Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: This paper proposes a new framework (LEADS) for learning dynamical systems (ODEs, PDEs) with the aim of generalising across related, but slightly different environments. It uses an additive two-component model, comprising a shared model and an environment-specific model. The paper presents some generic sample complexity results, then looks at the special case of a linear ODE. The paper claims this motivated a choice for an objective in the nonlinear NN case. This is then experimentally investigated in the small data regime, with just a few example trajectories, and just a few different environments. The paper claims the proposed framework significantly outperforms other baselines and ablations.
main_review: **Originality**

I’m not an expert in this area, but the theoretical propositions seem to be generic, from their reference [4], with some mild adaptation to their setting - this doesn’t seem that original in my opinion. Further, they don’t properly reference this with respect to [4], stating only that the theory was inspired by [4]. More detail is needed to properly assess the originality of this section. I would have also liked to see more detail in the related work section. To the best of my knowledge, the proposed method seems to be original. 

**Quality**

I couldn’t follow the proof of Prop 2 in supplemental, I don’t understand why the factor of 4/delta appears in c_g term, but not in c_f term. I haven’t tried to verify the proof of other statements. I didn’t understand why the linear case motivated taking equation (7) for the NN case. I would like a scale for the MSE error maps in fig 2 to assess quality. Otherwise, qualitative results in figs2,3 seem good quality. Table 1: Needs more detail on how significance window was calculated. I felt for LV and NS that LEADS wasn’t significantly better than the ablation LEADS no min. For GS, I felt LEADS wasn’t significantly better than FT-NODE (note typo in table 1, FT-ODE -> FT-NODE). To complement fig 5, I would like to see results on GS and NS. I would have also liked to see performance differences in the regime of larger number of environments and trajectories. 

**Clarity**

The prose in most of the paper was acceptable, but I found the math difficult to read, especially prop 2-4. Are these propositions really necessary for the main paper? They didn’t help me as a non-expert reader to understand what was going on, or why this was important. All figures would benefit from being larger. 

**Significance**

As noted in the limitations section, this only investigated simulated, deterministic data. I found this surprising because when considering generalization, this is typically done in the setting where the data contains some noise, and one tries not to overfit to that noise.  
Moreover, considering this paper is using neural networks, which are typically better in the larger data regime, this makes me question the significance of the results, compared to the baselines (especially those that use neural networks). 
As also noted in limitations section, the complexity analysis is only relevant in simple hypothesis spaces, such as the linear case. But already this wasn’t enough for the simple NN experiments in this paper. 

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Edit:

Following discussion period, I have now increased my score from 4 to 7, based on author's better explanation of equation (7), and their other proposed changes.
limitations_and_societal_impact: Some technical limitations were discussed in the work. No societal impact was discussed.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The paper proposes an algorithm which learns to model dynamical systems
consisting of multiple environments that exhibit shared and
environment-specific dynamics. The key idea is to learn separate approximation
functions for shared and environment-specific components. To prevent a
trivial solution that circumvents the approximation of shared dynamics, the
proposed objective regularizes environment-specific terms.

On the theoretical side, the paper quantifies the improvement in sample
complexity for novel environments in a simplified (linear) setting for which a
suitable regularization term is derived. Empirically, the method is evaluated
on three datasets with nonlinear dynamics (all datasets have shared and
environment-specific components) and the results support the claims about
improved generalization on known and novel environments, especially when data
is scarce (i.e., limited number of trajectories).

main_review: Even though the paper is outside my domain of expertise, it was mostly easy
to follow, due to the clear and unambiguous writing style and notation, as well
as the excellent explanations throughout the paper. The authors are transparent
about the limitations of the proposed approach. The distinction from previous
work in related fields appears somewhat superficial (see Comments and Questions). 

The presented empirical results show consistent and significant improvements
compared to all baselines, supporting the claims about improved generalization
on known and novel environments. However, I think that certain ablations (see
Comments and Questions) could provide a more complete picture of the results
and the suitability of the employed heuristics (e.g., the regularization term
for neural networks). Due to my lack of domain knowledge, I cannot evaluate the
originality and significance of this work in the context of dynamical systems.


**Comments and Questions:**

- Could you give an example for a real-world dynamical system that exhibits the
  properties assumed in this work (i.e., shared and
  environment-specific mechanisms across a bunch of environments)? Is there a
  particular system that you imagine in practice or are the three presented
  datasets the closest examples that you have for such a dynamical system?

- In the related work section, the distinction from work in related fields
  seems somewhat hand-wavy. With respect to meta-learning approaches, even if
  they do not focus on generalization on training environments, that does not
  imply that they do not generalize well on training environments. Even if
  meta-learning approaches or ODE2VAE (Yildiz et al., 2019) focus on a single
  model or a single dynamical system, it could still be interesting to include
  the methods as a benchmark.  For example, does their test error also increase
  with the number of training environments (similar to *One-For-All* in Figure
  4)?

- Did you consider running an ablation for the hyperparameter lambda? It could
  be instructive to see the generalization performance for different
  values of lambda. 

- From Appendix B.4 and Appendix C, I understand that the regularizer from
  Equation (7) is approximated via heuristics that introduce additional
  hyperparameters. For example, the choice of $p$-norm (Table S4) seems to have
  a non-negiglible effect on the results of the LV experiment. In light of this
  variablility, did you consider to include the results for different $p$-norms
  in Table 1 directly?

- For Figure 4, do you also have results for FT-RNN and FT-ODE? It would be
  interesting to see how robust the methods perform with respect to the number
  of training environments.

