# Fast dynamical similarity analysis

arXiv: https://arxiv.org/abs/2511.22828 · HTML: https://mlsanity.com/paper/2511.22828/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 88 references that are in the map.
- First posted on arXiv 28 Nov 2025.

## Builds on (most cited first, 20 of 20)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Generalized Shape Metrics on Neural Representations](https://mlsanity.com/paper/2110.14739/) (2021)
- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Fast and accurate optimization on the orthogonal manifold without retraction](https://mlsanity.com/paper/2102.07432/) (2021)
- [A Purely Algebraic Justification of the Kabsch-Umeyama Algorithm](https://mlsanity.com/paper/1902.03138/) (2019)
- [Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis](https://mlsanity.com/paper/2306.10168/) (2023)
- [Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)](https://mlsanity.com/paper/2106.01510/) (2021)
- [Latent Diffusion for Neural Spiking Data](https://mlsanity.com/paper/2407.08751/) (2024)
- [Identifying Equivalent Training Dynamics](https://mlsanity.com/paper/2302.09160/) (2023)
- [Algorithmic (Semi-)Conjugacy via Koopman Operator Theory](https://mlsanity.com/paper/2209.06374/) (2022)
- [Koopman-informed recurrent neural networks](https://mlsanity.com/paper/2410.23467/) (2024)
- [InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamics](https://mlsanity.com/paper/2510.25943/) (2025)
- [From univariate to multivariate coupling between continuous signals and point processes: a mathematical framework](https://mlsanity.com/paper/2005.04034/) (2020)
- [Delay Embedding Theory of Neural Sequence Models](https://mlsanity.com/paper/2406.11993/) (2024)
- [The kernel perspective on dynamic mode decomposition](https://mlsanity.com/paper/2106.00106/) (2021)
- [Koopman Operator Theory for Nonlinear Dynamic Modeling using Dynamic Mode Decomposition](https://mlsanity.com/paper/2110.08442/) (2021)
- [Neural timescales from a computational perspective](https://mlsanity.com/paper/2409.02684/) (2024)
- [Brain as a complex system, harnessing systems neuroscience tools & notions for an empirical approach](https://mlsanity.com/paper/2312.13478/) (2023)
- [Dynamical similarity analysis can identify compositional dynamics developing in RNNs](https://mlsanity.com/paper/2410.24070/) (2024)
- [Perceptual multistability: a window for a multi-facet understanding of psychiatric disorders](https://mlsanity.com/paper/2506.18176/) (2025)

## Abstract (arXiv)

Understanding how nonlinear dynamical systems (e.g., artificial neural networks and neural circuits) process information requires comparing their underlying dynamics at scale, across diverse architectures and large neural recordings. While many similarity metrics exist, current approaches fall short for large-scale comparisons. Geometric methods are computationally efficient but fail to capture governing dynamics, limiting their accuracy. In contrast, traditional dynamical similarity methods are faithful to system dynamics but are often computationally prohibitive. We bridge this gap by combining the efficiency of geometric approaches with the fidelity of dynamical methods. We introduce fast dynamical similarity analysis (fastDSA), a computationally efficient and accurate metric for measuring (dis)similarity between nonlinear dynamical systems. FastDSA leverages modern computational tools, including random matrix theory to determine optimal system rank, novel optimization pipelines for aligning system flow fields, and Koopman embeddings. Across benchmark nonlinear systems and recurrent network models, fastDSA is robust to arbitrary coordinate choices while remaining sensitive to meaningful dynamical differences, capturing variations in system evolution that geometric methods may miss and traditional methods detect only at high computational cost. To our knowledge, fastDSA is the fastest method that retains accuracy in comparing nonlinear dynamical systems. It enables scalable, statistical analyses across diverse systems, significantly expanding the practical applicability of dynamical similarity analysis.