Title: Characterizing control between interacting subsystems with deep Jacobian estimation
Venue: NeurIPS 2025
Accepted: yes
Keywords: control theory

dynamical systems

neuroscience

deep learning

modular systems
Abstract: 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 JacobianODE models 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.

Characterizing control between interacting subsystems with deep Jacobian estimation

Abstract

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 JacobianODE models 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.

1 Introduction

Complex systems are ubiquitous in nature. These systems exhibit a wide range of behavior and function, in large part through the dynamic interaction of multiple component subsystems within them. One approach to understanding such complex systems is to build detailed models of their underlying dynamics. An alternative and simpler yet powerful approach is offered by control theory, focusing instead on how subsystems influence and regulate one another, and how they can be controlled.

Control theory thus offers a complementary approach to both understanding and manipulating biological systems. The theory describes how inputs must be coordinated with system dynamics to achieve desired behaviors, and can be applied across domains ranging from robotics to biology (Figure 1A). The brain coordinates neural activity across multiple interconnected brain areas, dynamically modulating which regions receive information from which others depending on need and context [1, 2]. Interareal interactions play central roles in cognition and consciousness [3-7], in selective attention [8-16], decision making [17, 18], working memory [19, 20], feature binding [21, 22], motor control [23-27], and learning and memory [28-32].

A common approach to characterizing interareal interactions is to quantify communication between them, using methods such as reduced-rank regression to define 'communication subspaces'. These subspaces determine low-dimensional projections that maximally align high-dimensional states of the input area with high-dimensional states of the target area [33-35]. However, effective control not only involves alignment of high-variance input states with high-variance target states, but also appropriate alignment of the inputs with the dynamics of the target area. Given connected subsystems A and B, an identical signal from B will have dramatically different control effects on A, depending on whether the signal aligns with stable or unstable directions of A's dynamics: projections onto more unstable eigenvectors can much more readily drive the system to novel states. (For more detail see Appendix C.1.)

Accordingly, recent work in neuroscience has espoused control-theoretic perspectives on interareal interactions (Figure 1B) [24, 36-42]. The dominant approach has involved linear control [43-53]. However, the inherently nonlinear dynamics of the brain enable richer contextual control than possible to fully model with linear systems, necessitating a nonlinear control approach.

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One approach to extend control-theoretic analyses to nonlinear systems is by linearizing the nonlinear dynamics through Taylor expansion, which involves the Jacobian. This converts the nonlinear system into a linear state- and time-dependent one [40, 54-56], allowing for simple control. Jacobian linearization for control is straightforward with access to analytical expressions for the non-linear system, but it becomes non-trivial in purely data-driven scenarios. Estimating the Jacobian involves conjunctively inferring both a function and its derivative, yet good function approximation need not yield good approximations of its derivatives (see Section 4 and Appendix C.2).

This paper introduces several key contributions:

Robust data-driven Jacobian estimation. We present JacobianODE, a deep learning-based method for estimating Jacobians from noisy trajectories in high-dimensional dynamical systems. We demonstrate the validity of this approach in several sample systems that include high-dimensional chaos.

A framework to characterize control between interacting subsystems via data-driven Jacobian estimation. Harnessing our Jacobian estimation method, we devise a rigorous,

data-driven approach for nonlinear control-theoretic analysis of how paired interacting systems, including brain areas, drive and regulate each other across different contexts.

Data-driven inference of control dynamics in trained recurrent neural networks. We apply our data-driven framework to a recurrent neural network (RNN) trained on a working memory task. We show that, purely from data, we can identify key control-theoretic interactions between the areas, and that these interactions crucially evolve over the course of learning to produce the desired behavior.

Demonstration of accurate control of rich interacting high-dimensional coupled dynamical subsystems. We demonstrate high accuracy in a challenging high-dimensional data-driven nonlinear control task, enabling precise control of the behavior of the RNN.

Overall, our work lays the foundation for data-driven control-theoretic analyses in complex highdimensional nonlinear systems, including the brain.

2 Related work

Interareal communication A wide range of tools have been developed and harnessed to study interareal communication in neural data [34, 57]. This includes, but is not limited to, methods based on reduced rank regression [33, 35], recurrent neural network models of neural dynamics [40, 58-63], Gaussian process factor analysis [64, 65], canonical correlation analysis [66-68], convergent cross mapping [69-71], switching dynamical systems [72, 73], granger causality [74, 75], dynamic causal mapping [76], point process models [77], and machine learning methods [78, 79].

Nonlinear controllability Classical results assess the controllability of nonlinear control systems via the Lie theory [80-84]. Another approach to nonlinear network controllability is based on attractor strength [85]. Liu et al. [54] and Parkes et al. [55] note that the large literature of linear controllability analyses could be extended locally to nonlinear systems with an appropriate linearization method.

Neural network controllability Linear structural network controllability has been implicated across a wide range of contexts, tasks, and neuropsychiatric conditions [36, 42, 44-47, 49, 50]. This approach has been extended to functional brain networks [51-53]. Recent work has characterized the subspaces of neural activity that are most feedback controllable as opposed to feedforward controllable using linear methods [43]. Other approaches to neural control analyses consider the identification of structural driver nodes [86], and input novelty [87].

Data-driven Jacobian estimation Besides approaches estimating Jacobians through weighted linear regressions [88, 89], some methods have used direct parameterization via neural networks to learn Jacobians of general functions [90, 91]. These approaches inform our method but do not explicitly address dynamical systems. Applying path-integral-based Jacobian estimation to dynamical systems is challenging, as the target function (the system's time derivative) is typically unobserved. Beik-Mohammadi et al. [92] utilized this idea in dynamical systems to learn contracting latent dynamics from demonstrations.

3 JacobianODE: learning Jacobians from data

3.1 Jacobian linearization

We consider nonlinear dynamical systems in R n , defined by ˙ x ( t ) = f ( x ( t )) . The Jacobian of the dynamics is a matrix-valued function J f : R n → R n × n (henceforth, J ) given by

At each time t , the Jacobian defines a linear subspace relating input and output changes, capturing how perturbations to the system will propagate. This recasts nonlinear dynamics as linear time-varying dynamics in the tangent space locally along trajectories (formally, δ ˙ x ( t ) = J f ( x ( t )) δ x ( t ) , see Figure 2 left, also see Lohmiller and Slotine [93] for a discussion in the context of virtual displacements).

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+αx(t + △t)

3.2 Parameterizing differential equations via the Jacobian

We now turn to the problem of how to estimate the Jacobian J from data. We assume that we have access only to observed trajectories of the system, of the form x ( j ) ( t ( j ) 0 + k ∆ t ) , k = 1 , 2 , ..., j = 1 , 2 , ... where ∆ t is a fixed sampling interval, j indexes the trajectory, and t ( j ) 0 is a trajectory-specific start time. Crucially, we do not assume access to the function f . Our method estimates the Jacobian directly via a neural network. To do this, we parameterize a neural network function ˆ J θ with learnable parameters θ that is then trained to approximate J .

Path integration Following Lorraine and Hossain [91] and Beik-Mohammadi et al. [92], we exploit the fact that the path integral of the Jacobian is path independent in the construction of the loss function. This is because the rows of the Jacobian are conservative vector fields (i.e., they are the

gradients of scalar functions). For an intuitive picture, consider the work done by the force of gravity as you climb a mountain. Regardless of the path you take to climb, the resulting work is dependent only on the start and end points of the path. Formally, for the time derivative function f we have that

where C is a piecewise smooth curve in R n and c : [ t i , t f ] → C is a parameterization of C with c ( t i ) = x ( t i ) and c ( t f ) = x ( t f ) (Figure 3A). Given estimates of f ( x ( t i )) and the Jacobian J , we can then use Equation 2 to approximate f ( x ( t f )) at any time t f . For these integrals, we use a line between the endpoints as a simple choice of path (Figure 3A). Then, to generate an estimate ˆ x ( t f +∆ t ) of the next step, we can use a standard ordinary differential equation (ODE) integrator (e.g., Euler, fourth-order Runge-Kutta, etc.) to integrate the estimated time derivative ˆ f ( x ( t )) (see Appendix A for more detail). To avoid the need to represent f directly (thereby enabling all gradients to backpropagate through the Jacobian network) we note that we parameterize an estimate of f ( x ( t i )) in Equation 2 in terms of the Jacobian (see Appendix A.1).

3.3 Loss functions

Trajectory reconstruction loss Given an observed trajectory x ( t 0 + k ∆ t ) , k = 0 , ..., T -1 of length T , we can compute the trajectory reconstruction loss, L traj ( θ ; x ) , between the true trajectory and the estimated trajectory using an appropriate distance measure d (e.g., mean squared error, see Appendix A.2 for more detail on generating predictions and the trajectory prediction loss).

Generalized Teacher Forcing To avoid trajectory divergence in chaotic or noisy systems, we employ Generalized Teacher Forcing, generating recursive predictions partially guided by true states (Figure 3B, and Appendix D.8.1) [94].

