Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: This is a system identification method for nonlinear dynamical systems represented as ODEs based on input and output signals. Authors use neural network for such a system identification. The novelty of the paper is that the identification also comes with the guarantee on the stability. The enabling technique is to enforce/projection onto Hamiltonian-Jacobi inequality manifold for L-2 stability. 
strengths_and_weaknesses: Strengths:
System identification with neural networks has the potential to better generalize than existing methods. However, there is no guarantee that such methods are stable. This paper brings ideas from control and dynamical systems community and apply them to the learning-based system identification to enforce L2 stability. The resulting networks can be used for prediction of dynamical systems in the long-term. 

Weaknesses:
The method can be applied to nonlinear dynamical systems with fixed points or limit cycles. It is not clear if the method is applicable to more generic dynamical systems with strange attractors.
The method described is still a bit unclear. 
Although authors compare with several DNN architectures, more comparison should be made with traditional system identification methods rather than deep-learning based methods (e.g. ERA, SINDY, DMD, etc.). 
questions: Can you clarify on the application of this method to chaotic ODEs? Is it a limitation? What challenges do you foresee? 

Is it fair to say, this work is an extension of Neural ODE to enforce stability? If not, can authors briefly comment on such a possibility? Seems like this method, since it uses Euler method, can be faster than Neural ODE. What about generalization? Also what are the impact of $\Delta T$ on the convergence and numerical stability of learning. 

It's a bit confusing that \hat{y} is generated by modified dynamics (f,G,h); while the nominal dynamics is denoted by (\hat{f},\hat{G},\hat{h}). Is there a reason authors used such notation?

Maybe a schematic of the method or an algorithm can help to better clarify the method. It is still unclear how do authors in general prefer to represent V. 

Is it possible that the original system itself is not stable and then this system stabilize the neural network representation of that?
limitations: Yes.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: They propose an approach that uses the Hamiltonian-Jacobi (HJ) inequality to guarantee that the learned nonlinear dynamics have input-output stability.

This problem is fundamental and important in the sense that it bridges the gap between theoretical control theory and data-driven control for unknown nonlinear systems using well-established tools from control theory like the HJ inequality.

strengths_and_weaknesses: The merits of the paper are as follows:
1) In contrast to the most recent approaches which simultaneously design a controller and learn the dynamics, this paper goes a step ahead to also ensure input-output stability for the learned dynamics using the well-established HJ inequality.

2) The results are extensive in the sense that they clearly validate their approach and compare it with other state-of-the-art and recent approaches.

The weakness of the paper is as follows:
1) Ensuring that the HJ inequality is satisfied over a finite dataset D does not guarantee the same is true over the entire feasible time
Assumptions of (f, G, h) are not clear

More questions are in the next section “Questions”

questions: 1) What are the assumptions for (f,G,h) after Eqn(1)? The HJ inequality does not guarantee input-output stability if (f, G,h) are not continuous or differentiable. Please recheck.

2) Please cite the relevant reference for Proposition 1

3) As I am aware of the HJ inequality, apart from V being a positive definite function (which is mentioned), V must also be continuously differentiable and the value of V(0) must be zero which I think is not mentioned in your manuscript.

4) Guaranteeing that the HJ inequality is satisfied over dataset D (by minimizing the loss function given in (9)) does not ensure that the HJ inequality is satisfied for all points in the feasible domain of (x0,u,y). This is a major challenge that I observed in many recent relevant papers. However, some papers effectively address this issue by using SMT solvers where the SMT solver generates counter-examples that do not satisfy the inequality and are therefore fed back to the neural network to train. Please see relevant refs titled “Neural Lyapunov Control”, “Neural Koopman Lyapunov Control”, and “Neural Lyapunov Redesign” and the relevant references therein.

5) In eqn after (9), what does R(HJ(..)) denote? It is not clear to me from the write-up.

6) Please give some insight on whether one can compute a valid V that satisfies the HJ inequality. I think that is not mentioned in the manuscript except in the part where you write “The function V can be designed as a trainable neural network [14] or as a fixed function that achieves the desired properties of the nonlinear dynamics [15]”. However, this is not very clear to me.

