Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper proposed an autoencoder-based model for system identification (estimating unknown parameters of a system defined by a dynamic equation). The model is shown to improve predicting long-term system dynamics with unknown physical parameters by predicting both the physical parameters and the system state.
strengths_and_weaknesses: ### Strengths
While incorporating physics into Deep RL is not novel, this paper proposes to explicit predict the physical parameter which has shown promising results for predicting system dynamics

Provided theoretical and empirical evidence for the benefit of Fourier Mapping of system states with a high-frequency component. As shown in the experiments, Fourier features boost the performance by a large margin, especially for State Error and Parameter Error.

### Weaknesses
Experiments for the full system (including the decoder) only include systems with one-dimensional system parameters. Experiments with 24 system parameters take the state measurement directly.

Code was included in the supplemental material. It is modified from other codebases however all READMEs are from the original code base and outdated. Thus it’s impossible to reproduce the results with the code in its current state.

questions: How is state error (SE) measured in the autoencoder baseline when it only reconstructs the observations without predicting states?

Fig 4 Two-Body Observation Reconstruction: why are the GTs for ALPS and CDM different?

Weighting on loss function: it’s mentioned that `In practice, we find using the same weight for each loss term works well. However, in Fig. 4, it’s shown that SE, OE, and PE are of very different magnitude. If the same loss weight is used for each term, OE would dominate the loss for ALPS. 

Scale differences in Fig 4 and Fig 6: 
E.g. For Two-body, the loss converged to ~4,000 in fig 6 while SE+OE ~= 2.59*10^8 in Figure 4, and loss = VAE loss + SE +OE according to Section 4

limitations: The authors are upfront about the following two limitations of the work:

It assumes prior knowledge of the system,e.g. dynamics equations.

It only works for linear systems with periodic or vibrational behaviors.

One additional limitation the reviewer would like to point out:
As mentioned in Weaknesses, it lacks experimental results of the full system with higher dimensional system parameters. Such results would make the paper much stronger for real system applications.
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: This manuscript proposes ALPS, Autoencoder with Latent Physics, which uses neural networks as the feature extractor and parameter regressor. A differentaible physics simulator is used to assign physical meanings to the latent space so that it follows physical laws. The manuscript compare their method on several diverse tasks including a complex real-world example.
strengths_and_weaknesses: The overall presentation is very clear to me. The method makes sense to me and the pipeline is clear. There are motivating examples and also advanced examples. After training on each environment, this parameter estimation network can be deployed without re-training or re-running any costly solver.

I would not call them weaknesses but I would like to know more about some choices the authors made in the manuscript. In my opinion, decoder $g(\cdot)$ can be replaced with a differentiable rendering and save the effort of learning a visualization system. I am aware that using a network here might be more generalizable to other scenarios which are hard to model, but I would like to know if it is a possibility.
questions: A major question I would raise is the difference between $\theta$ and $x$. Since $\theta$ can be used as the most accurate descriptor of the physical dynamics (because it supports physical simulation), it makes more sense to use the network to directly regress $\theta$ and skip $x$. Is there a reason behind it?
limitations: See weaknesses.
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: This paper proposes a method for estimating states of a physical system along with the system parameters. It can be seen as a VAE encoder-decoder set up where the decoder comprises a physics simulator (neural ODE solver) so that the latents (physical states and parameters) are forced to be interpretable. Important features of the encoder that let this method work well are 1. self-attention for estimating states from observations and 2. Fourier features for estimating parameters from states. Experiments confirm that these feature are crucial. The method is successfully applied to a real-world problem.
strengths_and_weaknesses: This paper tackles an important problem of learning how to estimate interpretable physical states and parameters from observations. It's clearly written. The experiments are well-selected and empirically verify the important components of the system (Fourier features and self-attention). Moreover, it is successfully applied to a real-world problem.

I didn't fully grasp the significance of applying the NTK theory.

I'm giving this paper a 7 (Accept) based on my understanding but I'm not an expert.
questions: What are the limitations of the physics simulator. Does it need to be representable as a differential equation (possibly non-linear)? What about a simulator written as a program? It seems like the only requirement is differentiability with respect to the initial state and the physical parameters.
limitations: N/A
review_title: null

final_score: rating: 3: Reject: For instance, a paper with technical flaws, weak evaluation, inadequate reproducibility and incompletely addressed ethical considerations.
review_content: summary: This is an interesting paper. It addresses the task of predicting states and physical parameters of a system from observations using dynamic equations and autoencoder methods.
strengths_and_weaknesses: Overall, it is difficult to understand what has been achieved. First, the empirical analysis does not provide clear evidence that the proposed approach improves quite basic engineering system identification methods, as provided in tools like MATLAB. The examples are trivial, and the firepower of the proposed approach needs more complex examples to show itself. Second, it is unclear what assumptions you make about the knowledge of physics in the basic design; the empirical analysis has very strong physics models, so reading just the empirical analysis seems to say that this is little different to LTI system identification methods.

