Research map / Probabilistic and scientific modeling / Dynamical systems
Characterizing control between interacting subsystems with deep Jacobian estimation
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes JacobianODE, a deep-learning method that estimates Jacobians of dynamical systems directly from trajectories via path integration, teacher forcing, and loop-closure loss, and uses the learned Jacobians for nonlinear control analysis and control of a task-trained multi-area RNN.
Problem
Estimating Jacobians of nonlinear dynamical systems from time-series data so that control-theoretic interactions between subsystems (e.g., brain areas) can be characterized and used for control.
Approach
Learn the Jacobian network directly and train it by path-integrating the Jacobian to reconstruct trajectories (f(x) expressed through J without knowing f), with a self-supervised loop-closure loss enforcing that Jacobian rows are conservative vector fields; then compute interareal reachability Gramians and ILQR control from the learned Jacobians.
Key results
- JacobianODE achieved lower mean Frobenius Jacobian error than NeuralODE and weighted linear regression on virtually all systems; e.g., Lorenz 96 (64 dim) at 1% noise 30.9 vs 45.5 (NeuralODE) and 66.39 (weighted linear).
- On the task-trained RNN, JacobianODE Jacobian error was ~188 vs ~294 for NeuralODE, while trajectory MSE was comparable between the two.
- JacobianODE-driven ILQR control of the RNN widely outperformed baselines, achieving accuracy near the ground-truth Jacobian system.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use JacobianODE instead of NeuralODE | Jacobian estimation on VanDerPol (2 dim), 5% noise · VanDerPol (2 dim), 5% noise | mean Frobenius norm error: 0.72 ± 0.08 -> 0.72 ± 0.05 | No clear effect |
| Use JacobianODE instead of Weighted Linear | Jacobian estimation on Lorenz (3 dim), 5% noise · Lorenz (3 dim), 5% noise | mean Frobenius norm error: 21.90 -> 5.1 ± 0.9 | Helped |
| Use JacobianODE instead of NeuralODE | Jacobian estimation on Lorenz 96 (12 dim), 5% noise · Lorenz 96 (12 dim), 5% noise | mean Frobenius norm error: 5.9 ± 0.2 -> 2.7 ± 0.2 | Helped |
| Use JacobianODE instead of NeuralODE | Jacobian estimation on Lorenz 96 (32 dim), 5% noise · Lorenz 96 (32 dim), 5% noise | mean Frobenius norm error: 17.7 ± 0.6 -> 7.8 ± 0.2 | Helped |
| Use JacobianODE instead of NeuralODE | Jacobian estimation on Lorenz 96 (64 dim), 5% noise · Lorenz 96 (64 dim), 5% noise | mean Frobenius norm error: 45.7 ± 0.1 -> 34.0 ± 0.2 | Helped |
| Use JacobianODE instead of NeuralODE | Jacobian estimation on task-trained RNN, 5% noise · Task-trained RNN, 5% noise | mean Frobenius norm error: 294.357 ± 0.003 -> 166.8 ± 3.6 | Helped |
| Add loop closure loss (ablation) | JacobianODE · Dynamical systems | Jacobian estimation accuracy | Helped |
Surprising findings
- Adding a small amount of training noise (5%) improved Jacobian estimation for multiple systems.
- Good trajectory reconstruction did not translate into accurate Jacobians: NeuralODEs reproduce dynamics but learn Jacobians insufficient for control.
Limitations
- Only fully observed, moderately high-dimensional systems tested; scalability to thousands of neurons unknown.
- Reachability estimates are local to a reference trajectory and sensitive to alignment and eigenstructure; fine time-point comparisons should be cautious.
- Partially observed dynamics not addressed.
Experimental setup
Datasets: Van der Pol oscillator (dysts) Lorenz system (dysts) Lorenz 96 (12, 32, 64 dims) working memory selection task RNN trajectories
Benchmarks: Van der Pol Lorenz Lorenz 96 (12, 32, 64 dim) task-trained RNN
Metrics: mean Frobenius norm Jacobian error spectral 2-norm Jacobian error trajectory MSE R^2 Lyapunov spectrum ILQR control accuracy reachability Gramian trace and minimum eigenvalue
Baselines: NeuralODE weighted linear regression Jacobian estimation ground truth Jacobians
Builds on: Neural ODE generalized teacher forcing conservative vector field Jacobian regularization iterative linear quadratic regulator
Closest papers by meaning
- Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems (NeurIPS 2024 · similarity 0.86)
- Recurrent Self-Attention Dynamics: An Energy-Agnostic Perspective from Jacobians (NeurIPS 2025 · similarity 0.86)
- Identifying interactions across brain areas while accounting for individual-neuron dynamics with a Transformer-based variational autoencoder (NeurIPS 2025 · similarity 0.85)
- Let's do the time-warp-attend: Learning topological invariants of dynamical systems (ICLR 2024 · similarity 0.85)
- Partial observation can induce mechanistic mismatches in data-constrained models of neural dynamics (NeurIPS 2024 · similarity 0.85)
- Feedback control guides credit assignment in recurrent neural networks (NeurIPS 2024 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.