Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 8: Top 50% of accepted NeurIPS papers, clear accept
review_content: summary: This work proposes a neural network architecture and training approach that leverage theories of dynamical systems for introducing a novel parametrization which splits the underlying time evolution dynamics into two parts – an energy conserving part, and as irreversible part. This allows one to use this framework for learning dynamics with dissipative states (for example, a damped oscillator). To accommodate systems with dissipative states that may not be directly measured, this work introduces a training scheme which alternates between system identification (assumes knowledge about both measured and non-measured states and tries to learn the underlying dynamics) and state estimation (assumes knowledge about the underlying dynamics and tries to estimate the non-measured states from the measured states). The proposed framework has been demonstrated on 3 example systems. To conclude, this paper addresses a very topical problem and the details have been provided with sufficient clarity. 
main_review: ### Originality: 

The overall idea is novel and very interesting. 

However, some relevant prior works are missing from the related work section. *Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning* (DeepDiffEq Worksho at ICLR’2020) and *OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle* (arXiv: 2009.02327) have focused on incorporating energy-dissipation into such physics-governed neural networks. The authors should properly refer to these papers and highlight the similarities/differences. *Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control* (ICLR’2020) and *Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints* (NeurIPS’2020) are also relevant references in this context. The authors may refer to *Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data* (arXiv:2012.02334) for comprehensive details about the various existing work that focus on incorporating Hamiltonian/Lagrangian-based inductive bias. 

Also, Lagrangian-based neural networks precede Hamiltonian-based neural networks; *Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning* was published at ICLR’2019, almost 8 months before *Hamiltonian Neural Network* was presented at NeurIPS’2019 – this chronology contradicts the phrase “followed by” on line 68. 

Lastly, references [21] and [36] refer to the same paper *Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning* – the authors should fix this.

### Quality: 
The experiments demonstrate that GNODE outperforms black-box Neural ODE as well as the penalty-based approach. However, in this context the update frequency for the unobserved state estimation appears to be a critical factor. Therefore, the authors should report on how the performance of GNODE and the penalty-based method varies with changes in that update frequency. Also, why $n_{update}$ was chosen to be the same for both of these methods? 

### Clarity: 
This paper is very well-written which helps in conveying the message in an effective way. Also, it provides all the necessary details. 

### Significance: 
In the last few years, a fast growing volume of work has focuses on incorporating Hamiltonian of Lagrangian formulation based inductive bias while learning dynamics of data. However, most of these papers (except *Dissipative-SymODEN* and *OnsagerNet*) encode *energy conservation* in the architecture of neural networks. This paper expands that line of work by incorporating energy-dissipation into a such energy conserving neural networks. As almost all real-world physical systems have some-form of energy dissipation, this reviewer believes that the proposed approach would be relevant for many practical applications.
limitations_and_societal_impact: The authors have adequately addressed the limitations of the submitted work.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The paper introduces a new framework for learning neural ODEs obeying metriplectic dynamics. These differ from previous work on Hamiltonian dynamics in that it includes both conservative and dissipative terms guaranteeing that system entropy is non-decreasing. Dissapative dynamics are extremely common in real-world systems and so this is an important inductive bias to be able to account for. The authors compare their proposed method with black box and penalty based approaches and find that their proposed method outperforms both.
main_review: The writing and motivation in the paper are very strong, though there are a few terms missing, namely the $s_i$ in the experiments. They seem to be specific entropy, but I didn't catch a definition in the paper. Overall on this front, the motivation is clear and the approach taken seems to be well founded. To my knowledge, the approach taken is unique though it seems well-grounded in previous work and a natural extension from previous work on learning Hamiltonian dynamics. 

The main weakness of the paper is in the experiments and presentation of the results. The method shows promise and it's positive to see that H and S do seem to enjoy the proposed theoretical benefits in practice. There are a few aspects of the experiments that seem under-explained and perhaps overfit to the experiments:
- D is defined explicitly to be rank 1. This is not a requirement in the general framework and the matrix is too small for this to seem motivated by storage costs, but it seems like it would reduce the influence of the disappative term. It's hard to tell if this is theoretically or experimentally motivated. 
- The configurations of the networks seem to vary significantly and this isn't explained. All tests seem to be done on small systems with correspondingly small networks. While the additional parameters of the penalty and GNODE methods may motivate smaller networks for fair comparison, the significant differences make it hard to tell if the better fit is due to the method or if it's just that neural ODEs are hard to train and the smaller networks led to better fits on relatively simple dynamics. 
- The penalty comparison does not feel very robust. One value of the multipliers is reported without indication as to how they were chosen. It seems odd that the penalty method would completely underperform the black box approach as is seen in figure one and without the error bars, I'd guess it was a bad optimization run. Due to the small number of runs, it still seems possible that they were influence by an outlier.
- These seem like one particle systems, so if the target of interest is mapping the trajectory, then I'd like to see results reported on the discrepency between the true and targeted trajectories and not just H as the black box method is not trained to optimize performance on H. 

