# An Operator Theoretic Approach for Analyzing Sequence Neural Networks

arXiv: https://arxiv.org/abs/2102.07824 · HTML: https://mlsanity.com/paper/2102.07824/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 46 references that are in the map.
- First posted on arXiv 15 Feb 2021.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Transformer Interpretability Beyond Attention Visualization](https://mlsanity.com/paper/2012.09838/) (2020)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Visualizing and Understanding Recurrent Networks](https://mlsanity.com/paper/1506.02078/) (2015)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (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)
- [Lipschitz Recurrent Neural Networks](https://mlsanity.com/paper/2006.12070/) (2020)
- [Trivializations for Gradient-Based Optimization on Manifolds](https://mlsanity.com/paper/1909.09501/) (2019)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [Learning Compositional Koopman Operators for Model-Based Control](https://mlsanity.com/paper/1910.08264/) (2019)
- [Reverse engineering recurrent networks for sentiment classification reveals line attractor dynamics](https://mlsanity.com/paper/1906.10720/) (2019)
- [Optimizing Neural Networks via Koopman Operator Theory](https://mlsanity.com/paper/2006.02361/) (2020)
- [Reverse engineering learned optimizers reveals known and novel mechanisms](https://mlsanity.com/paper/2011.02159/) (2020)
- [Applications of Koopman Mode Analysis to Neural Networks](https://mlsanity.com/paper/2006.11765/) (2020)
- [How recurrent networks implement contextual processing in sentiment analysis](https://mlsanity.com/paper/2004.08013/) (2020)
- [Weakly Supervised Deep Functional Map for Shape Matching](https://mlsanity.com/paper/2009.13339/) (2020)

## Built on by (most cited first, 12 of 12)

- [Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs](https://mlsanity.com/paper/2310.02619/) (2023)
- [Utilizing Image Transforms and Diffusion Models for Generative Modeling of Short and Long Time Series](https://mlsanity.com/paper/2410.19538/) (2024)
- [Multifactor Sequential Disentanglement via Structured Koopman Autoencoders](https://mlsanity.com/paper/2303.17264/) (2023)
- [One-Step Offline Distillation of Diffusion-based Models via Koopman Modeling](https://mlsanity.com/paper/2505.13358/) (2025)
- [Sequential Disentanglement by Extracting Static Information From A Single Sequence Element](https://mlsanity.com/paper/2406.18131/) (2024)
- [First-Order Manifold Data Augmentation for Regression Learning](https://mlsanity.com/paper/2406.10914/) (2024)
- [Analyzing Deep Transformer Models for Time Series Forecasting via Manifold Learning](https://mlsanity.com/paper/2410.13792/) (2024)
- [Unsupervised Representation Learning from Sparse Transformation Analysis](https://mlsanity.com/paper/2410.05564/) (2024)
- [Interpreting Temporal Graph Neural Networks with Koopman Theory](https://mlsanity.com/paper/2410.13469/) (2024)
- [Finding separatrices of dynamical flows with Deep Koopman Eigenfunctions](https://mlsanity.com/paper/2505.15231/) (2025)
- [Disentanglement Beyond Static vs. Dynamic: A Benchmark and Evaluation Framework for Multi-Factor Sequential Representations](https://mlsanity.com/paper/2510.17313/) (2025)
- [Glocal Hypergradient Estimation with Koopman Operator](https://mlsanity.com/paper/2402.02741/) (2024)

## Abstract (arXiv)

Analyzing the inner mechanisms of deep neural networks is a fundamental task in machine learning. Existing work provides limited analysis or it depends on local theories, such as fixed-point analysis. In contrast, we propose to analyze trained neural networks using an operator theoretic approach which is rooted in Koopman theory, the Koopman Analysis of Neural Networks (KANN). Key to our method is the Koopman operator, which is a linear object that globally represents the dominant behavior of the network dynamics. The linearity of the Koopman operator facilitates analysis via its eigenvectors and eigenvalues. Our method reveals that the latter eigendecomposition holds semantic information related to the neural network inner workings. For instance, the eigenvectors highlight positive and negative n-grams in the sentiments analysis task; similarly, the eigenvectors capture the salient features of healthy heart beat signals in the ECG classification problem.