Mechanical Field Networks: Structured Neural Dynamics for Multivariate Systems

# Introduction
Source: [https://arxiv.org/html/2606.11251](https://arxiv.org/html/2606.11251)
Mechanical Field Networks: Structured Neural Dynamics for Multivariate Systems

Xingji Cui

Xi’an Jiaotong University, China

[cuixingji@stu\.xjtu\.edu\.cn](https://arxiv.org/html/2606.11251v1/mailto:[email protected])

Abstract

Many multivariate dynamical systems are observed only through trajectories, leaving the mechanisms governing their joint dynamics hidden\. Existing approaches can impose interpretable dynamics or learn flexible state transitions, yet the resulting interaction structure is typically either specified in advance or left implicit within the learned dynamics\. We introduce MF\-Net, a recurrent dynamical model that represents all variables in a shared field state and updates this state through a learned relation law\. Each variable carries a field component, and these components evolve jointly through a learnable mechanical transition\. Here, mechanical refers to the relation\-to\-motion organization of the transition, where learned relations shape state\-dependent flows, field responses, and motion tendencies that move the field state forward\. The resulting structure is part of the rollout itself: learned relations influence how the field moves, and the same internal quantities support both forecasting and structural readout\. Across known\-law interaction systems, chaotic benchmarks, real neural recordings, and ecological time series, MF\-Net achieves competitive short\- and medium\-horizon forecasting while retaining inspectable structural readout\. On the 40\-dimensional Lorenz–96 testbed, MF\-Net achieves an eight\-stepR2R^\{2\}of0\.798±0\.0180\.798\\pm 0\.018; across five seeds, its learned relation matrix recovers the local coupling support with a local/nonlocal strength ratio of19\.80±1\.0019\.80\\pm 1\.00and Precision@KKof1\.000±0\.0001\.000\\pm 0\.000\. MF\-Net provides a structure\-readable dynamical modeling framework in which learned relations are trained through forward evolution and, on real data, interpreted as functional predictive couplings under appropriate observational limits\.

Key words\.multivariate dynamical systems; relation\-structured recurrent models; structure\-readable forecasting; interaction structure inference; neural dynamical modeling; functional coupling; mechanical field networks

Observed trajectories of multivariate dynamical systems show how variables evolve over time, but the trajectory alone does not identify the structural relations underlying that evolution\(Granger,[1969](https://arxiv.org/html/2606.11251#bib.bib9);Sugihara et al\.,[2012](https://arxiv.org/html/2606.11251#bib.bib35);Brunton et al\.,[2016](https://arxiv.org/html/2606.11251#bib.bib3);Kipf et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib17)\)\. This creates a gap between prediction and structural interpretation: a model may forecast future values without exposing the relations that participate in its rollout, whereas post\-hoc relation scores may identify plausible associations without showing that those relations participated in the forecasting computation\(Tank et al\.,[2022](https://arxiv.org/html/2606.11251#bib.bib36);Rudin,[2019](https://arxiv.org/html/2606.11251#bib.bib33)\)\. MF\-Net starts from this gap: the internal quantities exposed for structural readout should also be used to compute the forward rollout\.

The problem is that existing models tend to make a trade\-off at this point\. Methods such as sparse equation discovery\(Brunton et al\.,[2016](https://arxiv.org/html/2606.11251#bib.bib3)\)or local linearization can expose part of the dynamics, but what they expose depends heavily on the measured variables, basis functions, or local approximation scheme chosen before learning\. More flexible recurrent or latent\-state models\(Chen et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib4);Rubanova et al\.,[2019](https://arxiv.org/html/2606.11251#bib.bib32)\)remove much of this manual design, yet the dependencies they use for prediction are often folded into the transition itself, leaving no explicit structural interface\. Graph\-based relational models\(Kipf et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib17)\)are closer in spirit, since they introduce explicit relations between variables, but the graph typically routes messages rather than defining the evolving dynamical object itself\. The gap, then, is not simply the absence of relations\. It is the absence of a forward process in which relation structure, state evolution, and structural readout are trained as one object\.

We introduce the Mechanical Field Network \(MF\-Net\) as a recurrent dynamical model built around this idea\. MF\-Net maintains a joint field state composed of components for individual variables, so that each variable has an internal state within a shared dynamical field\. The term mechanical refers to the way this field is advanced: its components are moved forward by a structured recurrent transition, rather than only decoded from a fixed latent summary\. Coupling is encoded by a single relation law, and this relation shapes how the joint field state evolves during prediction\. The same internal quantities that support structural readout therefore also take part in producing the rollout\. In this way, MF\-Net treats structure as part of the dynamical object being advanced, rather than as a score attached to a completed predictor\(Rudin,[2019](https://arxiv.org/html/2606.11251#bib.bib33)\)\.

To test this idea, we use systems in which different kinds of evidence are available\. Lotka–Volterra and Lorenz–96 provide direct structural checks: the learned relation law can be compared with true interactions or local couplings\. Chaotic rollouts provide a stricter check, since a readable structure should remain organized even as prediction errors accumulate\. We compare MF\-Net with several kinds of references, including equation\-discovery methods, feature\-expansion dynamical models, neural forecasters, and graph\-based baselines, so that prediction accuracy, structural readout, and rollout stability can be evaluated separately\. For real neural and ecological time series, the criterion is necessarily more modest: without ground\-truth mechanisms, we ask whether the learned relations are load\-bearing for prediction and whether they agree with independent functional evidence\. We also include auxiliary and boundary cases to show where MF\-Net forecasts well, and where structural interpretation should be weakened\.

Overall, this work contributes a concrete formulation of structure\-readable recurrent dynamics\. MF\-Net places a learned directed relation inside the forward evolution of a shared field state, so that the relation is not only inspected after training but also used to generate the rollout\. Across controlled interaction systems, chaotic benchmarks, and real neural recordings, this design allows the same model to support forecasting and expose an interpretable relation law\. The central message is that structural readout need not be separated from prediction: it can be made part of the dynamical process that produces the forecast\.