Loop closure loss The Jacobian captures how perturbations to the system will propagate along any direction in state space. Estimating it purely from dynamics constrains only the direction of the flow, leaving the full solution underdetermined. To address this, we again exploit the fact that each row of the Jacobian is a conservative vector field. Specifically, we note that for any piecewise smooth loop C loop, we have ∥ ∥ ∥ ∮ C loop J ds ∥ ∥ ∥ 2 = 0 (see Figure 3C). Thus, by integrating along loops that contain directions orthogonal to the system's dynamics (and penalizing the deviation from zero), we encourage the estimated Jacobians to capture information about other directions in state space (see Appendices A.3 and D.8.2 for full technical details). To ensure broad coverage of tangent space directions, we form loops from concatenations of line integrals between randomly selected data points. This strategy samples diverse directions from the tangent space while remaining easy to compute. The resulting self-supervised loss term, L loop ( θ ; x ) , builds on the loss introduced by Iyer et al. [95]. It constrains ˆ J θ to satisfy both the dynamics and conservativity. This improves Jacobian estimation accuracy significantly (see Appendix C.4 for ablation studies).

Training loss We therefore minimize the following loss function with respect to the parameters θ :

where λ loop controls the relative weighting of the loop closure loss L loop ( θ ; x ) compared to the trajectory prediction, and is a hyperparameter of the learning procedure (Figure 3D).

4 Jacobian estimation in dynamical systems

Data To evaluate the quality of the Jacobian estimation procedure, we apply our approach to several example systems for which the dynamics are known. For this analysis, we used the Van der Pol oscillator [96], Lorenz system [97], and the Lorenz 96 system across three different system sizes (12, 32, and 64 dimensional) [98]. All systems were simulated using the dysts package, which samples dynamical systems with respect to the characteristic timescale τ of their Fourier spectrum [99, 100]. All training data consisted of 26 trajectories of 12 periods, sampled at 100 time steps per τ . All models were trained on a 10 time-step prediction task with teacher forcing.

To evaluate the performance of the methods in the presence of noise, we trained the models on data with 1%, 5%, and 10% Gaussian observation noise added i.i.d over time, where the percentage is defined via the ratio of the euclidean norm of the noise to the mean euclidean norm of the data.

JacobianODE model For the JacobianODE framework, loop closure loss weights were chosen via line search (Appendix D.8.8), where the neural network J θ was taken to be a four-layer multilayer perceptron (MLP) with hidden layer sizes 256, 1024, 2048 and 2048. Path integration was performed using the trapezoid method from the torchquad package, with each integral discretized into 20 steps [101]. ODE integration was performed using the fourth-order Runge-Kutta (RK4) method from the torchdiffeq package [102]. The JacobianODE models used 15 observed points to generate the initial estimate of f (see Section 3.2 and Appendices A.1 and A.2). All models were built in PyTorch [103]. Full implementation details are provided in appendices A and D.

Baselines We chose two different Jacobian estimation procedures for comparison. The first was a neural ordinary differential equation (NeuralODE) model trained to reproduce the dynamics [102]. The NeuralODE was implemented as a four-layer MLP with hidden layers of the same size as the one used for the JacobianODE model. Jacobians were computed via automatic differentiation. NeuralODEs were regularized via a penalty on the Frobenius norm of the estimated Jacobians to prevent the model from learning unnecessarily large negative eigenvalues (see Appendix D.3) [104106]. We also employed a baseline that estimates Jacobian via a weighted linear regression, which computes locally linear models at each point in the space [88, 89] (see Appendix D.4).

Performance We tested the approaches on Jacobian estimation on held-out trajectories without noise. The JacobianODE method outperforms the baseline methods in terms of mean Frobenius norm error for virtually all systems (Table 1). This was also true when considering the spectral matrix 2-norm (see Table S3 in Appendix C.3).

We plot performance in Figure 4. While both the JacobianODEs and the NeuralODEs reproduce the observed dynamics (Figure 4A-D), the JacobianODE learns a more accurate estimate of the Jacobian (Figure 4 E,F). In particular, looking at Lyapunov spectra learned by the models, we note that the JacobianODE exceeds the other methods in estimating the full Lyapunov exponent spectrum, indicating a better overall representation of how perturbations along different directions will evolve (Figure 4G,H).

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5 Control-theoretic analyses in a task-trained RNN

5.1 Characterizing control between subsystems with Jacobian linearization

Consider neural data recorded from two areas, A and B. Concatenating their data into a state vector x ∈ R n , composed of x A ∈ R n A and x B ∈ R n B , and assuming dynamics governed by f , we linearize around a reference trajectory ( δ ˙ x ( t ) = J ( x ( t )) δ x ( t ) ). Splitting the Jacobian into block matrices yields:

where diagonal blocks J A → A ∈ R n A × n A and J B → B ∈ R n B × n B represent within-area dynamics, and off-diagonal blocks J B → A ∈ R n A × n B , J A → B ∈ R n B × n A represent interareal interactions (Figure 2, right). Explicit separation of each area's control dynamics quantifies the direct influence each exerts on the other, and readily generalizes beyond two areas (see Appendix B.3).

Since Jacobians are time-dependent, we obtain a linear time-varying representation of control dynamics along the trajectory. This enables computation of time-varying reachability ease, capturing how readily each area drives the other toward novel states [107-109]. Reachability is quantified via the reachability Gramian, a matrix defining a local metric in tangent space. For the above control system capturing the influence of area B on area A, the time-varying reachability Gramian on the interval [ t 0 , t 1 ] is defined as

where Φ (computed from J A → A ) denotes the state-transition matrix of the intrinsic dynamics of subsystem A without any input (i.e., δ x A ( t ) = Φ ( t, t 0 ) δ x A ( t 0 ) ), and B ( τ ) = J B → A ( x ( τ )) (see Appendix B.1) [107]. The Gramian W r ( t 0 , t 1 ) is symmetric and positive semidefinite for every t 1 > t 0 [107]. Each eigenvalue of the reachability Gramian quantifies how easily the target system can be driven along its corresponding eigenvector. Thus, the trace of the reachability Gramian reflects average ease of reaching new states, and its minimum eigenvalue reflects the ease along the most challenging direction of control.

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5.2 Estimating the Jacobian of a task-trained RNN

Task To demonstrate how JacobianODE models could be used in neuroscience, we performed a control-theoretic analysis of a task-trained RNN. We used a working memory selection task from Panichello and Buschman [110] (Figure 5A). On each trial, the network is presented with two of four possible "colors", denoted by one-hot vectors. After a delay, the network is presented with a cue indicating which of the colors to select. The network is then asked to reach a state of sustained activation that corresponds to the selected color.

RNN model To perform this task, we trained a 128-dimensional continuous-time RNN. The RNN had hidden dynamics defined by

where W hh defines the internal dynamics, W hi maps the input into the hidden state, W oh maps the hidden state to a four-dimensional output o ( t ) , b is a bias term, and σ is the exponential linear unit activation. The RNN had two 64-neuron areas: a "visual" area (which received sensory input) and a "cognitive" (which output the RNN's color choice) (Figure 5B). To encourage multi-area structure, we initialized the within-area weights with greater connectivity strength than the across-area weights (Figure 5C). Since input comes only to the visual area and output only from the cognitive area, the two areas are forced to interact to solve the task (Figure 5D,E). The RNN solves this task with 100% accuracy.

Jacobian reconstruction quality We trained JacobianODEs on RNN trajectories from the post-cue delay and response portion of the trials. We used the same baselines as in Section 4. JacobianODEs are not only robust to noise, but can also benefit from it (for multiple systems in Table 1, 5% training noise improves estimation). Noise encourages the model to explore how perturbations around the observed trajectory evolve, which is crucial for learning accurate Jacobians in high-dimensional systems. Thus (for both the JacobianODE and NeuralODE) we add a small amount of additional noise to the data during learning. Although the models perform similarly on trajectory reconstruction (Figure 5F,G), on Jacobian estimation, JacobianODEs drastically outperform both baseline models (Figure 5H,I, and Table 1).

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5.3 Reachability in the task-trained RNN across contexts

Next, we used the Jacobians learned by the JacobianODE (trained on 5% noise) to evaluate reachability control in the RNN, and compared it to evaluations using ground truth Jacobians. All analyses were performed using the 10 time-step reachability Gramian. We first found that it was on average easier for the visual area to drive the cognitive area, both when considering overall ease (Gramian trace) and worst-case ease (Gramian minimum eigenvalue) (Figure 6A, larger values indicate greater reachability). We next considered how reachability ease varied throughout the delay period. We found that the visual area could drive the cognitive area more easily at the beginning of the delay, with ease decreasing into the middle of the delay period (Figure 6B, bottom). The cognitive area was able to drive the visual area more easily slightly later in the delay period (Figure 6B, top). Finally, we considered whether reachability changes over the course of learning. We found that both directions of reachability increased after learning (Figure 6C). The JacobianODE reproduced all results accurately (Figure 6A-C, comparison with ground truth). Our analysis reveals that reachability is crucial in the RNN's ability to perform the working memory task, with the visual area's ability to drive the cognitive area shortly after the cue being especially important.