- Lines 103f: could you point out the source or theorem for why the additive
  hypothesis is not restrictive given functional vector spaces?
limitations_and_societal_impact:
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: This proposes a simple but effective method to learn a dynamics prediction model that can generalize different environments. The authors also provide theoretical and experimental results to demonstrate the superiority of the proposed method over baselines.
main_review: Advantages:
   1. Learning a dynamics prediction model that can generalize across different environments is important and practical for many real-world applications, and the authors provide a simple and intuitive method to solve it with theoretical guarantees.
   2. The experimental results shown in the main paper and Appendix are satisfactory.

Disadvantage:
  1. I notice that the proposed method can only be applied to deterministic dynamics prediction, which limits its potential application scenarios.
  2. The hyperparameters \lambda and \alpha varies significantly across different environments. Can the authors provide the results about the sensitivity of hyperparameters?
limitations_and_societal_impact: See the section "Main Review"
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper presents an approach called LEADS to learn *decomposed* dynamics models of system instances ("environments") from a family of systems, e.g., Lotka-Volterra systems with randomly sampled parameters. The main idea is to additively decompose the dynamics into a shared component $f$ which does not depend on the specific environment and an environment-specific component $g_e$. It is shown both theoretically and experimentally that it is crucial to limit the complexity of $g_e$. This allows to adapt to new, unseen environments by leveraging "knowledge" absorbed in $f$ and additionaly using only a few interactions with the new environment to learn a new $g_{e'}$. The performed experiments qualitatively and quantitatively confirm the hypothesis that the proposed algorithm LEADS is more accurate than training a single model for all environments and even a single model for each environment.


main_review: **Originality**

The paper suggests an additive decomposition of environment dynamics into a *shared* and *environment-specific* part. To my understanding, the main methodological novelty is that *both* parts are learned while a minimization objective on the environment-specific part is imposed. With regard to this, some related work should be discussed more precisely in my opinion:

  * Another instance of dynamics models which are decomposed into a *shared* part and an *environment-specific* part are "context-aware dynamics models" or "context-conditional dynamics models", e.g. by (Lee et al., 2020). I think the paper would benefit from mentioning these approaches.

  * An additive decomposition of a gradient field of an ODE has been described in (Yin et al. 2021), who also highlight the need for a minimization objective on one of the components (Yin et al., 2021, eq. 2) and show existence and uniqueness properties (Yin et al., 2021, Prop. 1+2). I think the paper of Yin et al. (2021) should be, due to this overlap, discussed in more detail in the paper under review.

  * I think the statement in l. 75/76 "Up to our knowledge, this is the first time it is addressed from an ML viewpoint" needs clarification. To me it is not immediately clear what is meant by "this". Has the problem/question already been addressed from a different, non-ML viewpoint?


**Quality**

* The chosen approach is well-motivated by mathematical derivations, e.g. regarding limiting the complexity of the environment-specific dynamics components. 

* All experiments are performed on a set of ODE/PDE dynamics (Lotka-Volterra, Gray-Scott and Navier-Stokes), showing that the reported effects are valid for multiple systems.

* Reasonable ablation and comparison experiments are performed highlighting the importance of the minimization objective.


**Clarity**

* The paper is well-structured and clearly written. The relevant research questions are clearly outlined.

* The figures nicely illustrate the results obtained using the method and the main statements of the paper. Some minor comments:
  * Fig. 2: I would appreciate a colorbar for the MSE error maps to at least estimate the quantitative difference in the predictions between LEADS and its ablations / baseline methods.
  * Fig. 4: Coordinate axes ('lines') at the bottom and on the right (including 'ticks') would help to immediately see that the y coordinate of the curves is not determined by "Training trajectories per env.", which I first falsely assumed to be a second "axis" on the left.
  * For consistency, I suggest to introduce coordinate axes also in Fig. 1, 3, 5.

* The derivations and experiments for the 'simplified' linear case help in understanding the mechanics of the method before considering the more-complicated NN models.


**Significance and conclusion**:
In my opinion the proposed idea of decomposing dynamics into a "general" component and an "environment-specific" component is very interesting and could, for example, help the community to build more sample-efficient model-based RL algorithms which are able to quickly adapt to changing environments. Although the idea of additive decomposition seems simple, it requires significant theoretical effort to concretize this idea, especially regarding the required complexity regularization. This paper provides a strong theoretical formulation and experimental evaluation of the proposed method. Noting that I haven't checked the mathematical derivations in detail, I vote for acceptance.


**References:**

 (Lee et al., 2020): 
 Kimin Lee, Younggyo Seo, Seunghyun Lee, Honglak Lee, Jinwoo Shin  
 "Context-aware Dynamics Model for Generalization in Model-Based Reinforcement Learning"  
 ICML2020, https://arxiv.org/abs/2005.06800  
 
 (Yin et al., 2021):  
 Yuan Yin, Vincent Le Guen, Jérémie Donà, Emmanuel de Bézenac, Ibrahim Ayed, Nicolas Thome, Patrick Gallinari  
 "Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting"  
 ICLR2021, https://arxiv.org/abs/2010.04456
limitations_and_societal_impact: * The authors clearly mention the limitations of their work.

* Societal impacts are not explicitly addressed. In my opinion, this work does not exhibit extraordinary societal impacts in addition to those induced by basic reasearch on dynamics modelling methods in general.
review_title: null

Meta-review: From the SAC: This is an instance where a good rebuttal has helped. The original decision on this paper was a reject, but the SAC felt that your rebuttal was very thorough and detailed. Hence I am recommending moving this paper to an accept. Please do make sure in the next version of the paper to take all reviewer comments into account as well as to merge into the paper the additional discussion in your rebuttal (perhaps, due to space considerations, into the appendices). 
Decision: Accept (Poster)