## Methods

Observed historyState initializerCurrent field stateMF transitionblockCurrent influenceAggregated effectsMove field stateShared relation lawFuture field statesReadoutRollout loss

Figure 1:MF\-Net forward computation\. Observed history initializes the current field state\. A shared relation law enters the recurrent MF transition block, where current influences are aggregated and used to move the field state forward\. Predictions are read from future field states, and the rollout loss constrains the same process used for structural readout\.MF\-Net can be summarized as a recurrent field rollout\. A structured transition repeatedly moves the field state forward, and predictions are read from the observed coordinate of the rolled state:

Qt\+1=fθ,D​\(Qt\),𝐳^t\+1=\[Qt\+1\]z\.Q\_\{t\+1\}=f\_\{\\theta,D\}\(Q\_\{t\}\),\\qquad\\hat\{\\mathbf\{z\}\}\_\{t\+1\}=\[Q\_\{t\+1\}\]\_\{z\}\.Herefθ,Df\_\{\\theta,D\}denotes one MF\-Net transition step\. The following sections unpack this transition into its state variables, directed relation, realized flow, field response, mechanical vector field, and training objective\.

### State representation

Letzi​\(t\)z\_\{i\}\(t\)denote the observed coordinate of variableiiat timett\. The semantic meaning of this coordinate depends on datasets; within MF\-Net,zi​\(t\)z\_\{i\}\(t\)is the data\-supervised component of the field state and the coordinate on which rollout prediction is evaluated\.

MF\-Net represents each variable with an augmented field state\. In addition to the observed coordinatezi​\(t\)z\_\{i\}\(t\), the state includes a field coordinateui​\(t\)u\_\{i\}\(t\), an augmented tendency stateηi​\(t\)\\eta\_\{i\}\(t\), and a history contextχi​\(t\)\\chi\_\{i\}\(t\)\. The field coordinateui​\(t\)u\_\{i\}\(t\)gives variableiian internal configuration inside the learned field\. The tendency stateηi​\(t\)\\eta\_\{i\}\(t\)stores the local motion tendency of the field state; its components later enter the updates of both the observed coordinatezi​\(t\)z\_\{i\}\(t\)and the hidden field coordinateui​\(t\)u\_\{i\}\(t\)\. The context stateχi​\(t\)\\chi\_\{i\}\(t\)carries information inferred from recent observations and helps set the current rollout condition\. Thus,zi​\(t\)z\_\{i\}\(t\)anchors the state to data,ui​\(t\)u\_\{i\}\(t\)provides field configuration,ηi​\(t\)\\eta\_\{i\}\(t\)provides motion tendency, andχi​\(t\)\\chi\_\{i\}\(t\)provides recent\-history context\.

We write the full model state of variableiias

qi​\(t\)=\(zi​\(t\),ui​\(t\),ηi​\(t\),χi​\(t\)\),Qt=\{qi​\(t\)\}i=1N\.q\_\{i\}\(t\)=\\bigl\(z\_\{i\}\(t\),u\_\{i\}\(t\),\\eta\_\{i\}\(t\),\\chi\_\{i\}\(t\)\\bigr\),\\qquad Q\_\{t\}=\\\{q\_\{i\}\(t\)\\\}\_\{i=1\}^\{N\}\.Together, these variables define the internal field state on which MF\-Net performs recurrent rollout, with the latent components providing the configuration, motion, and context through which relation structure and field response can shape future trajectories\.

At the beginning of a rollout, the internal components are initialized from a causal history window\. Given past observations𝐳t−L:t\\mathbf\{z\}\_\{t\-L:t\}, MF\-Net encodes the available history into a current state summary and maps it to the internal rollout configuration:

\(u^t,η^t,χ^t\)=Eϕ​\(𝐳t−L:t\)\.\(\\hat\{u\}\_\{t\},\\hat\{\\eta\}\_\{t\},\\hat\{\\chi\}\_\{t\}\)=E\_\{\\phi\}\(\\mathbf\{z\}\_\{t\-L:t\}\)\.This initialization uses only information available up to timett\. After this step, future states are generated by repeatedly applying the MF transition, rather than by directly decoding future values from the history window\.

### Directed relation law and realized flow

MF\-Net separates a stable directed relation law from its state\-dependent realization during rollout\. Let

D∈ℝN×ND\\in\\mathbb\{R\}^\{N\\times N\}denote the learned directed relation matrix, whereDj,iD\_\{j,i\}represents the directed channel from source variablejjto target variableii\. Diagonal self\-edges are masked\. Thus,DDis shared across rollout steps and serves as the stable directed component of the transition\.

At each time step, this stable relation is converted into a realized directed flow\. Each source variable has a scalar source strength

sj​\(t\)=sθ​\(zj​\(t\)\),s\_\{j\}\(t\)=s\_\{\\theta\}\\bigl\(z\_\{j\}\(t\)\\bigr\),or a dataset\-specific source\-strength map with the same role\. The realized flow from sourcejjto targetiiis

Jj→i​\(t\)=Dj,i​sj​\(t\)\.J\_\{j\\to i\}\(t\)=D\_\{j,i\}\\,s\_\{j\}\(t\)\.The matrixJ​\(t\)J\(t\)is therefore state dependent, whileDDis reused across time\. This distinction is important:DDis the relation law that can be inspected after training, whereasJ​\(t\)J\(t\)is the current expression of that law during a particular rollout step\.

The realized flows serve two roles in the transition\. First, their scalar sum enters the observed\-coordinate vector field\. Second, they weight source\-level field messages before target\-side response, as defined below\. Thus, the structural readoutDDis connected directly to the recurrent field motion throughJ​\(t\)J\(t\)\.