5.4 Controlling the task-trained RNN

To further validate our approach, we used JacobianODE-learned Jacobians to control the task-trained RNN (Figure 6D). Given a trial in which the network was cued to respond with a particular color, we tested if we could induce a specific incorrect response by input to the visual area alone. To do so, we implemented Iterative Linear Quadratic Regulator (ILQR) control, which relies on knowledge of the Jacobian [111, 112]. The controller guided the network towards the mean hidden state of

training trials corresponding to the desired incorrect color. We defined accuracy as the percentage of time points during which the RNN output the desired color. We found that the JacobianODE widely outperformed the baseline models on this task, achieving an accuracy nearing that of the ground truth system (Figure 6E). The JacobianODE was furthermore able to achieve the lowest mean squared error on the desired control trajectory (Figure 6F). This illustrates that while both the JacobianODE and the NeuralODE can learn the dynamics, only the JacobianODE learns a representation of the Jacobian that is sufficient for control.

6 Discussion

Extended Jacobian estimation A natural extension of JacobianODE models would add inductive biases for particular classes of dynamics. For example, one could parameterize a negative definite matrix and thus ensure contracting dynamics [92]. Other extensions could include an L1 penalty to encourage sparsity, as well as the inclusion of known dynamic structure (e.g., hierarchical structure, low-rank structure, etc.). In neuroscience, connectomic constraints could be incorporated to capture interactions within a neural circuit.

Limitations A future challenge for JacobianODE models is partially observed dynamics. Recent work has identified that it is possible to learn latent embeddings that approximately recover the true state from partial observation [113-117]. Jacobian-based dynamics learning has been performed in a latent space [92, 118], however it is unclear whether this translates to accurate Jacobian estimation, given the challenges related to automatic differentiation and Jacobian estimation presented here. We also note that reachability estimates depend sensitively on several factors: the alignment of cross-subsystem interactions and within-subsystem dynamics, the eigenvectors and eigenvalues of within-subsystem Jacobians, and the way activity propagates within each subsystem (see Appendix C.1). While our method reliably captures broad trends in reachability over time, fine-grained, timepoint-specific comparisons should be interpreted with caution. Finally, although JacobianODE models scale well to moderately high-dimensional systems, their performance in systems that are orders of magnitude larger than those considered here (e.g., recordings of thousands of neurons via calcium imaging) remains to be tested. In many practical settings, this may not be a limitation: neural representations during tasks often exhibit intrinsic dimensionalities comparable to those studied here, enabling JacobianODE models to operate in reduced dimensionality (see Appendix C.7). Notably, most models converged in under 45 minutes on a single GPU, suggesting favorable scaling (see Table S7). Future work should explore the viability of the method in higher dimensions, and assess whether dimensionality reduction strategies (such as latent state models or low-rank Jacobian approximations) can further improve scalability and accuracy.

Broader impact on dynamical systems Wedemonstrated in this work that it is possible to learn high accuracy Jacobians of arbitrary dynamical systems from data. While in this work we focus on controltheoretic analyses, we emphasize that JacobianODE models can be trained in any dynamical setting. Jacobians are widely applicable, and are essential in stability analysis [80, 119-123], contraction theory [40, 80, 93, 124-126], and nonlinear control [56, 111, 112, 127-129], among many other settings.

Broader impact on control-theoretic analyses The control-theoretic analysis presented here is performed with respect to a particular reference trajectory. It therefore describes the local controllability properties along the reference trajectory, rather than global properties for the overall system [56]. This locality is desirable in biological settings due to the context-sensitive nature of the relevant processes. Applications of this approach could involve a comparison of the functioning of the autonomic nervous system, or gene regulatory networks, between control and disease states, given the association of these processes with disease [130-138]. Future work should investigate whether a more global Jacobian-based control analysis (e.g., contraction theory) is achievable in a data-driven manner via JacobianODE models. This would be useful especially in engineering settings where global controllability is desirable, such as robotics and prosthetic limbs.

Acknowledgments and Disclosure of Funding

The authors would like to thank Andrew Kirjner, Laureline Logiaco, Federico Claudi, Lakshmi Narasimhan Govindarajan, Jaedong Hwang, and Akhilan Boopathy for providing stimulating discussion about this work. The authors also acknowledge the MIT Office of Research Computing and Data

for providing high-performance computing resources that have contributed to the research results reported within this paper. I.R.F. acknowledges funding from The National Science Foundation Computer and Information Science and Engineering Directorate, The Simons Collaboration on the Global Brain, and The McGovern Institute at MIT. E.K.M. acknowledges funding from ONR MURI N00014-23-1-2768, NIMH 1R01MH131715-01, NIMH R01MH11559, The Simons Center for the Social Brain, The Freedom Together Foundation, and The Picower Institute for Learning and Memory. L.K. acknowledges funding from the Goldstine Postdoctoral Fellowship at IBM Research. M.O. acknowledges funding from NSF GRFP 2141064.

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A JacobianODE technical details

A.1 Parameterizing the time derivative via the Jacobian

Recall that, for the time derivative function f , we have that the path integral is independent of the choice of path:

where C is a piecewise smooth curve in R n and c : [ t i , t f ] → C is a parameterization of C with c ( t i ) = x ( t i ) and c ( t f ) = x ( t f ) (also see equation 2). Path integration notably provides only differences between f at distinct points. Thus, knowing the value of f at one point is needed for trajectory reconstruction, which we do not assume. To address this, we note it is possible to represent f solely through the Jacobian at a point x ( t ) as

and we have abbreviated c s,t ( r ) = c ( r ; s, t, x ( s ) , x ( t )) , a piecewise smooth curve on [ s, t ] parameterized by r and beginning and ending at x ( s ) and x ( t ) respectively. Intuitively, we can circumvent the need to know f by recognizing that integrating f between time points will produce the difference in the system states at those time points, which are known to us. With this formalism, we avoid the need to represent f directly, thereby enabling all gradients to backpropagate through the Jacobian network. The proof of this formalism is presented below.

Proposition 1 (Jacobian-parameterized ODEs) . Let ˙ x ( t ) = f ( x ( t )) and let J f ( x ( t )) = J ( x ( t )) = ∂ ∂ x f ( x ( t )) . Then given times t 0 , t we can express f parameterized by the Jacobian as

and we have abbreviated c s,t ( r ) = c ( r ; s, t, x ( s ) , x ( t )) , a piecewise smooth curve on [ s, t ] parameterized by r and beginning and ending at x ( s ) and x ( t ) respectively.

Proof. For given times t , t 0 , and s , from the fundamental theorem of calculus, we have that

where where

and

Letting

We then have that

Letting now

We can see that and thus

A.2 Path integrating to generate predictions

Given an initial observed trajectory x ( t 0 + k ∆ t ) , k = 0 , . . . , b of length at least two ( b ≥ 1 ), we compute an estimate of ˆ f ( x ( t 0 + b ∆ t )) of f ( x ( t 0 + b ∆ t )) by replacing J with ˆ J θ in equation 8. Specifically, we compute an estimate of ˆ f ( x ( t b )) of f ( x ( t b )) as

In practice, we compute this estimate by constructing a cubic spline on the initial observed trajectory using 15 points (i.e., b = 14 ). Given that the computation of G involves a double integral - one over states and over time - using a spline is computationally advantageous. This is because the path integral (and all intermediate steps) can be quickly computed along the full spline, with the result of each intermediate step along the path then being summed to approximate the time integral. The integral is computed by interpolating 4 points for every gap between observed points, resulting in a discretization of 58 points along the spline. Integrals are computed using the trapezoid method from torchquad [101].

Once we have constructed our estimate of ˆ f ( x ( t b )) we can estimate f at any other point x ( t ) as

where by convention we integrate forwards in time and H is the path integral defined above, with ˆ J in place of J . In practice, for the integration path c ( r ; s, t, x ( s ) , x ( t )) , we construct a line from x ( s ) to x ( t ) as

to maintain the interpretability of having r in the range [ s, t ] however it can be easily seen that setting r ′ = r -s t -s we recover the familiar line

with r ′ taking values on [0 , 1] . Line integrals are computed with 20 discretization steps using the trapezoid method from torchquad [101].

Using ˆ f ( x ( t )) , we can then generate predictions as

where the integral can be computed by a standard ODE solver (we used the RK4 method from torchdiffeq with default values for the relative and absolute tolerance).

We can then compute the trajectory reconstruction loss as

where d is a distance measure between trajectories (e.g. mean squared error).