Finally, there were also a few discrepancies between what was stated in the checklist and what appeared in the paper. First, I do not think it is reasonable to take credit for releasing code when the code is not included in the supplementary material. If the code isn't included in the submission, then the answer to "did we release the code?" is no regardless of future intentions. Error bars are present, but wide and it's unclear if these are the extremes or a number of standard deviations. These are also smaller experiments, so it seems as though more than 5 runs should be feasible particularly when the deviation seems to be so large. 

Overall, the paper is solid and proposed a promising direction, but in it's current version, the experiments are hard to put significant stock into. It would make for a stronger paper if larger systems were explored or more consistent configurations were used. Additional comparisons would also be valuable as vanilla neural ODEs are not a very strong baseline. It would also be interesting to see how the model preserves on non-metriplectic systems (Hamiltonian or unknown systems from collected data) to see if the technique is capable of adapting in those settings (ie, it should be able to represent Hamiltonian dynamics, does it outperform black box ODEs on those tasks?) though obviously it would not be expected to outperform methods designed for those settings.

EDIT: Updated score based on author comments. 
limitations_and_societal_impact: Limitations and societal impact are adequately addressed.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The authors introduce a hard-constrained, parameterized GENERIC framework for learning irreversible (dissipative) dynamics. To guarantee total energy conservation and entropy production, the authors use reformulation techniques for Poisson (reversible) and irreversible brackets that automatically satisfy the degeneracy conditions for GENERIC. As a result, the authors’ model can learn various dynamical systems more reliably and plausibly, compared to the baselines such as conventional neural ODE or soft-constrained GENERIC (that learns the degeneracy condition via penalty loss.) The authors validate their model for various benchmarks including damped nonlinear oscillators, two gas containers, and stochastic damped oscillators. 
main_review: Originality: The main contribution of this paper is that the proposed model guarantees the degeneracy conditions (that should be satisfied for obeying the laws of thermodynamics) automatically while the prior work only guides such conditions via regularizers. This can be achieved by reformulating the reversible and irreversible brackets (Section 4.1 and Section 4.2). However, these reformulations are identical to the work by Oettinger ([42] in the paper). The authors pick some appropriate formulas from Oettinger’s paper, and then directly parameterize the formulas to enable them to learn. As a result, this paper can have originality from the perspective of bridging/combination of well-known techniques, but not definitely novel. 

One interesting point is that the authors distinguish the observable (position and momentum) and non-observable (entropy, …) states, and then learn the non-observables unsupervisely. I think this is an original point of this paper that might be important for real-world problems. 

Overall, I think this paper has moderate novelty/originality.

Quality: I think this paper is technically adequate. GENERIC framework and modified reversible/irreversible brackets are principled structures to learn and extrapolate dissipative dynamical systems. Experimental results seem to be solid, except for some minor concerns (see below comments 1 and 2).

Clarity: This paper is extremely well-written and easy to follow. The motivation is clearly presented. Prior works are well summarized (while some works are missing. See below comment 3). The preliminary and specific techniques are appropriately introduced. The experimental session sufficiently provides details on settings and hyper-parameters. Figures effectively show the experimental result.

Significance: Learning irreversible dynamical systems is definitely important for solving real-world problems. This paper provides a solid contribution to such important problems. 

I think this paper is good, especially in terms of clarity. However, the lack of originality and some concerns (see below) prevent me from recommending clear acceptance for this paper. My current evaluation of this paper is borderline.

Some concerns on the paper:

1. The authors used $10^{-4}$ penalty loss weight when they train the penalty-based GENERIC (line 191 and line 233). It seems $10^{-4}$ is too small to learn the degeneracy conditions. Why did you use such small penalty strength? What if one increases the penalty weight, e.g., $10^{-1}$ or 1?

2. For the two gas containers experiment (Section 5.2), the error bar / standard deviation is not reported, unlike the damped oscillator experiment (Section 5.1). Why?

3. There are some uncited prior works that try to model dissipative or chaotic dynamics, such as [1] Y. D. Zhong, B. Dey, and A. Chakraborty. Dissipative SymODEN: Encoding Hamiltonian dynamics with dissipation and control into deep learning. ICLR Workshop on Integration of Deep Neural Models and Differential Equations, 2020, [2] K. Course, T. Evans, and P. Nair. Weak form generalized Hamiltonian learning. NeurIPS, 2020.

***
Post-rebuttal: After reading the authors' response, now I think the paper’s contribution outweighs my initial concerns (especially regarding the originality). I increased my review score accordingly.
limitations_and_societal_impact: The authors mention the limitation of this work, the scalable issue, in the conclusion.
review_title: null

Meta-review: This is a very good paper and a clear accept. All reviewers agree that there is a novel methodological contribution that extends neural ODE approaches to irreversible dynamical systems. The parametrization of brackets cleverly preserve structures of the dynamical system, leading to exact preservation of thermodynamical consistency and better generalization. The numerical experiments are good, but can be further improved. The paper is written nicely and the main ideas are explained well. 
Decision: Accept (Poster)