### Field aggregation and response

Given the realized flowsJj→i​\(t\)J\_\{j\\to i\}\(t\), MF\-Net computes the input received by each target from source\-level messages\. Each source variable produces a message,

mj​\(t\)=Sθ​\(qj​\(t\)\)\.m\_\{j\}\(t\)=S\_\{\\theta\}\\bigl\(q\_\{j\}\(t\)\\bigr\)\.HereSθS\_\{\\theta\}is a learned map\. The messages arriving at targetiiare aggregated using the realized flows as weights:

hi​\(t\)=∑j≠iJj→i​\(t\)​mj​\(t\)\.h\_\{i\}\(t\)=\\sum\_\{j\\neq i\}J\_\{j\\to i\}\(t\)\\,m\_\{j\}\(t\)\.A target\-side response map then converts this aggregated input into the field response acting on variableii:

Gi​\(t\)=Rθ​\(qi​\(t\)\)​hi​\(t\)\.G\_\{i\}\(t\)=R\_\{\\theta\}\\bigl\(q\_\{i\}\(t\)\\bigr\)\\,h\_\{i\}\(t\)\.HereRθ​\(qi​\(t\)\)R\_\{\\theta\}\(q\_\{i\}\(t\)\)denotes a target\-conditioned response map, andGi​\(t\)G\_\{i\}\(t\)is the field response passed to the mechanical tendency update\. Although this aggregation has a message passing form, it should be read as a graph\-like operation inside the field dynamics rather than as a standalone GNN layer\. The learned relation law provides a structured routing pattern for field messages, while the recurrent state being advanced is still the MF\-Net field state\.

### Mechanical vector field

The field responseGi​\(t\)G\_\{i\}\(t\)shapes the mechanical tendencyηi​\(t\)\\eta\_\{i\}\(t\), which serves as the local motion variable of MF\-Net\. Its components drive both the observed coordinatezi​\(t\)z\_\{i\}\(t\)and the hidden field coordinateui​\(t\)u\_\{i\}\(t\)\. MF\-Net advances the joint field state with the following structured vector field\.

z˙i​\(t\)=ri\+αi​zi​\(t\)\+∑j≠iJj→i​\(t\)\+ηiz​\(t\)\.\\dot\{z\}\_\{i\}\(t\)=r\_\{i\}\+\\alpha\_\{i\}z\_\{i\}\(t\)\+\\sum\_\{j\\neq i\}J\_\{j\\to i\}\(t\)\+\\eta\_\{i\}^\{z\}\(t\)\.
Hererir\_\{i\}andαi\\alpha\_\{i\}are learned variable\-specific local trend parameters\. The observed coordinatezi​\(t\)z\_\{i\}\(t\)evolves through three terms: a variable\-specific local trendri\+αi​zi​\(t\)r\_\{i\}\+\\alpha\_\{i\}z\_\{i\}\(t\), the total realized flow received from other variables, and the observed\-coordinate componentηiz​\(t\)\\eta\_\{i\}^\{z\}\(t\)of the mechanical tendency\. This is the part of the vector field directly tied to the predicted signal\.

u˙i​\(t\)=bθ​\(zi​\(t\),ui​\(t\),χi​\(t\)\)\+ηiu​\(t\)\.\\dot\{u\}\_\{i\}\(t\)=b\_\{\\theta\}\\bigl\(z\_\{i\}\(t\),u\_\{i\}\(t\),\\chi\_\{i\}\(t\)\\bigr\)\+\\eta\_\{i\}^\{u\}\(t\)\.
The hidden field coordinateui​\(t\)u\_\{i\}\(t\)evolves through a local field\-drift mapbθb\_\{\\theta\}and the hidden\-coordinate componentηiu​\(t\)\\eta\_\{i\}^\{u\}\(t\)of the same mechanical tendency\. This lets the internal field state move together with the observed coordinate rather than remaining a static embedding\.

η˙i​\(t\)=λi​\(Gi​\(t\)−ηi​\(t\)\),λi\>0\.\\dot\{\\eta\}\_\{i\}\(t\)=\\lambda\_\{i\}\\bigl\(G\_\{i\}\(t\)\-\\eta\_\{i\}\(t\)\\bigr\),\\qquad\\lambda\_\{i\}\>0\.
The mechanical tendencyηi​\(t\)\\eta\_\{i\}\(t\)is pulled toward the field responseGi​\(t\)G\_\{i\}\(t\)through a relaxation update\. The responseGi​\(t\)G\_\{i\}\(t\)therefore shapes the local motion variable, whose components later enter the dynamics of bothzi​\(t\)z\_\{i\}\(t\)andui​\(t\)u\_\{i\}\(t\)\.

χ˙i​\(t\)=−λχ​χi​\(t\),λχ\>0\.\\dot\{\\chi\}\_\{i\}\(t\)=\-\\lambda\_\{\\chi\}\\chi\_\{i\}\(t\),\\qquad\\lambda\_\{\\chi\}\>0\.
The context stateχi​\(t\)\\chi\_\{i\}\(t\)decays during rollout\. It provides history information for the current rollout condition, while the future trajectory is still generated by recurrent field dynamics rather than by directly copying the input history\.

Together, these equations define the MF\-Net transition used during rollout\. In experiments, the vector field is integrated step by step to obtain multi\-horizon predictions\. The numerical integrator is an implementation choice; the structural definition of MF\-Net is the vector field above\.

### Rollout and training objective

Starting from the initialized stateQtQ\_\{t\}, MF\-Net repeatedly applies the mechanical transition to obtain future field states:

Qt\+1:t\+H=Rollout​\(Qt,D;θ\)\.Q\_\{t\+1:t\+H\}=\\mathrm\{Rollout\}\(Q\_\{t\},D;\\theta\)\.Predictions are read from the observed coordinates of the rolled states:

z^i​\(t\+k\)=\[zi​\(t\+k\)\]roll,k=1,…,H\.\\hat\{z\}\_\{i\}\(t\+k\)=\[z\_\{i\}\(t\+k\)\]\_\{\\mathrm\{roll\}\},\\qquad k=1,\\ldots,H\.The training objective is a multi\-horizon rollout loss,

ℒroll=∑k=1Hwk​ℓ​\(𝐳^​\(t\+k\),𝐳​\(t\+k\)\),\\mathcal\{L\}\_\{\\mathrm\{roll\}\}=\\sum\_\{k=1\}^\{H\}w\_\{k\}\\,\\ell\\\!\\left\(\\hat\{\\mathbf\{z\}\}\(t\+k\),\\mathbf\{z\}\(t\+k\)\\right\),whereℓ\\ellis a prediction loss such as squared error, andwkw\_\{k\}controls the relative weight of different horizons\. The key point is that future values are produced by repeated application of the same MF transition\. The encoder initializes the current state, but it does not directly output future trajectories\.