A.3 Loop closure loss

While the space of Jacobian functions ˆ J θ that solve the dynamics problem may be quite large, we are interested only in functions ˆ J θ that both solve the dynamics problem and generate matrices with rows that are conservative vector fields. To effectively restrict our optimization to this space, we employ a self-supervised loss building on the loss introduced by Iyer et al. [95]. This loss imposes a conservative constraint on the rows of the Jacobian matrix. Specifically, for any piecewise smooth curve C loop starting and ending at the same state x 0 , we have that ∥ ∥ ∥ ∫ C loop J ds ∥ ∥ ∥ 2 = 0 .

Thus, given n loops sets of any L (not necessarily sequential) points { x ( l ) ( t 1 ) , x ( l ) ( t 2 ) , ..., x ( l ) ( t L ) } we can form a loop C ( ( l ) ) loop consisting of the sequence lines from x ( l ) ( t i ) to x ( l ) ( t i +1 ) , i = 1 , ..., L -1 , followed by a line from x ( l ) ( t L ) to x ( l ) ( t 1 ) . We then define the following regularization loss to enforce this conservation constraint:

We note that while other choices for loops would have been possible (for instance, forward and backward passes along observed trajectories, or arbitrary circles beginning and ending at the same point), the approach presented here has several advantages (as discussed in Section 3.3). Namely, it balances computational tractability with uniform sampling of the directions in the tangent space. Enforcing that all the points that comprise the loop are on the data manifold ensures that the loops are integrated in directions that are as informative as possible for estimating the Jacobian of the given dynamical system.

B Control-theoretic analysis details

B.1 Gramian computation

We begin with equation 4 from Section 5, in which we separate the locally-linearized dynamics in the tangent space into separate pairwise inter-subsystem control interactions. These pairwise control interactions are in general of the form

where A is the within-subsystem Jacobian, B is the across subsystem Jacobian and x A , x B are the states of subsystems A and B, respectively. For a given control system, the time-varying reachability Gramian on the interval [ t 0 , t 1 ] is defined as

where Φ denotes the state-transition matrix of the intrinsic dynamics of subsystem A without any input (i.e., δ x A ( t ) = Φ ( t, t 0 ) δ x A ( t 0 ) ) [107]. Differentiating with respect to t , we obtain δ ˙ x A ( t ) = ∂ ∂t Φ ( t, t 0 ) δ x A ( t 0 ) . Noting also that, in the absence of input from subsystem B, we have

Thus setting the equations equal to each other and canceling δ x A ( t 0 ) from both sides we obtain

Note that Φ ( t 0 , t 0 ) = I . Now, letting Γ ( t, τ ) = Φ ( t, τ ) B ( τ ) B T ( τ ) Φ T ( t, τ ) and differentiating the Gramian expression with respect to the second argument (and using the Leibniz integral rule) yields

and observing

we can continue

which illustrates that the reachability Gramian can be solved for using an ODE integrator (with initial condition W r ( t 0 , t 0 ) = 0 ) [56]. In practice, to compute the reachability Gramians using the trained JacobianODE models, we fit a cubic spline c ( t ) to the reference trajectory x ( t ) and compute J ( t ) = J ( c ( t )) . We can then parse the Jacobian matrix into its component submatrices and compute the Gramians accordingly.

The reachability Gramian is a symmetric positive semidefinite matrix [107]. The optimal cost of driving the system from state δ x 0 to state δ x 1 on time interval [ t 0 , t 1 ] can be computed as

Thus each eigenvalue of W r reflects the ease of control along the corresponding eigenvector [107, 109]. Eigenvectors with larger corresponding eigenvalues will have smaller inverse eigenvalues, and thus scale the cost down along those directions when computing the cost as above.

B.2 Extension to non-autonomous dynamics

While we deal with autonomous systems in this work, we note that this construction can easily be extended to include non-autonomous dynamical systems, of the form ˙ x ( t ) = f ( x , t ) by constructing an augmented state ˜ x ∈ R n +1 which is simply the concatenation of x with t . This yields the autonomous dynamics ˙ ˜ x = ˜ f (˜ x ) , where ˜ f : R n +1 → R n +1 is the concatenation of f with a function that maps all states to 1 .

B.3 Extension to more than two subsystems

Consider a nonlinear dynamical system in R n , defined by ˙ x ( t ) = f ( x ( t )) . Suppose now that the system is composed of K subsystems, where the time evolution of subsystem k is given by x ( k ) ( t ) ∈ R n k . x ( t ) is then comprised of a concatenation of the x ( k ) ( t ) , with ∑ K k =1 n k = n . Given a particular reference trajectory, x ( t ) , with J as the Jacobian of f , then the tangent space dynamics around the reference trajectory for subsystem α ∈ { 1 , ..., K } are given by

̸

where J k → α ( x ( t )) ∈ R n α × n k is the submatrix of J ( x ( t )) in which the columns correspond to subsystem k and the rows correspond to subsystem α . Now, for a given subsystem β ∈ { 1 , ..., K } with β = α , we wish to analyze the ease with which β can control α locally around the reference trajectory, without intervention from other subsystems. Discounting the interventions from other subsystems equates to setting δ x ( k ) ( t ) = 0 for k = α, β , leaving the expression

which quantifies the influence β can exert over α in the absence of perturbations from any other subsystem. Using this representation, the reachability Gramian can be computed as described above. Performing this procedure for all pairs of subsystems α, β ∈ { 1 , ..., K } thus characterizes all pairwise control relationships between subsystems along the reference trajectory.

C Supplementary results

C.1 Control and communication capture different phenomena

In the Introduction (Section 1) we note that measuring communication between two systems is different than measuring control. To illustrate this, consider two brain areas, A and B, whose dynamics are internally linear. The areas are connected only through a linear feedforward interaction term from area B to area A (Figure S1A). Concretely, we consider the dynamics:

Here, the J matrices are time-invariant. When considering communication between brain areas, one might aim to find the subspaces in which communication between B and A occurs, as well as the messages passed [33, 73]. In this construction, the subspace in which area B communicates with area A is explicitly given by J B → A , and thus the message, or input, from area B to area A at any time t is simply given by J B → A x ( t ) . We pick the dimensions of each region to be n A = n B = 4 and let the eigenvectors of the (negative definite matrix) J A → A be given by v i , i = 1 , 2 , 3 , 4 . The eigenvectors are numbered in order of decreasing real part of their corresponding eigenvalue. Now, suppose we construct the interaction matrix J B → A in two different ways: If we let (1) J B → A 1 = v 1 1 T (Figure S1B, left) , then the signal from B is projected onto the most stable mode of region A, whereas if we let (2) J B → A 2 = v 4 1 T (Figure S1B, right), the signal is projected onto the least stable mode. While interaction 1 and interaction 2 communicated messages with identical magnitudes, interaction 2 led to much lower cost reachability control (Figure S1C). This illustrates that control depends not only on the directions along which the areas can communicate, but also on how aligned the communication is with the target area's dynamics.

C.2 Derivative estimation is not implied by function estimation

An alternative approach to directly estimating the Jacobians would be to learn an approximation ˆ f of the function f , and then approximate the Jacobian via automatic differentiation (i.e., an estimate ˆ J = ∂ ∂x ˆ f , as with the NeuralODEs). While this can be effective in certain scenarios, it is not generally the case that approximating a function well will yield a good approximation of its derivative. To illustrate this, we recall an example from Latrémolière et al. [90], in which functions

̸

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fn(x)=- sin(nx)

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-1

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interaction2

̸

f n = 1 n cos( nx ) are used to approximate the function f ( x ) = 0 (Figure S1D). While these approximations improve with increasing n (i.e., lim n →∞ f n = f ), this is not the case for the derivative ( lim n →∞ f ′ n = -sin( nx ) = f ′ ). This demonstrates the necessity of learning J directly (rather than first approximating f ), which we also demonstrate empirically. In the context of machine learning, this setting could be interpreted as overfitting [90]. As long as function estimates match at the specific points in the training set, how the function fluctuates between these points is not constrained to match the true function. For this reason, we included the Frobenius norm Jacobian regularization in our implementation of the NeuralODEs (Appendix D.3).

C.3 Full benchmark dynamical systems results

We here present the full results for all considered example dynamical systems. 10 time-step trajectory predictions along with Jacobian estimation and estimated Lyapunov spectra are displayed in Figure S2, along with MSE and R 2 in Table S1 (trajectory prediction) and Table S2 (Jacobian estimation). Jacobian estimation errors using the 2-norm are presented in Table S3.

C.4 Ablation studies

To determine the value of the different components of the JacobianODE learning framework, we performed several ablation studies. We chose to evaluate on the Lorenz system and the task-trained RNNs, as these together provide two common settings of chaos and stability, as well as low- and high-dimensional dynamics.