7) Please give a brief remark on how you compute yhat from the nominal dynamics (f, G,h) given (x0,u,y) in the simulations.
I am wondering why is “Relevant work” in Section 4 and not after Section 1?

limitations: The main limitation of the approach is point 4 under “Questions” and it is also not clear to me on know do they compute a valid V.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: Noticing the input-output stability in dynamical systems, the paper proposed a projection method to enforce the Hamilton-Jacobi inequality in learning such dynamics with neural network. The projection problem was converted into a quadratic constraint quadratic programming problem and solved analytically. The paper also provided experimental results to show the effectiveness of such methods.
strengths_and_weaknesses: Strengths: 
1. The paper structure is easy to follow and the hierarchy of math derivation is clear. The experiments are well organized and helpful for theorem illustration.
2. Despite many recent papers to enforce certain properties in learning dynamics, the input-output stability idea is novel.


Weakness

1. The problem formulation (eqn 1) with $ \dot{x} $ linearly dependent on u limits the application to certain systems. But this is common treatment in dynamical system literature and is not a big concern.

2. The projection idea can be seen as a continuum of [A],[B], with modification on projection methods to enforce Lyapunov like properties. The setups in the above two papers are more flexible, but lack the input u and h() output function. The projection method in this paper essentially downgrades to the setup in [A] with y=x and u=0. I’m not confident but the similar model performance might also be achieved from previous works. Please correct me if I'm wrong or provide comparison experiments.

[A]: Gaurav Manek and J. Zico Kolter. Learning stable deep dynamics models. In Advances in Neural 298 Information Processing Systems (NeurIPS), volume 32, 2019.
[B]Nathan Lawrence, Philip Loewen, Michael Forbes, Johan Backstrom, and Bhushan Gopaluni. Almost 300 surely stable deep dynamics. In Advances in Neural Information Processing Systems (NeurIPS), volume 33, 301 2020.

3. In the main experimental results(figure 1 and 6), I’m concerned if the bad performance of the step input response is mainly due to model never been unexposed to such input during the training process. The results' improvement from the proposed method could be trivial if adding similar examples into training data could address such issue.

4. the experiments could be stronger to illustrate the application of the method on more complex and diverse examples. The toy example is too simple, and the general audience are not familiar with the latter glucose-insulin system in terms of complexity, modeling difficulty, system stability etc. The current examples are not representative. This is my major concern.

5. More experimental details are needed, such as neural network structure, total parameter number in both NN and other systemID methods, training hyperparameters etc.
questions: 1. I’m not convinced by the introductory examples in figure 1. The author claims the reason is “magnitude of this input signal was larger than that of the input signals in the training dataset”. However, it looks like the output signal is unbounded (thus not satisfying I/O stability under finite step activation) regardless of input signal magnitude.
2. As for notation, the author denotes $(\hat{f},\hat{G},\hat{h})$ as nominal dynamics and $(f,G,h)$ as the modified dynamics in 3.1. It is confusing since $(f,G,h)$ denotes the true dynamics in equation 1. I suggest using other styles.
3. In the loss function, why the 2nd term is needed here? Since $\hat{y}$ is calculated by modified $(f,G,h)$, then such trajectory must satisfy the Hamilton Jacobian inequality. nominal $(\hat{f},\hat{G},\hat{h})$ is not directly used in prediction, why do we need to penalize nominal $(\hat{f},\hat{G},\hat{h})$?
4. Regarding the computational time, I’m curious what is bottleneck for such long hours (60 GPU day in total). The toy example is simply single variable ODE, the Glucose-insulin examples only have 1000 input and output signals and seems to be low dimensional as well.
limitations: There is no potential negative societal impact for this work.
review_title: null

Meta-review: metareview: This paper proposes a method to learn nonlinear dynamic systems with certain theoretical guarantees.

The reviews express a positive evaluation for the research question and that the methods proposed generalise better than current SOTA. There are some concerns as to whether some of the guarantees proposed were truly verifiable, but these were resolved in the discussion phase.

The ethics review makes recommendations around including potential negative societal impacts, which we hope the authors would take on board. 
confidence: Certain
Decision: Accept