### Pseudocode

Algorithm 1\. MF\-Net rollout and training with fixed\-step Heun integration

1:Normalized series

𝐳0:T\\mathbf\{z\}\_\{0:T\}, history length

LL, horizons

ℋ\\mathcal\{H\}, step size

Δ​t\\Delta t
2:Encoder

EϕE\_\{\\phi\}, relation parameter

DparamD\_\{\\mathrm\{param\}\}, source\-strength map

sθs\_\{\\theta\}
3:Source emitter

SθS\_\{\\theta\}, reception map

RθR\_\{\\theta\}, local field\-drift map

bθb\_\{\\theta\}
4:Mask diagonal self\-edges:

5:

D←Dparam⊙\(1−I\)D\\leftarrow D\_\{\\mathrm\{param\}\}\\odot\(1\-I\)
6:foreach minibatch of causal prediction origins

ttdo

7:Form the causal history window

𝐳t−L:t\\mathbf\{z\}\_\{t\-L:t\}\.

8:Initialize hidden field components from history:

9:

\(ut0,ηt0,χt0\)←Eϕ​\(𝐳t−L:t\)\(u\_\{t\}^\{0\},\\eta\_\{t\}^\{0\},\\chi\_\{t\}^\{0\}\)\\leftarrow E\_\{\\phi\}\(\\mathbf\{z\}\_\{t\-L:t\}\)
10:Set the initial rollout state:

11:

Q0←\{qi0=\(zi​\(t\),ui0,ηi0,χi0\)\}i=1NQ^\{0\}\\leftarrow\\\{q\_\{i\}^\{0\}=\(z\_\{i\}\(t\),u\_\{i\}^\{0\},\\eta\_\{i\}^\{0\},\\chi\_\{i\}^\{0\}\)\\\}\_\{i=1\}^\{N\}
12:for

k=1,…,max⁡\(ℋ\)k=1,\\ldots,\\max\(\\mathcal\{H\}\)do

13:Compute first vector\-field estimate:

14:

K1←VectorField​\(Qk−1,D\)K\_\{1\}\\leftarrow\\textsc\{VectorField\}\(Q^\{k\-1\},D\)
15:Take a provisional Euler step:

16:

Q~←Qk−1\+Δ​t​K1\\widetilde\{Q\}\\leftarrow Q^\{k\-1\}\+\\Delta t\\,K\_\{1\}
17:Compute second vector\-field estimate:

18:

K2←VectorField​\(Q~,D\)K\_\{2\}\\leftarrow\\textsc\{VectorField\}\(\\widetilde\{Q\},D\)
19:Update by Heun integration:

20:

Qk←Qk−1\+Δ​t2​\(K1\+K2\)Q^\{k\}\\leftarrow Q^\{k\-1\}\+\\frac\{\\Delta t\}\{2\}\(K\_\{1\}\+K\_\{2\}\)
21:if

k∈ℋk\\in\\mathcal\{H\}then

22:Read prediction from the observed coordinate:

23:

𝐳^t\+k←\[Qk\]z\\hat\{\\mathbf\{z\}\}\_\{t\+k\}\\leftarrow\[Q^\{k\}\]\_\{z\}
24:endif

25:endfor

26:Compute multi\-horizon rollout loss:

27:

ℒroll←∑h∈ℋwh​ℓ​\(𝐳^t\+h,𝐳t\+h\)\\mathcal\{L\}\_\{\\mathrm\{roll\}\}\\leftarrow\\sum\_\{h\\in\\mathcal\{H\}\}w\_\{h\}\\,\\ell\(\\hat\{\\mathbf\{z\}\}\_\{t\+h\},\\mathbf\{z\}\_\{t\+h\}\)
28:Add regularization:

29:

ℒ←ℒroll\+ℒreg\\mathcal\{L\}\\leftarrow\\mathcal\{L\}\_\{\\mathrm\{roll\}\}\+\\mathcal\{L\}\_\{\\mathrm\{reg\}\}
30:Update

θ,ϕ,Dparam\\theta,\\phi,D\_\{\\mathrm\{param\}\}by backpropagation through the full rollout\.

31:endfor

Algorithm 2\. MF\-Net vector\-field evaluation

1:procedureVectorField\(

Q,DQ,D\)

2:Unpack the field state:

3:

Q=\{qi=\(zi,ui,ηi,χi\)\}i=1NQ=\\\{q\_\{i\}=\(z\_\{i\},u\_\{i\},\\eta\_\{i\},\\chi\_\{i\}\)\\\}\_\{i=1\}^\{N\}
4:foreach source variable

jjdo

5:Compute source strength:

6:

sj←sθ​\(zj\)s\_\{j\}\\leftarrow s\_\{\\theta\}\(z\_\{j\}\)
7:Compute source field message:

8:

mj←Sθ​\(qj\)m\_\{j\}\\leftarrow S\_\{\\theta\}\(q\_\{j\}\)
9:endfor

10:foreach directed pair

j→ij\\to i,

j≠ij\\neq ido

11:Compute realized flow:

12:

Jj→i←Dj,i​sjJ\_\{j\\to i\}\\leftarrow D\_\{j,i\}\\,s\_\{j\}
13:endfor

14:foreach target variable

iido

15:Aggregate incoming source messages:

16:

hi←∑j≠iJj→i​mjh\_\{i\}\\leftarrow\\sum\_\{j\\neq i\}J\_\{j\\to i\}\\,m\_\{j\}
17:Compute target\-side field response:

