does not capture some finer details of the velocity field (e.g. vortex structures in the last row of Fig. 3 and right column of Fig. 4; wakes behind small obstacles in Fig. 5-6, long, narrow, and fast jets in Fig. 6), but it is unclear what impact the proposed extensions (DenseBlock, losses, and stream function branch) have on these structures. Similarly, interpolating missing data points with a fairly dense input where no points are more than a few pixels apart seems like a much easier problem than filling large empty spaces, and it would make sense to do separate analyses for these cases.
Questions & suggestions for improvements:
* Has the impact of different weight combinations in A.2. been investigated? How was the ratio of 6:1 for empty:valid determined?
* The text says that in the stream function pathway, the features are passed "through 4 densely connected convolution layers". I was confused the first time I read that sentence, and only later I realized that this refers to a DenseNet-like pattern of connectivity. A reference to Huang et al. here or some other clarification would help.
* In the figures, please consider also showing the difference between the predictions and ground truth to make it easier to see which features of the flow field are predicted accurately, and which are not.
* Is there an L1 loss applied as well directly to the output of the stream prediction branch?
* What are the Reynolds numbers used in the simulations? How far from the original Re does the network generalize?
* When computing MAE, what are the units? What are the velocity magnitudes? Consider also reporting mean relative error.
* Impact of the various L1 deriv. losses seems negligible when the stream function is used, but is more visible when only velocity is being directly predicted. Please comment on why that might be.
* Why does (f) (Jacobian only) work better than (e) and (g)?
* It is not clear that that the effect of the additional L1 losses is truly cumulative. Please consider testing just (u, DenseBlock, L_div).
* It would be an interesting extension to include a non-ML baseline, which could be compared against the current results in terms of flow field quality and computational cost. 
* Can the network predictions deviate from the known (non-masked) values in the input? If so, please consider including a breakdown of the evaluation that shows how much the error varies between the "valid" and "empty" areas.