In this work, we take the first steps in making use of this connection.

Koopman operator theory, a powerful framework for discovering the underlying dynamics of nonlinear dynamical systems, was recently shown to be intimately connected with neural network training. In this work, we take the first steps in making use of this connection. As Koopman operator theory is a linear theory, a successful implementation of it in evolving network weights and biases offers the promise of accelerated training, especially in the context of deep networks, where optimization is inherently a non-convex problem. We show that Koopman operator theoretic methods allow for accurate predictions of weights and biases of feedforward, fully connected deep networks over a non-trivial range of training time. During this window, we find that our approach is >10x faster than various gradient descent based methods (e.g.

As discussed in Sec. 3 of the main text, the computational complexity of Koopman training is We assume that both standard training and Koopman training use simple matrix computation methods. We note that none of these factors are relevant for Koopman training. The finite section method, Eq. 4, implies the run time complexity would be The authors contributed equally 34th Conference on Neural Information Processing Systems (NeurIPS 2020), V ancouver, Canada. Koopman operator(s) and evolve each partition separately from the others. In Sec. 3, we discussed when we think this "patching" approach should give small errors.

Our approach leverages Koopman operator theory, which enables a linear representation of nonlinear dynamics by lifting trajectories into a higher-dimensional space. The framework follows a two-stage design: first, a probabilistic neural goal estimator predicts plausible long-term targets, specifying where to go; second, a Koopman operator-based refinement module incorporates intention and history into a nonlinear feature space, enabling linear prediction that dictates how to go. This dual structure not only ensures strong predictive accuracy but also inherits the favorable properties of linear operators while faithfully capturing nonlinear dynamics. As a result, our model offers three key advantages: (i) competitive accuracy, (ii) interpretability grounded in Koopman spectral theory, and (iii) low-latency deployment. We validate these benefits on ETH/UCY, the Waymo Open Motion Dataset, and nuScenes, which feature rich multi-agent interactions and map-constrained nonlinear motion. Across benchmarks, KoopCast consistently delivers high predictive accuracy together with mode-level interpretability and practical efficiency.

Trajectory planning for robotic manipulators operating in dynamic orbital debris environments poses significant challenges due to complex obstacle movements and uncertainties. This paper presents Deep Koopman RRT (DK-RRT), an advanced collision-aware motion planning framework integrating deep learning with Koopman operator theory and Rapidly-exploring Random Trees (RRT). DK-RRT leverages deep neural networks to identify efficient nonlinear embeddings of debris dynamics, enhancing Koopman-based predictions and enabling accurate, proactive planning in real-time. By continuously refining predictive models through online sensor feedback, DK-RRT effectively navigates the manipulator through evolving obstacle fields. Simulation studies demonstrate DK-RRT's superior performance in terms of adaptability, robustness, and computational efficiency compared to traditional RRT and conventional Koopman-based planning, highlighting its potential for autonomous space manipulation tasks.

Additional Feedback: As noted above, one of the biggest drawbacks of this very interesting work at present is the very limited scope of the demonstrations. I believe this should be easy to address, and were this done I would feel comfortable increasing my score. It would also be useful to see more detailed empirical study regarding the choice of the window (t1-t2) used to collect the data to inform the operator approximation.) There are a couple of details that I would like to see to help improve reproducibility. In terms of related work, there are a couple of more tangential directions that come to mind where connections could potentially be made / that may be interesting for the authors to consider. There may be connections related to initial stages of gradient descent identifying subspaces in which most of the parameter evolution will occur (i.e. containing the lottery ticket weights).

This paper provides a new perspective on neural network training based on Koopman operator theory (KOT). The paper received mixed reviews (top 50% - marginally above, reject, marginally above - accept, marginally below). On the positive side, and despite KOT being very old, the new perspective has a lot of potential: since the KOT is linear, if one can find (or approximate) eigenfunctions, one could compute and analyze training dynamics more easily and make optimization more efficient. On the negative side, the paper is a first step, and needs further development and experimental evaluation to demonstrate the value. Some reviewers also expressed the paper lacks clarity.

Koopman operator theory, a powerful framework for discovering the underlying dynamics of nonlinear dynamical systems, was recently shown to be intimately connected with neural network training. In this work, we take the first steps in making use of this connection. As Koopman operator theory is a linear theory, a successful implementation of it in evolving network weights and biases offers the promise of accelerated training, especially in the context of deep networks, where optimization is inherently a non-convex problem. We show that Koopman operator theoretic methods allow for accurate predictions of weights and biases of feedforward, fully connected deep networks over a non-trivial range of training time. During this window, we find that our approach is 10x faster than various gradient descent based methods (e.g.

The strong performance of simple neural networks is often attributed to their nonlinear activations. However, a linear view of neural networks makes understanding and controlling networks much more approachable. We draw from a dynamical systems view of neural networks, offering a fresh perspective by using Koopman operator theory and its connections with dynamic mode decomposition (DMD). Together, they offer a framework for linearizing dynamical systems by embedding the system into an appropriate observable space. By reframing a neural network as a dynamical system, we demonstrate that we can replace the nonlinear layer in a pretrained multi-layer perceptron (MLP) with a finite-dimensional linear operator. In addition, we analyze the eigenvalues of DMD and the right singular vectors of SVD, to present evidence that time-delayed coordinates provide a straightforward and highly effective observable space for Koopman theory to linearize a network layer. Consequently, we replace layers of an MLP trained on the Yin-Yang dataset with predictions from a DMD model, achieving a mdoel accuracy of up to 97.3%, compared to the original 98.4%. In addition, we replace layers in an MLP trained on the MNIST dataset, achieving up to 95.8%, compared to the original 97.2% on the test set.

Machine learning methods allow the prediction of nonlinear dynamical systems from data alone. The Koopman operator is one of them, which enables us to employ linear analysis for nonlinear dynamical systems. The linear characteristics of the Koopman operator are hopeful to understand the nonlinear dynamics and perform rapid predictions. The extended dynamic mode decomposition (EDMD) is one of the methods to approximate the Koopman operator as a finite-dimensional matrix. In this work, we propose a method to compress the Koopman matrix using hierarchical clustering. Numerical demonstrations for the cart-pole model and comparisons with the conventional singular value decomposition (SVD) are shown; the results indicate that the hierarchical clustering performs better than the naive SVD compressions.