Please read the report in pdf.

This work presents a data-driven method to infer a linear stochastic

model from a partially observed system. 

This work is well-written and contains interesting parts, especially the not-so-common effort to explain the dynamics of the latent (embedding) space. Nevertheless

simplifications made in the work do reduce a lot the impact of this paper. 

Also, there is very little novelty in

the approach. The principle of alternating between DA and a data-driven model has been already applied, in more challenging settings

(noisy/sparse observations, model with more dimensions). The fact to

have a variable that is never-observed has also already been tested. The

originality of the approach to have a stochastic model and to explain

the latent space is not very developed.

Other general comments:

- the justification of the setting and the approach is not convincing

  to me (see my comments about the abstract and the introduction), and

  I fail to foresee the real application of the approach. Maybe

  rephrasing the last part of the conclusion and putting it in the

  the introduction instead could help regarding that matter.

- The data-driven model used is linear. It is acknowledged by the

  authors in the conclusion, but it is one limit of the

  approach. Maybe the linear approach works because the setting is

  simple enough (low dimension, weakly non-linear). But also, I wonder

  if the interpretability of the latent space is precisely related to

  the choice of the linear model (maybe with a non-linear model, there is

  no need for a latent space to emulate observed variables...)

- The experiment is done on the Lorenz 63 model, which is very

  low-dimensional (3) and weakly non-linear. See for example:

  https://raspstephan.github.io/blog/lorenz-96-is-too-easy/#

  There are toy models (L96, QG) that could display more interesting

  behaviors for this methodology.

- The forecast is evaluated only in the next time step, which is again

  a very easy case. How would behave the forecast over several time

  steps?

- The method is interestingly stochastic, but no ensemble metrics are

  used to evaluate the work which would have been interesting. 

Specific comments:

Abstract: The 2 first sentences of the abstract is a justification of

the approach. Due to the limited size of the abstract, this

justification cannot be extended making it too simplistic: 1) It is true

that defining a set of equations is difficult, but I would say that a

bigger issue is the resolution of the existing set of equations given

that coefficients are unknowns and that a discretization is needed

for the numerical resolution, which introduces some errors. 2) If we

follow the narrative, it is well justified that we should cope with "imperfect

equations". But here, the choice is to assume that no equation is known.

The fact that those two points are overlooked makes the narrative a

bit too simple to be convincing for me. I would suggest starting right

away with what you want to achieve in the abstract and having an extended justification in the introduction.

L13: "governing differential equations are not necessarily known": I

would like to see examples of that. I think that, even if some equations

are known, a fully data-driven system can be justified, but here, this core question is eluded: What is the range of applications of a purely

data-driven model from partial observations?

L21: "All the approaches cited above are assuming that the full state

of the system is observed, which is a strong assumption."

This is misleading. The papers above (at least Fablet, Bocquet and

Brajard) assume that observations are

noisy and sparse, but indeed each variable has a non-null probability

to be observed. Is it what you mean by "the full state is observed?" There are also many works done in the case a variable

is never observed, e.g.:

https://arxiv.org/pdf/2102.07819.pdf

Figure 1. My understanding is that the paper aims at going one step

forward into learning a data-driven model from a realistic setting (by assuming that the

state is not fully observed), but it assumes later on that a part of

the state is always observed with a very small error. To me, this is a

very strong assumption, even stronger than assumptions made by the existing

cited papers. So again, I don't see what application is targeted by

this work.

L110: the "sequential methodology": Is there a theoretical reason to

add sequentially the hidden components or is this mainly practical? How

do you see that applied with high-dimensional systems in which, e.g.,

10^5 variables are non-observed?

L140: This part is, in my opinion, the most interesting part. But I

miss some details to fully understand what is done (see below)

Eq 7: How do you derive those equations? Is it by trials/error or is

does it correspond to theoretical reasons?

L150 "correspond to a3 ≈ 0 and to a2 ≈ 0, respectively": sorry I don't

get the "respectively" here, in which case a3 is 0 and in which case

a2 is 0?

L150-151:  "This suggests that xdot3 is more important than xdot2":

Why is that? you still have b2 coefficient associated with xdot2...

L153-155: sorry I have read this part several times, and I still don't

understand. What does it mean that "the algorithm focuses on the

estimation of a_2" I don't see where is the estimation of a_2 in the

algorithm and I don't understand what is meant by "focus".

L158: The term "model-driven" is misleading. The data-driven model is

also a model.

L175: "the dynamical evolution of the system is retrieved with our

methodology. "This is a strong assertion since by construction the

evolution of x2 and x3 are observed and you test the forecast

skill over only one time-step.

End of the introduction: I think it would be nice to have part of

these comments in the introduction, justify the approach.