Ablating the Jacobian-parameterized ODEs The JacobianODE framework constructs an estimate of the initial time derivative f via a double integral of the Jacobian as described in Section 3.2. To briefly recall, Jacobian path integration is described as

When we do not have access to the initial derivative and base point f ( x ( t i )) , x ( t i ) , we use the following formulation

10 time-step

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Weighted linear Jacobians

Ground truth system

where

Alternatively, one could use Equation 14 and learn f ( x ( t i )) and x ( t i ) as learnable parameters ( x ( t i ) is needed as the first point of the path, i.e. c ( t i ) = x ( t i ) inside the integral) [92]. Here, the path integral

between x ( t i ) and x ( t f ) is a linear interpolation, as before. These ablated models were trained on 10 time-step prediction, as with the original JacobianODEs. For these models, we completed a full sweep over the loop closure loss weight λ loop in order to determine the best hyperparameter.

Ablating teacher forcing We trained models without any teacher-forcing. That is, models were able to generate only one-step predictions, without any recursive predictions. Again we did a full hyperparameter sweep to pick λ loop .

Ablating loop closure loss We ablated the loop closure loss in two ways. The first was to set λ loop = 0 to illustrate what would happen if there were no constraints placed on the learned Jacobians. The second was to instead use the Jacobian Frobenius norm regularization that was used for the NeuralODEs (details are in Appendix D.3). We did a full sweep to pick λ jac, the Frobenius norm regularization weight.

Table S4: Mean Frobenius norm error ⟨∥ J -ˆ J ∥ F ⟩ for different model ablations on the Lorenz system with 10% observation noise. Errors are reported as mean ± standard deviation, with statistics computed over 8 test trajectories, each consisting of 1200 points.

Ablation results The performance of the ablated models on Jacobian estimation in the Lorenz system are presented in Table S4. The original JacobianODE outperforms all ablated models, indicating that all components of the JacobianODE training framework improve the model's performance in this setting. Ablating the Jacobian-parameterized initial derivative estimate resulted in a slight decrease in the estimation loss. This is potentially because the network could offload some of the

responsibility for generating correct trajectory predictions onto the estimated base point x ( t i ) and derivative estimate ˆ f ( x ( t i )) , slightly reducing the necessity of estimating correct Jacobians. Ablating the teacher forcing annealing predictably led to worse Jacobian estimation, as the network no longer has to consider how errors will propagate along the trajectory. The most dramatic increase in error was with the ablation of the loop closure loss. Without this important regularization, the learned Jacobians reproduced the dynamics but were not constrained to be conservative, resulting in poor Jacobian estimation. The inclusion of the Frobenius penalty on the Jacobians mitigated this, although it did not encourage accurate Jacobian estimation to the same degree as the loop closure loss.

Table S5: Mean Frobenius norm error ⟨∥ J -ˆ J ∥ F ⟩ for different model ablations on the task-trained RNN with 10% observation noise. Errors are reported as mean ± standard deviation, with statistics computed over 409 test trajectories, each consisting of the 49 points from the second delay and response epochs.

We then tested the ablated the models on Jacobian estimation in the task-trained RNN, with results presented in Table S5. Again, ablating the Jacobian-parameterized derivative estimates, teacher forcing, and loop closure resulted in worse Jacobian estimation. Interestingly, in this setting, the inclusion of a penalty on the Frobenius norm of the Jacobians outperformed the use of the loop closure loss. This could potentially be because the loop closure loss is more difficult to drive to zero in high dimensional systems, or because the loop closure loss is more important in chaotic systems like the Lorenz system considered above. Future work should consider in what contexts each kind of regularization is most beneficial to JacobianODE models.

C.5 NeuralODEs achieve improved performance at the cost of increased inference time

In the main paper, we implemented both the JacobianODEs and the NeuralODEs as four-layer MLPs, with the four layers having sizes of 256, 1024, 2048, and 2048 respectively. This was done for the fairest architectural comparison between the models, to ensure that both models had the same representational capacity when generating their respective outputs. However, there are many architectural changes that we could make to this setup that impact performance. We hypothesized based on the discussion in Appendix C.2 that increasing the hidden layer size of the NeuralODEs would improve Jacobian estimation, as larger models have been known to learn smoother representations. Furthermore, we wondered whether including residual blocks in place of the standard MLP implementation would improve Jacobian estimation.

To test this, we implemented the NeuralODEs as four-layer residual networks and tested three different sizes of hidden layer: 1024, 2048, and 4096. Results are in Table S6. For nearly all models, these changes yielded only marginal improvements over the original NeuralODE model. Only the model with 4096-dimensional hidden layers under 10% training noise achieves a performance near the original JacobianODEs. However, the performance limit is still below that of the JacobianODEs, even with a large increase in the representational capacity of the model. It is furthermore of note that the NeuralODEs were able to significantly improve performance only in the high noise setting. This suggests that high noise is necessary for the model to be forced to learn the response of the system to perturbation. In contrast, JacobianODEs perform similarly across all noise levels, indicating a stronger inductive bias to learn the response of the system to perturbation.

While the increased hidden layer size seems to induce smoother hidden representations (as expected), it comes with increasing computational cost. Recall that the NeuralODEmodels directly parameterize the dynamics as ˙ x = ˆ f θ ( x ) . The Jacobian at state x is computed by directly backpropagating through the Neural ODE ˆ f θ , thereby significantly increasing compute. On the other hand, the JacobianODEs only require a forward pass. We evaluated the Jacobian inference time of JacobianODE

and NeuralODE models with four hidden layers of the same size on an H100 GPU. Each model was timed on inferring the Jacobians of 100 batches of the 128-dimensional task-trained RNN, with each batch containing 16 sequences of length 25. Timings were repeated ten times for each model. Results are plotted in Figure S3.

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The JacobianODE achieves much faster inference times than the NeuralODE - approximately two orders of magnitude faster at large hidden dimension sizes. Furthermore, as shown in Table S6, the JacobianODE with a maximum hidden layer size of 2048 outperforms the NeuralODE with a maximum hidden layer size of 4096 on Jacobian estimation, and does so with orders of magnitude faster inference (Figure S3, green and red circles, Table S6). This suggests that while both architectures appear to have inference times that scale approximately exponentially, the JacobianODE achieves more favorable scaling across every hidden layer size we tested. Our analysis therefore illustrates that it is possible to improve the NeuralODEs' Jacobian estimation with larger models, but the inference time scaling renders these models ill-equipped for important settings such as real-time control and the analysis of high-volume neural data.

C.6 Comparison to echo state networks

Echo state networks (ESNs, or reservoir networks) are tools for time-series forecasting and analysis. ESNs are typically implemented as RNNs with fixed, sparse, hidden connectivity. The output weights

of the ESNs can then be trained via linear regression to reproduce specific temporal patterns. ESNs have been shown to be able to forecast the Lorenz 63 system for a particular choice of hyperparameters [139]. We note that Pathak et al. [139] implements the same method as Pathak et al. [140] . We implemented the ESNs exact hyperparameters reported in Pathak et. al. 2017 for the Lorenz system, which involved a reservoir of 300 nodes. Our implementation of the Lorenz system was simulated with a sampling interval of ˜ 0.015 s, very similar to the 0.02 s used in the paper. Thus, for consistency of comparison with JacobianODEs, we fit ESN models to data sampled every 0.015 s.

To enable Jacobian estimation, we changed one small implementation detail from Pathak et al. [139]. Specifically, we trained the network output weights to predict the third variable z using only the network state r ( t ) , as opposed to the vector ˜ r ( t ) for which the node state is squared ( ˜ r i ( t ) = r 2 i ( t ) ) for half of the nodes i . This is because squaring the reservoir state complicates Jacobian computation, and was stated by Pathak et al. [139] and Lu et al. [141] to only be necessary for symmetry breaking if only the z variable of the Lorenz system were provided. We trained the ESNs on the same volume of data as the JacobianODEs and NeuralODEs, with 1% observation noise. We swept the regularization parameter β to ensure a good fit. We furthermore generated predictions using a 150 time-step lead-up to ensure the ESNs converged to a good internal representation. The ESNs predicted the Lorenz system well, achieving an MSE of 0.03 on the 10-step prediction task from our paper (on test data). This was very similar to the reported 10-step prediction MSEs of 0.07 and 0.04 for the JacobianODEs and NeuralODEs, respectively.

In Pathak et al. [139], the Lyapunov exponents of the system were computed using the Jacobian of the interior dynamics of the ESN, as the recursion necessary to compute the Jacobian can be formulated uniquely only in regard to this state representation. Thus, each of these Jacobians would be a 300 × 300 matrix. While this suffices to compute the Lyapunov exponents, it does not suffice for our purposes of obtaining Jacobian estimates of the dynamics of the original system. For this purpose, it has been shown that it is possible to obtain an estimate of the Jacobian of the original dynamical system (i.e., not the echo state space) using ESNs [142]. However, as Banerjee et al. [142] noted, it is necessary to invert the non-square ESN output matrix in order to obtain these Jacobians, thus they are non-uniquely determined, and are used in the paper "only for causality estimation purposes". Nevertheless, we computed ESN-estimated Jacobians on the 1% observation noise test data. We used a large (7,200 time-step) lead-up and ensured coherence to the trajectory by feeding in the true state as input. The Frobenius norm error over five random seeds was 85 . 8 ± 3 . 1 , much larger than the reported errors of 3 . 3 ± 0 . 2 and 8 . 7 ± 0 . 3 for the JacobianODEs and NeuralODEs respectively. Thus, similar to the NeuralODE, while ESNs can achieve comparable prediction on chaotic systems, they do so without enabling an accurate reconstruction of the Jacobian of the original dynamical system.

