# ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals

arXiv: https://arxiv.org/abs/2501.00701 · HTML: https://mlsanity.com/paper/2501.00701/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 45 references that are in the map.
- First posted on arXiv 1 Jan 2025.

## Builds on (most cited first, 15 of 15)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [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)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Variational approach for learning Markov processes from time series data](https://mlsanity.com/paper/1707.04659/) (2017)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [The Barron Space and the Flow-induced Function Spaces for Neural Network Models](https://mlsanity.com/paper/1906.08039/) (2019)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [The Dynamic Sensorium competition for predicting large-scale mouse visual cortex activity from videos](https://mlsanity.com/paper/2305.19654/) (2023)
- [Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2403.02524/) (2024)
- [Retrospective for the Dynamic Sensorium Competition for predicting large-scale mouse primary visual cortex activity from videos](https://mlsanity.com/paper/2407.09100/) (2024)

## Built on by (most cited first, 6 of 6)

- [ReLaX: Reasoning with Latent Exploration for Large Reasoning Models](https://mlsanity.com/paper/2512.07558/) (2025)
- [Information Shapes Koopman Representation](https://mlsanity.com/paper/2510.13025/) (2025)
- [Hierarchical Koopman Diffusion: Fast Generation with Interpretable Diffusion Trajectory](https://mlsanity.com/paper/2510.12220/) (2025)
- [Factorized Neural Implicit DMD for Parametric Dynamics](https://mlsanity.com/paper/2603.10995/) (2026)
- [Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation](https://mlsanity.com/paper/2606.29083/) (2026)
- [Conformal Online Learning of Deep Koopman Linear Embeddings](https://mlsanity.com/paper/2511.12760/) (2025)

## Abstract (arXiv)

Analyzing the long-term behavior of high-dimensional nonlinear dynamical systems remains a significant challenge. While the Koopman operator framework provides a powerful global linearization tool, current methods for approximating its spectral components often face theoretical limitations and depend on predefined dictionaries. Residual Dynamic Mode Decomposition (ResDMD) advanced the field by introducing the spectral residual to assess Koopman operator approximation accuracy; however, its approach of only filtering precomputed spectra prevents the discovery of the operator's complete spectral information, a limitation known as the `spectral inclusion' problem. We introduce ResKoopNet (Residual-based Koopman-learning Network), a novel method that directly addresses this by explicitly minimizing the spectral residual to compute Koopman eigenpairs. This enables the identification of a more precise and complete Koopman operator spectrum. Using neural networks, our approach provides theoretical guarantees while maintaining computational adaptability. Experiments on a variety of physical and biological systems show that ResKoopNet achieves more accurate spectral approximations than existing methods, particularly for high-dimensional systems and those with continuous spectra, which demonstrates its effectiveness as a tool for analyzing complex dynamical systems.