# Information Shapes Koopman Representation

arXiv: https://arxiv.org/abs/2510.13025 · HTML: https://mlsanity.com/paper/2510.13025/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 72 references that are in the map.
- First posted on arXiv 14 Oct 2025.

## Builds on (most cited first, 30 of 32)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [WeatherBench 2: A benchmark for the next generation of data-driven global weather models](https://mlsanity.com/paper/2308.15560/) (2023)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors](https://mlsanity.com/paper/2305.18803/) (2023)
- [Rethinking Minimal Sufficient Representation in Contrastive Learning](https://mlsanity.com/paper/2203.07004/) (2022)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [On Mutual Information in Contrastive Learning for Visual Representations](https://mlsanity.com/paper/2005.13149/) (2020)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)
- [CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamics](https://mlsanity.com/paper/2310.05963/) (2023)
- [Robust Locally-Linear Controllable Embedding](https://mlsanity.com/paper/1710.05373/) (2017)
- [Information Theory with Kernel Methods](https://mlsanity.com/paper/2202.08545/) (2022)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Optimal construction of Koopman eigenfunctions for prediction and control](https://mlsanity.com/paper/1810.08733/) (2018)
- [Koopman Q-learning: Offline Reinforcement Learning via Symmetries of Dynamics](https://mlsanity.com/paper/2111.01365/) (2021)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Prediction, Consistency, Curvature: Representation Learning for Locally-Linear Control](https://mlsanity.com/paper/1909.01506/) (2019)
- [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023)
- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)
- [Consistent Long-Term Forecasting of Ergodic Dynamical Systems](https://mlsanity.com/paper/2312.13426/) (2023)
- [Latent Representation and Simulation of Markov Processes via Time-Lagged Information Bottleneck](https://mlsanity.com/paper/2309.07200/) (2023)
- [ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals](https://mlsanity.com/paper/2501.00701/) (2025)
- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)

## Built on by (most cited first, 4 of 4)

- [Temporal Straightening for Latent Planning](https://mlsanity.com/paper/2603.12231/) (2026)
- [Physics-Constrained Deep Learning Model for Contactless Blood Pressure Monitoring from Triaxial Bodyseismography](https://mlsanity.com/paper/2608.23562/) (2026)
- [Diffusion Distillation for Efficient Weather Ensembles](https://mlsanity.com/paper/2608.27728/) (2026)
- [Kapture: Capturing Cardiac Dynamics with Koopman-Governed Learning for Efficient Radar-Based Electrocardiogram Recovery](https://mlsanity.com/paper/2610.04955/) (2026)

## Abstract (arXiv)

The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods. The implementation is publicly available at https://github.com/Wenxuan52/InformationKoopman.