C.7 Analyzing the dimensionality of Jacobian manifolds

RNN activity is often confined to low-dimensional manifolds, potentially simplifying Jacobian estimation. To address the extent to which this influences our results, we performed the following analyses.

Jacobian rank : We first note that both the RNN hidden weight matrix and nearly all Jacobians had full rank (128; ~0.1% had rank 127). To assess the effective rank, we computed the participation ratio (PR), a singular value-based measure of intrinsic dimensionality. The RNN weight matrix had a PR of 123, and the Jacobians had a mean PR of ~106 (std 0.7).

Intrinsic-data dimensionality : Another possibility is that the data itself may be low-dimensional. Since the Jacobian is a function of the trajectory data, the Jacobian manifold's intrinsic dimension is at most that of the trajectory manifold. To estimate the intrinsic dimensions, we pooled time points from a large sample of trajectories. For RNN state vectors, we computed PR directly; for Jacobians, we flattened them into n 2 -dimensional vectors before computing PR. We found the RNN trajectory PR to be ~13.5, and the Jacobian PR to be ~6. Given the 128D extrinsic space, this confirms both RNN states and Jacobians lie on lower-dimensional manifolds.

Notably, prior work estimated the dimensionality of prefrontal cortex delay activity at 24 using ~4000 neurons, falling to ~6 when subsampled to 100 neurons [143]. These values were interpreted as high-dimensional. Thus, the dimensionalities we observe are consistent with biological systems performing complex tasks.

Furthermore, in the 64-dimensional Lorenz 96 system, the trajectory PR was ~47.5, and the Jacobians had a PR of ~50, which we believe to be high-dimensional enough to demonstrate the effectiveness of our method on complex systems.

We note that in larger systems, approximating the Jacobian as low rank may be more efficient. Rather than computing each of the outputs directly, we can constrain the Jacobian to be low-rank by designing the output as a low-rank combination of vectors (e.g., it learns to output a low-rank singular value decomposition).

D Experimental details

D.1 Dynamical systems data

We used the following dynamical systems for the testing and validation of Jacobian estimation.

Van der Pol oscillator We implement the classic Van der Pol oscillator [96]. The system is governed by the following equations

We pick µ = 2 in our implementation.

Lorenz We implement the Lorenz system introduced by Lorenz [97] as

with the typical choices for parameters ( σ = 10 , ρ = 28 , β = 8 / 3 ).

Lorenz 96 We implement the Lorenz 96 system introduced by Lorenz [98] and defined by

with F = 8 and N ∈ { 12 , 32 , 64 }

Simulation We use the dysts package to simulate all dynamical systems [99, 100]. The characteristic timescale of their Fourier spectrum τ is selected and the systems are sampled with respect to τ . For all systems, the training data consisted of 26 trajectories of 12 periods, sampled at 100 time steps per τ . The validation data consisted of 6 trajectories of 12 periods sampled at 100 time steps per τ . The test data consisted of 8 trajectories of 12 periods sampled at 100 time steps per τ . Trajectories were initialized using a random normal distribution with standard deviation 0.2. The simulation algorithm used was Radau. Batches were constructed by moving a sliding window along the signal. The sequence length was selected such that the generated predictions would generate 10 novel time points (i.e., 11 time steps for the NeuralODE, and 25 time steps for the JacobianODE, due to the 15 time steps used to estimate the initial time derivative f ).

Noise We define P % observation noise with P = 100 p in the following way. Let A signal = E [ ∥ x ( t ) ∥ 2 2 ] be the expected squared norm of the signal with x ( t ) ∈ R n . Then consider a noise signal η ( t ) ∈ R n where each component η i ( t ) ∼ N (0 , 1 √ n p √ A signal ) . Then

and thus the noise percent is

D.2 Task-trained RNN

The task used to train the RNN was exactly as defined in Section 5. The hidden dimensionality of the RNN was 128, and the input dimensionality was 10, where the first four dimensions represented the one-hot encoded "upper" color, the second four dimensions represented the one-hot encoded "lower" color, and the last two dimensions represented the one-hot encoded cue. The RNN used for the task had hidden dynamics defined by

with τ = 50 ms, which for the purposes of training was discretized with Euler integration with a time step of ∆ t = 20 ms. W hh is the 128x128 dimensional matrix that defines the internal dynamics, W hi is the 128 x 10 dimensional matrix that maps the input into the hidden state, W oh is the 4x128 dimensional output matrix that maps the hidden state to a four-dimensional output o ( t ) , and b is a static bias term. σ was taken to be an exponential linear unit activation with α = 1 . The RNN hidden state was split into two "areas" each with 64 dimensions. The input matrix W hi was masked during training so that inputs could only flow into the first 64 dimensions - the "visual" area. The same procedure was performed for the output matrix W oh , except the mask was such that outputs could stem only from the second group of 64 dimensions - the "cognitive" area. The within-area subblocks of the matrix W hh were first initialized such that the real part of the eigenvalues were randomly distributed on the interval [ -0 . 1 , 0] and the imaginary part of the eigenvalues were randomly distributed on the interval [0 , 2 π ] . The eigenvectors were random orthonormal matrices. We then computed the matrix exponential of this matrix. The across-area weights were first initialized to be random normal, then divided by the 2-norm of the resulting matrix and multiplied by 0.05. The input and output matrices were initialized as random normal and then scaled by the 2-norm of the resulting matrix. The static bias b was initialized at 0 . After initialization, all weights could be altered unimpeded (except for the masks). Notably, the inputs u ( t ) were present only during the stimulus and cue presentation epochs - otherwise the network evolved autonomously. The loss was computed via cross entropy loss on the RNN outputs during the response period (the final 250 ms of the trial).

For the training data, we generated 4096 random trials, and used 80% for training and the remainder for validation. The batch size used was 32. Training was performed for 40 epochs. The learning rate was 0.0005. For use with the Jacobian estimation models, data was batched and used for training exactly as was done with the other dynamical systems data (see Appendix D.1). Observation noise was also computed in the same way.

D.3 NeuralODE details

NeuralODE models directly estimate the time derivative f with a neural-network parameterized function ˆ f θ . Then the Jacobians can be computed as ˆ J = ∂ ∂ x ˆ f θ .

The NeuralODEs were implemented as described in Section 4 and ODE integration was done exactly as for the JacobianODE using the torchdiffeq package with the RK4 method [102]. To regularize the NeuralODE we implemented a Frobenius norm penalty on the estimated Jacobians, i.e.

where ˆ J is the estimated Jacobian computed via automatic differentation and λ jac is a hyperparameter that controls the relative weighting of the Jacobian penalty [104-106]. As mentioned in the main text, this penalty prevents the model from learning unnecessarily large eigenvalues and encourages better Jacobian estimation.

D.4 Weighted linear Jacobian details

We implemented a baseline Jacobian estimation method using weighted linear regression models as described in Deyle et al. [88]. Given a reference point x ( t ∗ ) at which the (discrete) Jacobian will be

computed, all other points are weighted according to

is the average distance from x ( t ∗ ) to all other points. We then perform a linear regression using the weighted points (and a bias term), the result of which is an estimate of the discrete Jacobian at x ( t ∗ ) , which can be converted to continuous time by subtracting the identity matrix and dividing by the sampling time step (i.e., ˆ J = ˆ J discrete -I ∆ t , where ˆ J discrete is the discrete Jacobian). The parameter θ tunes how strongly the regression is weighted towards local points. To pick θ , we sweep over values range from 0 to 10, and pick the value that yields the best one-step prediction according to

where ˆ J is the estimated Jacobian. This form of prediction has been previously been used to learn Jacobians in machine learning settings [90]. To test the method, we pick θ based on data with observation noise at a particular noise level, then add in the denoised data to the data pool in order to compute regressions and estimate Jacobians at the true points.

D.5 Model details

All models were implemented as four-layer MLPS, with the four layers having sizes of 256, 1024, 2048, and 2048 respectively. All models used a sigmoid linear unit activation. JacobianODE models output to the dimension n 2 which was then reshaped into the matrix of the appropriate dimension. NeuralODEs output to the dimension n .

D.6 Lyapunov spectrum computation

To compute the Lyapunov spectrum, we employ a QR based algorithm [121, 122]. We discretize the Jacobians using the matrix exponential (i.e., ˆ J discrete = e ˆ J ∆ t ) and then propagate a bundle of small vectors through the Jacobians, using QR to ensure the perturbations remain bounded.

