Paper:
Spatio-Temporal processes bear a central importance in many applied scientific fields. Generally, differential equations are used to describe these processes. In this work, we address the problem of learning spatio-temporal dynamics with neural networks when only partial information on the system's state is available. Taking inspiration from the dynamical system approach, we outline a general framework in which complex dynamics generated by families of differential equations can be learned in a principled way. Two models are derived from this framework. We demonstrate how they can be applied in practice by considering the problem of forecasting fluid flows. We show how the underlying equations fit into our formalism and evaluate our method by comparing with standard baselines.

Review:
I very much like the aim of this work. This is a problem of interest to a wide community, which as far as I'm aware hasn't yet had much focus from the deep learning community. However, perhaps in part because of this, the paper reads as naive in places. Pages 1-4 are all background saying nothing new, but ignoring the effort made on this problem by other communities. There has been some work done o this problem within statistics, and within the Gaussian process community, to which no reference is made at all by the paper.
There are two novelties as far as I can see (these may or may not be novel - but they were novel to me). The first is use of NNs to model the system. The second is the multiple state restimation (MRSE) on page 5. I struggled to get a feeling about how successful these two aspects of the work are. The results section is difficult to follow, and doesn't compare the method to existing methods and so there is no baseline to say that this is successful or not. Thus I find it hard to judge the execution of the idea. What I really want to know reading a paper like this is should I use this approach? Because there is no comparison to existing methods, it leaves me unsure.
Other comments:
- Is the title correct? I don't see how these are PDE guided NNs? You've used data from a PDE to train the network and as a test problem. A PDE guided NN would, for me, know something about the dynamics (compare with recently work in the GP community where kernels are derived that lead to GPs that analytically obey simple PDEs). 
- There is an obvious link to work in the uncertainty quantification community, particularly around the use of multi-fidelity/ multi-level simulation. This paper is likely to be of interest to them and the link could be more explicit.
- Page 3, after eq 2 - there is notation used here that is undefined Y_{t-k}^t
- The simplifying assumption on page 3 is very strong and unlikely to hold for many systems. But it isn't clear to me whether this is necessary or not? Presumably if it doesn't hold then we may still get an approximation that could be useful, but it is just that we lose any guarantee the method will work.
- I thought the MSRE idea was interesting. It wasn't very well explained or motivated, and it was unclear to me whether it works well or not from the results, or whether it is novel to this paper or not. But I'd like to have read more about it.
- Is the trick in Section 8.2 original to this paper? If so, it seems a nice idea (I've not checked the detail). 
- Most of section 8.1 strikes me as unnecessary.
- There are quite a few typos. In particular, words such as Markovian, Newtonian should be capitalised.