10/⚙️Loop Closure 🔁
We also ensure integrals around closed loops are zero. This forces the model to learn the tangent space dynamics over the whole data manifold, not just along the flow. Why? To control a system, you have to know what happens off its normal path.

9/⚙️Time-series Prediction ⛰️
The path integral of Jacobians depends on endpoints, not the path. Think of a mountain peak: your elevation is the same regardless of the trail taken. We parameterize the Jacobian with a deep network, and use this insight for time-series prediction.

8/⚡Controlling neural dynamics
We also used our framework to actively control the network based purely on observed data. By stimulating the sensory area in a targeted way, we precisely manipulated the RNN's behavior and forced it to make a specific incorrect choice.

7/🎛️ Control between areas
We applied our framework to a simplified model of interacting brain areas: a multi-area recurrent neural network (RNN) trained on a working memory task. After learning the task, its "sensory" area gained control over its "cognitive" area.

6/🎯 In rigorous tests, JacobianODEs accurately predicted dynamics and outperformed NeuralODEs on Jacobian estimation, even in noisy, high-dimensional chaotic systems. Accurate control starts with accurate Jacobians, so this was an important check.

Now what can we do with it?👇

5/🔎 Estimating the Jacobian from data is difficult. To do so, we developed JacobianODE, a deep learning framework that leverages geometric properties of the Jacobian to infer it from data.

Scroll down the thread to learn how it works. For now, does it work?

How do brain areas control each other? 🧠🎛️

✨In our NeurIPS 2025 Spotlight paper, we introduce a data-driven framework to answer this question using deep learning, nonlinear control, and differential geometry.🧵⬇️