While deep learning has shown tremendous success in a wide range of domains, it remains a grand challenge to incorporate physical principles in a systematic manner to the design, training and inference of such models. In this paper, we aim to predict turbulent flow by learning its highly nonlinear dynamics from spatiotemporal velocity fields of large-scale fluid flow simulations of relevance to turbulence modeling and climate modeling. We adopt a hybrid approach by marrying two well-established turbulent flow simulation techniques with deep learning. Specifically, we introduce trainable spectral filters in a coupled model of Reynoldsaveraged Navier-Stokes (RANS) and Large Eddy Simulation (LES), followed by a specialized U-net for prediction. Our approach, which we call Turbulent-Flow Net (TF-Net), is grounded in a principled physics model, yet offers the flexibility of learned representations. We compare our model, TF-Net, with state-of-theart baselines and observe significant reductions in error for predictions 60 frames ahead. Most importantly, our method predicts physical fields that obey desirable physical characteristics, such as conservation of mass, whilst faithfully emulating the turbulent kinetic energy field and spectrum, which are critical for accurate prediction of turbulent flows.
## Summary
The authors propose use learned spatio-temporal filtering and a convolutional model to predict the behavior of a turbulent fluid flow. Turbulent flow is a very difficult problem, of great interest for engineers and physical scientists, so the topic of the paper is certainly compelling.
This paper has several significant issues. The baselines the authors compare to are quite weak. Many of them were designed for other purposes, such as video prediction. The authors claim significant improvements ("64.2% reduction in divergence") that I believe are calculated in a (unintentionally) misleading way. In fact, I think they missed the most important baseline---the ground truth simulation itself. Unless they can argue that the learned model is superior to the classical simulation in a significant way, it is hard to see the benefit of using something like TF-NET.
## Specific Comments
* Page 2: The claim "64.2% reduction in flow divergence, compared to the best baseline" seems misleading. In Figure 5, the constrained TF-NET is as low as ~590 but the ResNet is between ~610 and ~810. I am guessing that the authors meant a 64.2% reduction in *difference* from the Target model, but this should be clarified.
* Page 4: Is 'T' supposed to appear in the denominator of Equation 3? I would have expected this to be a normalizing factor resulting from integrating the filter G over the whole domain.
* Page 4: I would expect filters used for LES are symmetric. Are any symmetry requirements being enforced on the learned filters?
* Page 4: It's not clear what TF-NET is outputting. I would expect it to output the time derivative of the velocity field, but this should be spelled out explicitly.
* Page 5: Since you are using incompressible Navier-Stokes equations, I don't think the Mach number is relevant. IIUC this is only relevant for the propogation of shock waves in compressible flows.
* Page 7: It looks as if the Target model has significant divergence. Why is this? Should we expect this much divergence in the ground truth data?
* Page 8: In homogeneous isotropic turbulent flow, the energy spectrum is governed by Kolmogorov, so we know what the spectrum should look like. Is there an analytic result for RB convection? If so, could you include that on the plot?
* Page 8: The lines for U-Net and TF-NET are nearly indistinguishable in Figure 7. Could you change them?
* Page 8: It seems like it would be worth including the ResNet in the energy spectrum plots as well.
* What did the learned spatial and temporal filters look like? How do they compare to typical 'hand-chosen' filters?
* IIUC, the training, validation and test data all have identical Rayleigh number. Does the learned model generalize to higher/lower energy? This seems critical to making this sort of model useful.
* The paper doesn't describe a tuning process for any of the models. I would expect this to lead to significant improvement. In particular, how do the models' performance change as the weight on the divergence loss term is increased?
### Baselines
* I think the objective should be to show that TF-NET is superior to the ground-truth method in some way. Can it match the results of the ground truth simulator but do it faster, or with fewer resources?
* The authors say "we compare our model with a series of state-of-the-art baselines for turbulent flow prediction." However, most of these models are intended for video prediction or other tasks unrelated to turbulent flows. I wouldn't expect any of the baselines considered to do perform well in this context.
* Since TF-NET gets the benefit of a loss related to divergence, I would have expected the other architectures to get the same treatment. In fact, without this extra loss term, the ResNet architecture is competitive with TF-NET.