18:

Gi←Rθ​\(qi\)​hiG\_\{i\}\\leftarrow R\_\{\\theta\}\(q\_\{i\}\)\\,h\_\{i\}
19:Compute observed\-coordinate velocity:

20:

z˙i←ri\+αi​zi\+∑j≠iJj→i\+ηiz\\dot\{z\}\_\{i\}\\leftarrow r\_\{i\}\+\\alpha\_\{i\}z\_\{i\}\+\\sum\_\{j\\neq i\}J\_\{j\\to i\}\+\\eta\_\{i\}^\{z\}
21:Compute hidden field\-coordinate velocity:

22:

u˙i←bθ​\(zi,ui,χi\)\+ηiu\\dot\{u\}\_\{i\}\\leftarrow b\_\{\\theta\}\(z\_\{i\},u\_\{i\},\\chi\_\{i\}\)\+\\eta\_\{i\}^\{u\}
23:Compute tendency velocity:

24:

η˙i←λη,i​\(Gi−ηi\)\\dot\{\\eta\}\_\{i\}\\leftarrow\\lambda\_\{\\eta,i\}\(G\_\{i\}\-\\eta\_\{i\}\)
25:Compute context velocity:

26:

χ˙i←−λχ​χi\\dot\{\\chi\}\_\{i\}\\leftarrow\-\\lambda\_\{\\chi\}\\chi\_\{i\}
27:endfor

28:return

Q˙=\{\(z˙i,u˙i,η˙i,χ˙i\)\}i=1N\\dot\{Q\}=\\\{\(\\dot\{z\}\_\{i\},\\dot\{u\}\_\{i\},\\dot\{\\eta\}\_\{i\},\\dot\{\\chi\}\_\{i\}\)\\\}\_\{i=1\}^\{N\}
29:endprocedure

### Structural readout

The main structural readout of MF\-Net is the learned relation lawDD\. This readout is part of the forward transition:DDis used to form the realized flowJ​\(t\)J\(t\), which then enters field aggregation, field response, the tendency update, and the recurrent rollout\. Thus, structural inspection is performed on a quantity that participates in the model dynamics, rather than on a diagnostic fitted after prediction\.

The interpretation ofDDdepends on the available evidence\. In systems with known interaction or coupling structure,DDcan be compared with the corresponding ground truth\. In real neural or ecological time series, where direct mechanisms are not observed,DDis treated as an inspectable functional coupling signal rather than as direct causal recovery\. Metrics and analyses based on datasets are described in the Results section\.

## Results

### MF\-Net recovers interpretable structure in identifiable and chaotic dynamics

We first evaluated MF\-Net in controlled systems with known dynamical structure\. Lotka–Volterra\(Lotka,[1925](https://arxiv.org/html/2606.11251#bib.bib21);Volterra,[1926](https://arxiv.org/html/2606.11251#bib.bib37)\)provides a direct interaction benchmark, since the true pairwise interaction matrix is part of the data\-generating model\. Lorenz–96\(Lorenz,[1996](https://arxiv.org/html/2606.11251#bib.bib20)\)provides a complementary chaotic benchmark, where the trajectories are harder to roll out but the local coupling pattern is still known\. Together, the two systems test whether the learned relation lawDDremains structurally readable across both identifiable pairwise interactions and chaotic local coupling systems\.

On the Lotka–Volterra benchmark, MF\-Net can recover the interaction structure from observed trajectories\. Across five runs, the off\-diagonal entries of the aligned learned relation matrix closely matched the true interaction matrix, with Pearson=0\.9937=0\.9937, Spearman=0\.9831=0\.9831, scaled RMSE=0\.00237=0\.00237, and sign accuracy=1\.000=1\.000\. Orientation alignment and scalar calibration were applied only after training for reporting evaluation metrics\. The scalar calibration fits a single global multiplier between the learned off\-diagonal matrix and the true interaction matrix; it does not enter the training objective, perform elementwise fitting, or alter the learned sign pattern, ranking, or support\. These results indicate that MF\-Net can infer a stable relation law without direct supervision on the interaction matrix\. The same runs also achieved near\-perfect short\-horizon rollout, withh=4h=4R2=0\.999960±0\.000005R^\{2\}=0\.999960\\pm 0\.000005\.

Table 1:Structural recovery on Lorenz–96 withN=40N=40\. The local/nonlocal ratio is the mean absolute relation strength on true local couplings divided by that on nonlocal pairs\. Precision is evaluated at the true edge budgetK=3​N=120K=3N=120\. MF\-Net is reported as mean±\\pmstd over five seeds\.Lorenz–96 provides a chaotic benchmark with a known directed local\-neighbor coupling pattern\. We evaluated structural recovery using the local/nonlocal strength ratio and Precision@KK, where the topKKdirected edges are selected by absolute relation strength andKKis matched to the number of true local couplings \(K=3​N=120K=3N=120for the 40\-dimensional Lorenz–96 system; Table[1](https://arxiv.org/html/2606.11251#S3.T1)\)\. Across five seeds, MF\-Net achieved a local/nonlocal ratio of19\.80±1\.0019\.80\\pm 1\.00and Precision@KKof1\.000±0\.0001\.000\\pm 0\.000\. These results show that MF\-Net recovers the Lorenz–96 local coupling support with perfect Precision@KK, while retaining a stable directed relation law that is used in its recurrent rollout\. The much larger SINDy\-poly2 ratio should be interpreted in light of its model\-matched quadratic library: Lorenz–96 is generated by a quadratic equation, so SINDy\-poly2 has access to the correct symbolic hypothesis class\. By contrast, MF\-Net recovers the same local support through a learned relation law embedded in the recurrent rollout, rather than through a hand\-specified polynomial library\.

