# Characterizing control between interacting subsystems with deep Jacobian estimation

arXiv: https://arxiv.org/abs/2507.01946 · HTML: https://mlsanity.com/paper/2507.01946/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 145 references that are in the map.
- First posted on arXiv 2 Jul 2025.

## Builds on (most cited first, 26 of 26)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Robust Learning with Jacobian Regularization](https://mlsanity.com/paper/1908.02729/) (2019)
- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [Generalized Teacher Forcing for Learning Chaotic Dynamics](https://mlsanity.com/paper/2306.04406/) (2023)
- [Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design](https://mlsanity.com/paper/1503.03144/) (2015)
- [Model scale versus domain knowledge in statistical forecasting of chaotic systems](https://mlsanity.com/paper/2303.08011/) (2023)
- [Discovering Sparse Interpretable Dynamics from Partial Observations](https://mlsanity.com/paper/2107.10879/) (2021)
- [Learning Latent Dynamics for Partially-Observed Chaotic Systems](https://mlsanity.com/paper/1907.02452/) (2019)
- [Beyond Convexity -- Contraction and Global Convergence of Gradient Descent](https://mlsanity.com/paper/1806.06655/) (2018)
- [Neural Contractive Dynamical Systems](https://mlsanity.com/paper/2401.09352/) (2024)
- [Controllability of Brain Networks](https://mlsanity.com/paper/1406.5197/) (2014)
- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [Using Machine Learning to Assess Short Term Causal Dependence and Infer Network Links](https://mlsanity.com/paper/1912.02721/) (2019)
- [Deep learning delay coordinate dynamics for chaotic attractors from partial observable data](https://mlsanity.com/paper/2211.11061/) (2022)
- [Reconstruction, forecasting, and stability of chaotic dynamics from partial data](https://mlsanity.com/paper/2305.15111/) (2023)
- [RNNs of RNNs: Recursive Construction of Stable Assemblies of Recurrent Neural Networks](https://mlsanity.com/paper/2106.08928/) (2021)
- [JacNet: Learning Functions with Structured Jacobians](https://mlsanity.com/paper/2408.13237/) (2024)
- [Empirical Differential Gramians for Nonlinear Model Reduction](https://mlsanity.com/paper/1902.09836/) (2019)
- [Attention for Causal Relationship Discovery from Biological Neural Dynamics](https://mlsanity.com/paper/2311.06928/) (2023)
- [Observability and Controllability of Nonlinear Networks: The Role of Symmetry](https://mlsanity.com/paper/1307.5478/) (2013)
- [Estimating the Jacobian matrix of an unknown multivariate function from sample values by means of a neural network](https://mlsanity.com/paper/2204.00523/) (2022)
- [Inferring the time-varying coupling of dynamical systems with temporal convolutional autoencoders](https://mlsanity.com/paper/2406.03212/) (2024)
- [Time-series attribution maps with regularized contrastive learning](https://mlsanity.com/paper/2502.12977/) (2025)
- [Input Novelty as a Control Metric for Time Varying Linear Systems](https://mlsanity.com/paper/1411.5892/) (2014)
- [Minimum energy control for complex networks](https://mlsanity.com/paper/1612.03082/) (2016)

## Built on by (most cited first, 2 of 2)

- [InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamics](https://mlsanity.com/paper/2510.25943/) (2025)
- [Watch your neighbors: Training statistically accurate chaotic systems with local phase space information](https://mlsanity.com/paper/2605.14405/) (2026)

## Abstract (arXiv)

Biological function arises through the dynamical interactions of multiple subsystems, including those between brain areas, within gene regulatory networks, and more. A common approach to understanding these systems is to model the dynamics of each subsystem and characterize communication between them. An alternative approach is through the lens of control theory: how the subsystems control one another. This approach involves inferring the directionality, strength, and contextual modulation of control between subsystems. However, methods for understanding subsystem control are typically linear and cannot adequately describe the rich contextual effects enabled by nonlinear complex systems. To bridge this gap, we devise a data-driven nonlinear control-theoretic framework to characterize subsystem interactions via the Jacobian of the dynamics. We address the challenge of learning Jacobians from time-series data by proposing the JacobianODE, a deep learning method that leverages properties of the Jacobian to directly estimate it for arbitrary dynamical systems from data alone. We show that JacobianODEs outperform existing Jacobian estimation methods on challenging systems, including high-dimensional chaos. Applying our approach to a multi-area recurrent neural network (RNN) trained on a working memory selection task, we show that the "sensory" area gains greater control over the "cognitive" area over learning. Furthermore, we leverage the JacobianODE to directly control the trained RNN, enabling precise manipulation of its behavior. Our work lays the foundation for a theoretically grounded and data-driven understanding of interactions among biological subsystems.