D.7 Iterative Linear Quadratic Regulator (ILQR)

We implement the standard algorithm for ILQR, the details of which can be found in Li and Todorov [111] and Tassa et al. [112]. In brief, the ILQR algorithm linearizes the system dynamics around a nominal trajectory using the Jacobian, and then iteratively optimizes the control sequence using forward and backward passes to minimize the total control cost. The state cost matrix Q was a diagonal matrix with 1.0 along the diagonal. The final state cost matrix Q f was a diagonal matrix with 1.0 along the diagonal. The control cost matrix R was a diagonal matrix with 0.01 along the diagonal. The control matrix was a 128 × 128 matrix in which the 64 × 64 block corresponding to the first 64 neurons (the "visual" area) was the 64-dimensional identity matrix. The control algorithm was seeded with only the initial state of the test trajectory with 5% noise. The control sequence was initialized random normal with standard deviation 0.001 and mean 0. The ILQR algorithm was run for a max of 100 iterations. The regularization was initialized at 1.0, with a minimum of 1 × 10 -6 and a maximum of 1 × 10 10 . ∆ 0 was set to 2, as in Tassa et al. [112]. If the backward pass failed 20 times in a row, the optimization was stopped. The list of values for the line search parameter α was 1 . 1 -k 2 for k ∈ 0 , ..., 9 (see Tassa et al. [112]). The linear model used for the linear baseline was computed via linear regression.

D.8 Training details

All models were implemented in PyTorch. The batch size used was 16. Gradients were accumulated for 4 batches. Training epochs were limited to 500 shuffled batches. Validation epochs were limited to 100 randomly chosen batches. Testing used all testing data. Training was run for a maximum of 1000 epochs, 3 hours, or until the early stopping was activated (see Appendix D.8.6), whichever came first.

where

D.8.1 Generalized Teacher Forcing

The Jacobian can be best learned when training predictions are generated recursively (i.e., replacing x ( t ) by ˆ x ( t ) ). However, in chaotic systems, and/or systems with measurement noise (as considered here), this could lead to catastrophic divergence of the predicted trajectory from the true trajectory during training. We therefore employ Generalized Teacher Forcing when training all models [94]. Generalized Teacher Forcing prevents catastrophic divergence by forcing the generated predictions along the line from the prediction to the true state. Specifically, for a given predicted state ˆ x ( t ) and true state x ( t ) , the teacher forced state is

with α ∈ [0 , 1] . This effectively forces the predictions along a line from the prediction to the true state, by an amount with proportion α . α = 1 corresponds to fully-forced prediction (i.e., one-step prediction) and α = 0 corresponds to completely unforced prediction (i.e., autonomous prediction). Hess et al. [94] suggested that a good estimate of α is

where J ( p ) T :2 are the Jacobians of the modeled dynamics computed at data-constrained states, p indicates the batch or sequence index, and

effectively computes the discrete maximum Lyapunov exponent. In our implementation, we compute ∥G ( J ( p ) T :2 ) ∥ using a QR-decomposition-based Lyapunov computation algorithm [121]. As the Jacobian of the dynamics is necessary to compute this quantity, the JacobianODEs enjoy an advantage over other models in that the Jacobians are directly output by the model, and do not have to be computed via differentiating the model itself.

We furthemore employ a slightly modified version of the suggested annealing process in Hess et al. [94], which sets α 0 = 1 and updates α n as

where α is computed according to equation 17. Following the suggested hyperparameters, we set γ = 0 . 999 and update α n every 5 batches. Once the teacher forced state ˜ x ( t ) is computed, it can simply replace x ( t ) in equation 11 to generate predictions.

D.8.2 Loop closure loss

We implemented a loop closure loss as discussed in Section 3.3 and Appendix A.3. For each loop, we used 20 randomly chosen points from the batch. For each batch, we constructed the same number of loops as there were batches. Path integrals were discretized in 20 steps and computed using the trapezoid method from torchquad [101].

D.8.3 Validation loss

All models were validated on 10 time-step prediction task with teacher forcing parameter α = 0 (i.e., autonomous prediction).

D.8.4 Learning rate scheduling

For all models, the learning rate was annealed in accordance with teacher forcing annealing. Given an initial and final learning rates η i and η f we compute the effective learning rate as

where α n is the current value of the teacher forcing parameter and

σ ( α n ) is a scaling function with σ (1) = 1 and σ (0) = 0 and for which the shape of the scaling is controlled by the parameter k . For positive values of k , the scaling is super-linear, and for negative values of k it is sub-linear. We use k = 1 , ensuring that the learning rate does not decrease too quickly at the start of learning. We set η i = 10 -4 and η f = 10 -6 for all models.

D.8.5 Optimizer and weight decay

All models were trained with PyTorch's AdamW optimizer with the learning rate as descried above, and weight decay parameter 10 -4 . All other parameters were default ( β 1 = 0 . 9 , β 2 = 0 . 999 , ϵ = 10 -8 ). We also used gradient norm clipping, with a clipping value of 1.0.

D.8.6 Early stopping

For all models, we implemented an early stopping scheme that halted the training if the validation loss improved by less than 1% for two epochs in a row.

D.8.7 Added noise during learning

For the models trained on the task-trained RNN dynamics, we added 5% Gaussian i.i.d. noise (defined relative to the norm of the training data with observation noise already added). Noise was sampled for each batch and added prior to the trajectory generation step of the learning process. Additional noise was not added for the loop closure computation.

D.8.8 Hyperparameter selection

For the JacobianODEs, the primary hyperparameter to select is the loop closure loss λ loop . To select this hyperparameter, we trained JacobianODE models with λ loop ∈ [0 , 10 -6 , 10 -5 , 10 -4 , 10 -3 , 10 -2 , 10 -1 , 1 , 10] . For each run, the epoch with the lowest trajectory validation loss ( L traj) is kept. Then, for this model, we compute the one-step prediction error on validation data, the validation loop closure loss ( L loop), and the percentage of Jacobian eigenvalues on all validation data that have a decay rate faster than the sampling rate 1 ∆ t . We exclude any models that meet any of the following criteria:

One-step prediction error greater than the persistence baseline . The persistence baseline is computed as the mean error between each time step k ∆ t and the subsequent time step ( k +1)∆ t across the dataset, and constitutes a sanity check for whether a model is capturing meaningful information about the dynamics.

Loop closure loss greater than √ n , where n is the system dimension (see Appendix D.11 for the derivation of this bound). As discussed in the main text, we are interested in Jacobians that not only solve the trajectory prediction problem, but that also are constructed so that the rows of the matrix are approximately conservative vector fields.

More than 0.1% of the Jacobian eigenvalues have a decay rate faster than the sampling rate 1 ∆ t . Since large negative eigenvalues do not impact trajectory prediction, the models may erroneously learn Jacobians with large negative eigenvalues. If the decay rate of these eigenvalues is faster than the sampling rate, we can infer that the eigenvalues are not aligned with the observed data.

If none of the models that meet criterion (2) meet criterion (1), we discount criterion (2), as this suggests that a loop closure loss below √ n bound is too strict to obtain good prediction on this system. Additionally, if none of the models that meet criterion (3) meet criterion (1), we discount criterion (1), as this suggests that noise is very high in the data, which leads to both high one-step prediction error, and large negative eigenvalues to compensate for the perturbations introduced by the noise. Of the remaining models, we select the one with the lowest trajectory validation loss L traj .

For the NeuralODEs, we needed to select the hyperparameter λ jac, which regularized the mean frobenius norm of the Jacobians computed through automatic differentiation. Again, to select this hyperparameter, we trained JacobianODE models with λ jac ∈ [0 , 10 -6 , 10 -5 , 10 -4 , 10 -3 , 10 -2 , 10 -1 , 1 , 10] . We followed exactly the above procedure with the exception of criterion (2), which was deemed unnecessary, as computing the Jacobians implicitly via a gradient of the model (using automatic differentiation) ensures that the rows of the matrix are conservative.

All other hyperparameters (model size, learning rate, length of initial trajectory, number of discretization steps, etc.) were fixed for all systems. Given the wide range of systems and behaviors and dimensionalities that the JacobianODEs are capable of capturing, this indicates that the method is robust given a reasonable choice of these hyperparameters.

D.8.9 Model hyperparameters and training details

Below are presented the details of all Jacobian estimation models considered in the main paper.

D.9 Information about computing resources and efficiency

All models were able to be trained on a single H100 GPU, with 80 GB of memory.

Jacobian inference times Jacobian inference times for the JacobianODE and NeuralODE models are discussed in Appendix C.5. As discussed, models were implemented with four hidden layers of the same size, and tested on 100 batches of the 128-dimensional task-trained RNN data, with each batch consisting of 16 sequences of length 25. Timings were repeated ten times for each model (see Figure S3 for details).

Training time Total training times for each of the chosen models are presented in Table S7. Furthermore, we include the training time (including backward pass) for 100 batches (with 16 sequences per batch), using 10 time-step prediction, in Table S8.

D.10 Statistical details

All statistics were computed using scipy . For the comparison between JacobianODE and NeuralODE trajectory and Jacobian predictions, as well as the comparison of Gramian traces and minimum eigenvalues, we used a two-sample t-test. For the comparison of ILQR control accuracies and errors, we used a Wilcoxon signed-rank test.