![Refer to caption](https://arxiv.org/html/2606.11251v1/L96LV.png)Figure 2:Core structural recovery results\. \(a\) Lotka–Volterra: learned relation lawDDand true interaction matrixAA\. \(b\) Lorenz–96: learned relation map, with black outlines marking the known local\-neighbor couplings\. \(c\) Top\-KKsupport of\|D\|\|D\|, whereKKis matched to the number of true local couplings \(K=3​NK=3N\), showing that the strongest learned relations recover the known local\-coupling structure\. Panels \(b\) and \(c\) show the five\-seed mean learnedDDmatrix for Lorenz–96N=40N=40\.Table 2:Forecasting performance on Lorenz–96 withN=40N=40\. MF\-Net is reported as mean±\\pmstd over five seeds\.*Note\.*SINDy\-poly3 and persistence are nearly identical after rounding to three decimals, but their unrounded RMSE values differ slightly, e\.g\.0\.9383030\.938303versus0\.9382860\.938286ath=4h=4\.

The Lorenz–96 forecasting results compare MF\-Net with sparse equation\-discovery, local and linear dynamical, feature\-expansion, neural forecasting, and structure\-learning baselines\(Brunton et al\.,[2016](https://arxiv.org/html/2606.11251#bib.bib3);Gauthier et al\.,[2021](https://arxiv.org/html/2606.11251#bib.bib8);Tank et al\.,[2022](https://arxiv.org/html/2606.11251#bib.bib36);Kipf et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib17);Nie et al\.,[2023](https://arxiv.org/html/2606.11251#bib.bib25);Liu et al\.,[2024](https://arxiv.org/html/2606.11251#bib.bib18);Luo and Wang,[2024](https://arxiv.org/html/2606.11251#bib.bib22);Oreshkin et al\.,[2020](https://arxiv.org/html/2606.11251#bib.bib28);Chen et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib4);Rubanova et al\.,[2019](https://arxiv.org/html/2606.11251#bib.bib32)\)and show strong open\-loop performance on the 40\-dimensional chaotic benchmark \(Table[2](https://arxiv.org/html/2606.11251#S3.T2)\)\. SINDy\-poly2 remains a very strong short\- and medium\-horizon baseline, which is expected because the Lorenz–96 equation is quadratic and therefore matches the polynomial library\. However, its long\-horizon rollout deteriorates sharply ath=24h=24\. In contrast, SINDy\-poly3 collapses to persistence\-level rollout performance, showing that a larger symbolic library can make the discovered dynamics less stable rather than more accurate\. MF\-Net achieves competitive short\- and medium\-horizon accuracy and retains the best long\-horizon performance among the tested methods\. This result is important because MF\-Net does not obtain its forecasts from a black\-box terminal decoder alone: the same recurrent rollout also exposes a stable directed relation law that recovers the known local\-neighbor structure\. Thus, Lorenz–96 supports the central tradeoff of MF\-Net: structure\-readable chaotic forecasting with the learned structure remaining part of the forward computation\.

Together, these two benchmarks show that MF\-Net learns a directed matrixDDthat is used during rollout and can still be read after training\. In Lotka–Volterra,DDclosely matches the true pairwise interaction matrix\. In Lorenz–96, the learned relation matrix recovers the known local coupling support at the matched edge budget\. MF\-Net also gives the strongest overall rollout performance among the tested neural baselines and retains positive long\-horizon skill ath=24h=24\. Thus, the same learned matrix supports prediction and provides a clear summary of directed structure when the observed system supports a stable coupling law\.

### Ablations show that the mechanical field path contributes to rollout

A readable directed structure carries more weight when it shapes the model’s prediction than when it is recovered only after training\. We therefore ran ablations to test whether MF\-Net depends on its mechanical field path\. On Lorenz–96, shuffling the field history caused the largest drop in accuracy, which shows that the learned field state cannot be replaced by raw observation history alone\. ShufflingDD, settingDDto zero, or removing theJ/G/ηJ/G/\\etapathway also increased rollout error \(Table[3](https://arxiv.org/html/2606.11251#S3.T3)\)\. The field history, the directed matrixDD, and the relation\-to\-motion variables are therefore functional parts of the rollout rather than quantities read off after the fact\.

Table 3:Ablation effects on Lorenz–96 withN=40N=40\. Values are RMSE ratios relative to the full MF\-Net rollout\.The ablations connect the structural results to the forecasting mechanism\. The largest degradation comes from destroying the learned field history, showing that the internal field state carries information beyond raw observation history\. Perturbing the directed relation matrix also substantially increases error, especially for medium\-horizon rollout\. Removing theJ/G/ηJ/G/\\etapathway has a smaller but still visible effect, indicating that the realized\-flow and tendency variables contribute to the transition rather than serving as passive diagnostics\. Together, these results show that MF\-Net uses its learned directed structure during prediction\.

### MF\-Net remains informative on real\-world data

We next tested MF\-Net on real non\-pharyngealC\. eleganscalcium recordings\(Atanas et al\.,[2023](https://arxiv.org/html/2606.11251#bib.bib1);WormWideWeb,[2023](https://arxiv.org/html/2606.11251#bib.bib39)\)\. For this dataset, we evaluated both forecasting performance and the learned directed matrixDD\. The forecasting experiment tests whether MF\-Net can model neural activity in open loop, while the analysis ofDDasks whether the learned matrix contains consistent directed temporal information across recordings\.

The forecasting results are shown in Table[4](https://arxiv.org/html/2606.11251#S3.T4)\(Zeng et al\.,[2023](https://arxiv.org/html/2606.11251#bib.bib42);Rubanova et al\.,[2019](https://arxiv.org/html/2606.11251#bib.bib32);Liu et al\.,[2024](https://arxiv.org/html/2606.11251#bib.bib18);Wu et al\.,[2020](https://arxiv.org/html/2606.11251#bib.bib40),[2019](https://arxiv.org/html/2606.11251#bib.bib41);Hochreiter and Schmidhuber,[1997](https://arxiv.org/html/2606.11251#bib.bib13);Gauthier et al\.,[2021](https://arxiv.org/html/2606.11251#bib.bib8)\)\. MF\-Net does not consistently beat the strongest linear baselines, which are well suited to the smooth and slowly varying structure of calcium activity\. Nevertheless, MF\-Net remains very close to these baselines at the four\-step horizon and achieves the best result at the sixteen\-step horizon\. It also outperforms the tested recurrent, latent, graph\-based, and transformer\-style neural baselines at both reported horizons\. Thus, the subsequent analysis ofDDis based on a model that is already predictive on real neural recordings\.

