Predicting outcomes and planning interactions with the physical world are longstanding goals for machine learning. A variety of such tasks involves continuous physical systems, which can be described by partial differential equations (PDEs) with many degrees of freedom. Existing methods that aim to control the dynamics of such systems are typically limited to relatively short time frames or a small number of interaction parameters. We present a novel hierarchical predictorcorrector scheme which enables neural networks to learn to understand and control complex nonlinear physical systems over long time frames. We propose to split the problem into two distinct tasks: planning and controlling. To that end we introduce a predictor network that plans optimal trajectories and a controller network that infers the corresponding control parameters. Both stages are trained end-to-end using a differentiable PDF solver. We demonstrate that our method successfully develops an understanding of complex physical systems and learns to control them for tasks involving multiple PDEs, including the incompressible Navier-Stokes equations.
[Summary]
This paper proposes to combine deep learning and a differentiable PDE solver for understanding and controlling complex nonlinear physical systems over a long time horizon. The method introduces a predictor-corrector scheme, which employs a hierarchical structure that temporally divides the problem into more manageable subproblems, and uses models specialized in different time scales to solve the subproblems recursively.
For dividing the problem into subproblems, they use an observation predictor network to predict the optimal center point between two states. To scale the scheme to sequences of arbitrary length, the number of models scales with O(log N). For each subproblem, the authors propose to use a corrector network to estimate the control force to follow the planned trajectory as close as possible. 
They have compared their method with several baselines and demonstrated that the proposed approach is both more effective and efficient in several challenging PDEs, including the incompressible Navier-Stokes equations.
[Major Comments]
Predicting the middle point between two states for modeling the dynamics via deep neural networks is not new, but I did not know any other works that use this idea for controlling PDEs.
I like the idea of splitting the control problem into a prediction and a correction phase, which leverages the power of deep neural networks and also incorporates our understanding of physics. The introduction of the hierarchical structure alleviates the problem of accumulating error in single-step forwarding models and significantly improves the efficiency of the proposed method. The videos for fluid control in the supplement materials also convincingly demonstrate the effectiveness of the technique.
I still have a few questions regarding the applicability and the presentation of the paper. Please see the following detailed comments.
[Detailed Comments]
In Section 3, the authors claim that their model "is conditioned only on these observables" and "does not have access to the full state." However, the model requires a differentiable PDE solver to provide the gradient of how interactions affect the outcome. These seem to contradict each other. Doesn't the solver require full-state information to predict the behavior of the system?
Related to the previous question, how can we make use of the differentiable PDE solver if we are uncertain or unknown of the underlying physics, i.e., partially observable scenarios.
The algorithm described in Section 5 seems to be the core contribution of this work. Instead of describing the algorithm in words, I think it would make it more clear if the authors can add an algorithm block in the main paper. It would also be better if the authors can include a few sentences describing the algorithm in the abstract to inform the readers of what to expect.
Figure 4 is a bit confusing, and it would be better if the authors can include the label for the x-axis. Besides, in the caption, the authors said that they show "the target state in blue." However, there are a lot of blue lines in the figure, and it is hard to know, at first glance, which one of them is the target.
In Table 1, the bottom two methods are using the same execution scheme and training loss, but the results are different. Is there a typo? Also, it would be better to bold the number that has the best performance.