D.11 Derivation of loop closure loss bound

We consider the loop closure loss as defined in 3.3. We are interested in estimating a bound on the error

where n is the system dimension. While in theory this quantity should be 0, in practice due to numerical estimation error, it will not be. First recall, that

where L is the number of loop points and c is a line from x ( t i (mod L ) ) to x ( t ( i +1) (mod L ) ) . We assume that

∀ i, j ∈ [1 , .., L ] , which is justified as the numerical error accrued will likely be similar along different lines for the same system. Thus

where we have used the fact that the errors are assumed to be equivalent for each of the line segments comprising the overall loop path. Recall now that for the trapezoid integration rule, the error E in

integrating ∫ b a f ( x ) dx can be computed as E = -( b -a ) 3 12 M 2 f ′′ ( u ) for some u ∈ [ a, b ] . Thus the squared error is bounded as

In our case, when path integrating lines, we are effectively integrating from a = 0 to b = 1 . Furthermore, let g ( r ) = J ( c ( r )) c ′ ( r ) , the integrand of our line integrations. We assume that E [ ∥ max r g ′′ ( r ) ∥ 2 2 ] ∝ n · √ n where the first n comes from the fact that the norm involves summing over the n components of the vector g ′′ ( r ) and the second √ n involves an assumption that loop closure loss will be more difficult to compute accurately in higher dimensions, though this will be more pronounced as dimensionality initially starts increasing. Thus E [ ∥ max r g ′′ ( r ) ∥ 2 2 ] = kn · √ n for some k ∈ R + . Now, continuing on,

Finally, assuming that the number of discretization steps M was chosen to be large enough such that M 4 ≥ 1 12 2 Lk we finally obtain

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nAuthors introduce a modeling approach for predicting next states in an observed dynamical system. They specifically utilize the linearized dynamics and learn the Jacobian to both achieve an interpretable modeling of the observed dynamics (though locally) that later enables control of the system and be able to predict next states for a given initial condition and inferred set of time-varying jacobians. They design a novel learning algorithm to train these Jacobians and demonstrate these two specific claims with simulations.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n- The focus on predicting not just dynamics, but also the jacobian is novel and quite interesting. This is particularly relevant since the multiplication J(t) x is under constrained by just observed dynamics, but Jacobian is needed for defining optimal control dimensions.\n\n- The authors demonstrate both of their claims with simulations and RNN training, which is almost sufficient (see below).\n\n## Weaknesses\n\n- The writing is quite dense, which leads to many questions in the mind of (this) reader. In several cases, I felt that the results were too good to be true (which can also become a strength once authors clearly set the boundaries of their method).\n\n- I believe a different RNN architecture may be more appropriate to test these claims (see below).\n\nQUESTIONS:\n1) It is still not clear to me how Jacobian at a given time point can be approximated without repetition of the same trials/time windows/states? One would ideally need to sample around a particular state to be correctly estimate the jacobian? There is some coarse-graining here that I am not seeing/not being discussed explicitly. Can you comment on this?\n\n2) The model of the RNN should be shown in the main text. In my opinion, this work would significantly benefit from using more complex RNN architectures. Specifically, the RNN architecture used here is known to have neural activities confined to low-dimensional hyperplanes, hence the jacobian estimation is actually quite simpler and maybe effectively low-dimensional. You can check this by computing the PC dimensionality in learned RNNs, which would be important to note here. This may be ok, since most behavioral tasks animals perform in the labs are low-dimensional anyways, but this point should be discussed and raised in the limitations. In my opinion, using LSTMs trained on slightly more complex tasks (and validated to have high PC dimensionality) may be a good control for this. \n\n3) The discussion on Gramian is too quick. Could you please expand on this? The reader cannot really appreciate what is exactly being computed, how it relates to the experiments etc. \n\n\nHere are some simple extra suggestions:\n\n- Line 150: waiting -> weighting?\n\n- Title was a bit misleading for me. This seems to be a neuroscience-relevant work, though only looking at the title, I would have likely not read this work. (and it would have been a mistake on my part, since this is quite interesting!)\n\n- The transition to Eq. (1) is weird. At first, I thought that was part of the figure.\n\n- Section 6 should likely come way earlier, before the results section.\n\nLIMITATIONS:\nSee above.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors provide a novel method of local deep Jacobian estimation for non-linear dynamical systems, involving optimising a neural network jointly on trajectory prediction and adherence to conservativity. The authors used this method to model the interarea controllability of a two-area task-trained RNN, showing that their method outperforms other Jacobian estimation methods in predicting and exerting control on local dynamics.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- The authors provide a well-motivated Jacobian estimation technique, providing extensive results on synthetic data. \n- The authors are careful to provide details throughout the paper about the technical and intuitive details of their method e.g. appendix C.1, C.2 provide welcome information and results to reinforce the method motivation, and ablation studies justify each component of the model\n\nWeaknesses\nThis work presents an interesting method and theoretically strong results. The primary weaknesses of this paper are around the communication and format of information:\n- Section 2.1 seems out of place. While the authors do emphasise the importance of modelling interareal/intersystem communication in the introduction, this does not play a role in the training of JacobianODE, or understanding of the method as a whole. Introducing these concepts in any technical detail may be better placed near section 5, where interarea systems’ data are first introduced.\n- Overall, there is a poor separation of information in the main text and in the appendix. Section 3.1 provides much information that is not strictly necessary for understanding the training procedure in Figure 3, or at least requires an understanding of the appendix material to take advantage of this level of explanation. This could be moved to the appendix to create a clearer narrative in the main text. This would mirror other methods that were key to appreciating the results of this paper, e.g. ILQR, being delegated to the appendix.\n- Despite being an innovation for this work, section D.8.2 seems like insufficient exposition on the formulation and importance of the loop closure loss. Table 3 shows that this is critical for performance, but no investigation or context is provided on how one could curate optimal loops for training.\n\nLess importantly:\n- Images of trajectories provide little information in their current form. Dimensionality reduction, e.g. by tSNE, could provided a better qualitative visualisation of the quality of predictions by each model.\n- A final minor weakness is that spacing is limited throughout the paper, table captions are difficult to discern, and information on figures are often too small\n\nQUESTIONS:\n- Why was this method of loop curation chosen? Do you expect other design choices to significantly change performance, beyond a simple binary ablation?\n- Figure 6B suggests that the model is systematically delayed in predicting changes to area reachability in opposite directions. Is there an intuitive explanation for why this is? Did you consider any investigations into this offset?\n- Did the authors consider training on other laboratory tasks where control theory has previously been applied, e.g. motor control? Comparison of insights provided by JacobianODE compared to previous analyses would provide powerful justification for the method\n\nLIMITATIONS:\nThere are no relevant negative societal impacts associated with this work.\n\nThe authors sufficiently discuss limitations to their work in the final section of the paper. In particular, they highlight the need for an extension to partially observable dynamical systems. This would be a necessary step towards applying this method to real neural dynamics", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nAuthors are modeling dynamical systems by learning from observational data of trajectories. Instead of learning the vector field of the dynamical system directly, they reformulate the problem to learn the system's Jacobian matrix.\n\nThe authors recognize an important fact: each row of the Jacobian corresponds to the gradient of a scalar function, and therefore represents a conservative vector field. This allows them to impose structure in the learning process — specifically, that line integrals over these fields are path-independent.\n\nAlthough the full vector field f(x) does not need to be conservative, its Jacobian rows must be, since they are gradients by definition.\n\nBy applying a generalized form of the fundamental theorem of calculus, they express the function f(x(t)) — which is not modeled directly — as a path integral over the Jacobian.\n\nThis approach enables them to reuse standard ODE solvers to generate trajectories of the full nonlinear system — not just a linearized approximation.\n\nIn the loss function, they combine a trajectory reconstruction loss (to ensure consistency with observed dynamics) with a loop closure loss that enforces path independence, reflecting the conservative structure of Jacobian rows.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- elegant theoretical framework that is quite applicable for range of dynamical systems\n- casting the problem from learning vector field of dynamics to jacobian learning\n- efficient way of learning, without explicit need to model directly vector field\n\nNo major weaknesses.\n\nQUESTIONS:\nFor the case of chaotic dynamical systems, I would like to see comparison how this approach compares to echo state networks:\n\nsee: Doan, Nguyen Anh Khoa, Wolfgang Polifke, and Luca Magri. \"Physics-informed echo state networks for chaotic systems forecasting.\" Computational Science–ICCS 2019: 19th International Conference, Faro, Portugal, June 12–14, 2019, Proceedings, Part IV 19. Springer International Publishing, 2019.\n\nPathak, Jaideep, et al. \"Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach.\" Physical review letters 120.2 (2018): 024102.\n\nPlease state limitations more explicitly.\n\nLIMITATIONS:\nPlease state limitations more explicitly.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-04"}