Table 4:Open\-loop forecasting on non\-pharyngealC\. eleganscalcium recordings\.After evaluating forecasting performance, we examined the learned directed matrixDDin the same neural recordings\. For each directed pairj→ij\\to i, we measured a future calcium effect by comparing the source activityzj​\(t\)z\_\{j\}\(t\)with the target changezi​\(t\+h\)−zi​\(t\)z\_\{i\}\(t\+h\)\-z\_\{i\}\(t\)\. We then tested whether entries ofDDalign with these source\-to\-target temporal effects across horizons and recordings\. This comparison provides a direct test of whetherDDreflects directed temporal effects in neural activity\.

![Refer to caption](https://arxiv.org/html/2606.11251v1/function.png)Figure 3:Directed temporal evidence in non\-pharyngealC\. eleganscalcium recordings\. \(a\) The learned matrixDDaligns with future source\-to\-target calcium effects\. For each directed pairj→ij\\to i, the future effect is defined asEh​\[j,i\]=partial​corr​\(zj​\(t\),zi​\(t\+h\)−zi​\(t\)∣zi​\(t\),z¯​\(t\)\)E\_\{h\}\[j,i\]=\\mathrm\{partial\\ corr\}\(z\_\{j\}\(t\),z\_\{i\}\(t\+h\)\-z\_\{i\}\(t\)\\mid z\_\{i\}\(t\),\\bar\{z\}\(t\)\), wherez¯​\(t\)\\bar\{z\}\(t\)is the global mean activity\. Curves show the alignment between the off\-diagonal entries ofDDandEhE\_\{h\}across horizons, measured by Pearson correlation and sign agreement\. \(b\) Directional asymmetry comparesDj,i−Di,jD\_\{j,i\}\-D\_\{i,j\}with the corresponding asymmetryEh​\[j,i\]−Eh​\[i,j\]E\_\{h\}\[j,i\]\-E\_\{h\}\[i,j\]\. \(c\) Recording\-level consistency of the same alignment analysis\. Each point denotes one recording, and the black line shows the mean across recordings\.Figure[3](https://arxiv.org/html/2606.11251#S3.F3)summarizes this analysis\. The learned matrixDDshowed positive alignment with future calcium effects across all tested horizons\. After controlling for the target self state and global mean activity, the partial Pearson correlation increased from0\.3050\.305ath=1h=1to0\.4100\.410ath=8h=8, and remained positive at longer horizons\. Sign agreement followed the same pattern, reaching0\.6450\.645ath=8h=8and staying above0\.590\.59ath=24h=24\. These values were well above the instantaneous coactivity baseline, whose signed correlation withDDwas only0\.0770\.077\. The directionality ofDDwas also consistent with the data: the asymmetryDj,i−Di,jD\_\{j,i\}\-D\_\{i,j\}aligned with the corresponding asymmetry of future calcium effects, with correlations around0\.560\.56–0\.580\.58across horizons\. Finally, the alignment was positive in all six recordings, showing that the signal was not driven by a single recording\. Together, these results indicate thatDDreflects directed temporal structure in calcium dynamics, rather than simple coactivity measured at the same time\.

Overall, theC\. elegansresults provide a real\-data test of the MF\-Net mechanism\. The model gives strong forecasts of neural activity, and the learned matrixDDshows a consistent relation to directed future changes in the same recordings\. We therefore readDDas a functional predictive coupling matrix for calcium activity: it summarizes which source neurons are associated with later changes in target neurons after controlling for target self state and global activity\. We do not use this analysis to infer anatomical connectivity\.

### Additional dynamical probes support broader applicability

We finally used several auxiliary systems to test whether the rollout remains effective beyond the main structural benchmarks and theC\. elegansrecordings\. These experiments are secondary evidence: they do not replace the known\-structure tests above, but they check whether the same MF\-Net formulation can handle periodic, excitable, oscillator, and small real consumer\-resource dynamics\.

Table 5:Additional dynamical probes under rollout\. Values are mean±\\pmstd over three runs\.MF\-Net retained accurate long\-horizon rollout on all four additional probes \(Table[5](https://arxiv.org/html/2606.11251#S3.T5)\)\. The SinCos result shows that the model can preserve phase structure over long horizons\. The FitzHugh–Nagumo result\(FitzHugh,[1961](https://arxiv.org/html/2606.11251#bib.bib7);Nagumo et al\.,[1962](https://arxiv.org/html/2606.11251#bib.bib24)\)shows that the same rollout remains accurate on excitable dynamics with hidden recovery variables\. The Sakaguchi–Kuramoto result\(Sakaguchi and Kuramoto,[1986](https://arxiv.org/html/2606.11251#bib.bib34)\)is nearly saturated, so we treat it as a regular coupled\-oscillator sanity check rather than as central structural evidence\. The NRI\-Springs\(Kipf et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib17)\)result provides an additional rollout probe from the neural relational inference tradition: MF\-Net learns accurate spring\-particle dynamics, but we do not use this result as evidence of direct recovery of the undirected spring graph\.

We also tested a small real two\-species algae–rotifer chemostat system\(Blasius et al\.,[2020](https://arxiv.org/html/2606.11251#bib.bib2)\)\. This dataset is too small to support strong mechanism recovery, but it provides a useful sign check for the learned directed matrix\. Across three seeds, the learned relation from algae to rotifers was positive in every run, while the relation from rotifers to algae was negative in every run \(Table[6](https://arxiv.org/html/2606.11251#S3.T6)\)\. This matches the expected consumer\-resource direction: resource availability supports consumer growth, while consumers suppress the resource\. We therefore treat this result as sign\-stable functional predictive coupling, not as causal mechanism recovery\.

