Course Correcting Koopman Representations
Mahan Fathi, Clement Gehring, Jonathan Pilault, David Kanaa, and Pierre-Luc Bacon, Ross Goroshin
TL;DR
This paper investigates autoencoder-based Koopman representations for modeling nonlinear dynamical systems, identifying limitations in long-term prediction and proposing a periodic reencoding method to improve accuracy.
Contribution
It introduces a novel inference-time mechanism called Periodic Reencoding to enhance long-term predictions in Koopman-based models, supported by analytical and empirical validation.
Findings
Periodic Reencoding improves long-term prediction accuracy.
Koopman representations can be effectively learned via autoencoders.
Limitations in latent space predictions are addressed by the proposed method.
Abstract
Koopman representations aim to learn features of nonlinear dynamical systems (NLDS) which lead to linear dynamics in the latent space. Theoretically, such features can be used to simplify many problems in modeling and control of NLDS. In this work we study autoencoder formulations of this problem, and different ways they can be used to model dynamics, specifically for future state prediction over long horizons. We discover several limitations of predicting future states in the latent space and propose an inference-time mechanism, which we refer to as Periodic Reencoding, for faithfully capturing long term dynamics. We justify this method both analytically and empirically via experiments in low and high dimensional NLDS.
Peer Reviews
Decision·ICLR 2024 poster
- The paper is well-written, featuring a fascinating main idea, and employing innovative and effective methods with good results. - The overview provided in the appendix is very helpful and is written in a precise and technical mathematical style. - The visualization provided in Fig. 1 is very helpful in providing a clear illustration of their method.
**A minor point**: Regarding the sentence "This characteristic becomes especially significant when the dimensionality of the latent space is substantially larger than that of the original space, i.e., n >> d, ...", why is this case considered more important? While it's true that a mapping to a larger dimensional space is not surjective, as discussed in the paper, it seems that the lack of injectivity of the map is more critical. So, since a mapping to a smaller dimensional space is not injectiv
- This paper is mostly well-written. - The proposed method demonstrates promising empirical performance.
- I feel the theoretical benefits of re-encoding are not well explained. It appears that re-encoding should be most useful in eliminating undesirable local optimas; if the model is correctly specified, the AE objective should have the same global optima with or without re-encoding. It is thus not clear whether it is a principled solution to the switching dynamics issue, which is one of the two motivations for this work; if the dynamics cannot be (globally) linearized the model is misspecified
- the concept of periodic re-encoding is clear and simple - it leads to across-the-board improvements in performance
- any underlying mechanism governing periodic re-encoding is not discussed. For example, are there deficiencies in the encoder or decoder that, if adjusted, remove the need for reencoding, or change the optimal period? - how is the optimum period chosen? Since every step reencoding is worse in terms of performance than periodic reencoding, this is not a simple tradeoff in terms off costs - what causes do we hypothesise for "long term drift"? Perhaps during training the encoder does not learn to
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsModel Reduction and Neural Networks · Gaussian Processes and Bayesian Inference · Computational Physics and Python Applications