# InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamics

arXiv: https://arxiv.org/abs/2510.25943 · HTML: https://mlsanity.com/paper/2510.25943/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 89 references that are in the map.
- First posted on arXiv 29 Oct 2025.

## Builds on (most cited first, 23 of 23)

- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [The Platonic Representation Hypothesis](https://mlsanity.com/paper/2405.07987/) (2024)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Generalized Shape Metrics on Neural Representations](https://mlsanity.com/paper/2110.14739/) (2021)
- [Universality and individuality in neural dynamics across large populations of recurrent networks](https://mlsanity.com/paper/1907.08549/) (2019)
- [Generalizing Koopman Theory to allow for inputs and control](https://mlsanity.com/paper/1602.07647/) (2016)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis](https://mlsanity.com/paper/2306.10168/) (2023)
- [The Topology and Geometry of Neural Representations](https://mlsanity.com/paper/2309.11028/) (2023)
- [PyDMD: A Python package for robust dynamic mode decomposition](https://mlsanity.com/paper/2402.07463/) (2024)
- [Interpretable statistical representations of neural population dynamics and geometry](https://mlsanity.com/paper/2304.03376/) (2023)
- [An overview of Koopman-based control: From error bounds to closed-loop guarantees](https://mlsanity.com/paper/2509.02839/) (2025)
- [Identifying Equivalent Training Dynamics](https://mlsanity.com/paper/2302.09160/) (2023)
- [Meta-Dynamical State Space Models for Integrative Neural Data Analysis](https://mlsanity.com/paper/2410.05454/) (2024)
- [Analyzing Populations of Neural Networks via Dynamical Model Embedding](https://mlsanity.com/paper/2302.14078/) (2023)
- [Delay Embedding Theory of Neural Sequence Models](https://mlsanity.com/paper/2406.11993/) (2024)
- [Comparing noisy neural population dynamics using optimal transport distances](https://mlsanity.com/paper/2412.14421/) (2024)
- [DFORM: Diffeomorphic vector field alignment for assessing dynamics across learned models](https://mlsanity.com/paper/2402.09735/) (2024)
- [Control-Coherent Koopman Modeling: A Physical Modeling Approach](https://mlsanity.com/paper/2403.16306/) (2024)
- [Characterizing control between interacting subsystems with deep Jacobian estimation](https://mlsanity.com/paper/2507.01946/) (2025)

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

- [Block-Recurrent Dynamics in Vision Transformers](https://mlsanity.com/paper/2512.19941/) (2025)
- [Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifolds](https://mlsanity.com/paper/2603.28764/) (2026)
- [Active Electrosensing and Communication in MARL-trained Weakly Electric Fish Collectives](https://mlsanity.com/paper/2511.08436/) (2025)
- [Fast dynamical similarity analysis](https://mlsanity.com/paper/2511.22828/) (2025)
- [Interpreting Reinforcement Learning Model Behavior via Koopman with Control](https://mlsanity.com/paper/2603.19968/) (2026)
- [Analyzing the Dynamics of WSBM-based Echo State Networks using InputDSA](https://mlsanity.com/paper/2610.15380/) (2026)

## Abstract (arXiv)

In control problems and basic scientific modeling, it is important to compare observations with dynamical simulations. For example, comparing two neural systems can shed light on the nature of emergent computations in the brain and deep neural networks. Recently, Ostrow et al. (2023) introduced Dynamical Similarity Analysis (DSA), a method to measure the similarity of two systems based on their recurrent dynamics rather than geometry or topology. However, DSA does not consider how inputs affect the dynamics, meaning that two similar systems, if driven differently, may be classified as different. Because real-world dynamical systems are rarely autonomous, it is important to account for the effects of input drive. To this end, we introduce a novel metric for comparing both intrinsic (recurrent) and input-driven dynamics, called InputDSA (iDSA). InputDSA extends the DSA framework by estimating and comparing both input and intrinsic dynamic operators using a variant of Dynamic Mode Decomposition with control (DMDc) based on subspace identification. We demonstrate that InputDSA can successfully compare partially observed, input-driven systems from noisy data. We show that when the true inputs are unknown, surrogate inputs can be substituted without a major deterioration in similarity estimates. We apply InputDSA on Recurrent Neural Networks (RNNs) trained with Deep Reinforcement Learning, identifying that high-performing networks are dynamically similar to one another, while low-performing networks are more diverse. Lastly, we apply InputDSA to neural data recorded from rats performing a cognitive task, demonstrating that it identifies a transition from input-driven evidence accumulation to intrinsically-driven decision-making. Our work demonstrates that InputDSA is a robust and efficient method for comparing intrinsic dynamics and the effect of external input on dynamical systems.