Other Issues
============
One main issue that I have is the fundamental mismatch between the input data (images) and the empirical validation (wheel suspension system, Mass-spring-damper, pendulum). In real-world systems of this kind, typically the data will be from cheap sensors, e.g., accelerometer, position, etc.). This type of time-series input is easier to deal with than image data, and so the approach should be more successful.

Is this meant to work with systems like autonomous vehicles, where image data is integrated with lidar, accelerometer, etc. data? What is the target application domain? It is extremely unlikely that a car manufacturer would install a camera to monitor a wheel suspension system, due to cost above all.

It would be helpful to have a diagram of your full pipeline--it's hard to piece this together from the fragmented descriptions of pieces of inference.

A second fundamental issue is that the Simulations, section 6, only show system identification examples, with the help of autoencoders. The pendulum example has a well-defined ODE, and just needs to identify angle and friction coefficient. In fact, you only predict friction coefficient.
Analogous issues are with the other 2 examples. For mass-spring damper, the physics is pretty comprehensive, and in fact all you estimate is \theta = [a(2)].





 Other issues:

line 41: claim "not guaranteed to have the Markov property, i.e., be a system state."----why is this true? Do you require ONLY Markov models?


Your approach is significantly more complex than a basic autoencoder. Please address the relative computation complexity wrt a standard baseline, and describe the additional data and time (training, online) as compared to the baseline.
Obviously, there will be a tradeoff between relative computation complexity and accuracy, and this should be reported.

Figure 5:a)  image is blurred since it is proprietary. Please delete it as it adds zero content.
questions: 1. Why not compare your approach with standard system identification methods from engineering: MATLAB will do this automatically for the types of examples you use. 
2. equation 1: is the system time-invariant (LTI)?
3. line 171ff: what assumptions do you make about the knowledge of physics?

limitations: A better empirical analysis is needed. You must fix the fundamental mismatch between the input data (images) and the empirical validation (wheel suspension system, Mass-spring-damper, pendulum). 

Is this approach to be deployed on-line? Will the training be done off-line only?


line 71, section 2 "Related Work" is highly limited. There is a huge literature on learning dynamical systems that is much more relevant than PINNs. 
 --System identification is one area (you reference [50-52] later. 
 --Nagel and Huber[2021] "Autoencoder Identification of LTI systems", ECC-2021.

Also see:

@article{markovsky2021behavioral,
  title={Behavioral systems theory in data-driven analysis, signal processing, and control},
  author={Markovsky, Ivan and D{\"o}rfler, Florian},
  journal={Annual Reviews in Control},
  volume={52},
  pages={42--64},
  year={2021},
  publisher={Elsevier}
}
@article{padoan2022behavioral,
  title={Behavioral uncertainty quantification for data-driven control},
  author={Padoan, Alberto and Coulson, Jeremy and van Waarde, Henk J and Lygeros, John and D{\"o}rfler, Florian},
  journal={arXiv preprint arXiv:2204.02671},
  year={2022}
}
review_title: null

Meta-review: metareview: The paper builds a deep learning-based framework for estimating states and physical parameters by embedding a differential physics simulator into an autoencoder architecture. Integrating physics into neural networks is an interesting research area, and this paper proposes some interesting and novel ideas such as explicitly predicting the physical parameter of a given problem from data. The reviewers acknowledged the relevance of the proposed method and generally appreciated the results. The paper is nicely written, but the experiments are somewhat limited. Also the related work section is very shallow, ignoring to discuss recent work on Koopman-inspired autoencoders for scientific problems that has appeared at ICLR, ICML and NeurIPS in recent years. This said, I want to thank the authors for their detailed responses that helped in answering some of the reviewers' questions. The reviewers have provided detailed feedback in their reviews, and we strongly encourage the authors to incorporate this feedback when preparing a revised version of the paper. I particularly stress this, because the authors have not sufficiently incorporated the feedback of previous reviews at ICML. (In general, I feel that this is bad practice!! In this particular case I will have an additional look at the camera ready version of this paper.) Nevertheless, the overall feedback of the reviewers is positive (reviewer dT3B noted that he will raise his score), and I feel that this paper has the potential to motivate future research in this area. Thus, I am leaning toward suggesting a (weak) accept of this paper. 
confidence: Less certain
Decision: Accept