Table 6:Three\-seed direction check on the real algae–rotifer system\.
### History\-only wind power as a boundary case

We also tested MF\-Net on the SDWPF wind\-power dataset\(Zhou et al\.,[2024](https://arxiv.org/html/2606.11251#bib.bib45)\)in a history\-only setting, where the model receives only past turbine power traces without wind speed, wind direction, weather fields, or turbine\-layout features\. As shown in Figure[4](https://arxiv.org/html/2606.11251#S3.F4), MF\-Net improves over persistence at medium and long horizons, fromR2=0\.148R^\{2\}=0\.148to0\.2880\.288ath=12h=12and fromR2=−0\.575R^\{2\}=\-0\.575to−0\.013\-0\.013ath=24h=24\. However, the learned relation matrix remains weak and does not align with physical turbine distance: the off\-diagonal mean\|D\|\|D\|is0\.005100\.00510, the\|D\|\|D\|–distance Pearson correlation is only0\.1610\.161, and the top\-3​N3Ndistance ratio is0\.99950\.9995\. Moreover,DD\-shuffle and zero\-DDablations cause little damage ath=24h=24, with RMSE ratios of1\.0421\.042and1\.0101\.010\. This result shows that MF\-Net can extract useful history\-driven temporal dynamics while showing that the predictive gain does not come from an interpretable turbine\-to\-turbine relation law\.

![Refer to caption](https://arxiv.org/html/2606.11251v1/wind.png)Figure 4:History\-only wind\-power boundary case\. MF\-Net improves medium\- and long\-horizon forecasting over persistence, but the learned relation matrix remains weak and does not align with physical turbine distance\. Perturbing or removingDDcauses little damage, indicating that the predictive gain mainly comes from history\-driven latent dynamics rather than an interpretable turbine\-to\-turbine relation law\.Together, these auxiliary and boundary experiments support the broader behavior of MF\-Net without changing the main evidence hierarchy of the paper\. The central claims still rest on known\-system structure recovery, ablation of the mechanical field path, and theC\. elegansdirected temporal analysis\. The additional probes show that the same rollout formulation remains accurate on several simpler dynamical settings and gives sign\-stable functional readout in a small real biological system\. The wind\-power boundary case further shows that useful forecasting does not by itself justify a strong structural interpretation ofDD\.

## Discussion

### Key Contribution

Structurelearned directed matrixDDMotionfield update throughJ,G,ηJ,G,\\etaPredictionreadout from rolled statesMF\-Net organizes forecasting as structured field motion, rather than a direct history\-to\-future map\.Figure 5:The central modeling view of MF\-Net\. A learned directed structure shapes field motion during rollout, and predictions are read from the evolved field state\.The central contribution of MF\-Net is a modeling view for multivariate forecasting\. MF\-Net organizes prediction through a recurrent field state and a directed mechanical transition\. The matrixDDsupplies a persistent directed component of the transition; the realized effects, field responses, and tendency variables specify how this component is expressed in the current state and how the field moves forward\. The specific equations instantiate this view by forcing the transition to pass through stable structure, current effect, and motion before producing a forecast\. This makes the directed component functional and inspectable: it contributes to the rollout and remains available for analysis after training\.

### Relation to Modern Dynamical and Relational Models

MF\-Net lies at the intersection of several recent lines of work\. Neural differential equation models learn continuous or latent state evolution through vector fields such asQ˙=fθ​\(Q,t\)\\dot\{Q\}=f\_\{\\theta\}\(Q,t\)\(Chen et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib4);Rubanova et al\.,[2019](https://arxiv.org/html/2606.11251#bib.bib32);Kidger et al\.,[2020](https://arxiv.org/html/2606.11251#bib.bib16);Oh et al\.,[2025](https://arxiv.org/html/2606.11251#bib.bib27)\)\. Augmented and second\-order Neural ODEs expand this view by adding hidden dimensions or motion\-like state structure\(Dupont et al\.,[2019](https://arxiv.org/html/2606.11251#bib.bib6);Norcliffe et al\.,[2020](https://arxiv.org/html/2606.11251#bib.bib26)\), while Hamiltonian, Lagrangian, and symplectic neural models impose stronger physical or geometric inductive biases\(Greydanus et al\.,[2019](https://arxiv.org/html/2606.11251#bib.bib10);Cranmer et al\.,[2020](https://arxiv.org/html/2606.11251#bib.bib5);Zhong et al\.,[2020](https://arxiv.org/html/2606.11251#bib.bib44)\)\. These methods are close to MF\-Net in their concern with learned state evolution, but they usually do not expose a stable directed relation law between observed variables\.

MF\-Net is also related to graph neural differential equations, dynamic graph neural networks, graph neural networks for time series, and neural relational inference\(Poli et al\.,[2019](https://arxiv.org/html/2606.11251#bib.bib30);Liu et al\.,[2025](https://arxiv.org/html/2606.11251#bib.bib19);Jin et al\.,[2024](https://arxiv.org/html/2606.11251#bib.bib14);Zheng et al\.,[2024](https://arxiv.org/html/2606.11251#bib.bib43);Kipf et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib17)\)\. These methods combine graph structure with temporal or dynamical modeling, and recent relational\-inference work continues to study how hidden interaction graphs can be identified from observed trajectories\(Pan et al\.,[2024](https://arxiv.org/html/2606.11251#bib.bib29);Kang et al\.,[2024](https://arxiv.org/html/2606.11251#bib.bib15)\)\. The difference is the role assigned to the learned relation\. In many graph\-dynamical models, edges primarily route messages or parameterize a graph decoder\. In MF\-Net, the directed matrixDDis a stable relation law inside the transition: it is realized as state\-dependent flow, converted into field response and motion tendency, and reused across rollout steps\.

Recent work on neural structure learning and dynamical causal discovery also highlights the difficulty of inferring structure from temporal processes under continuous\-time dynamics, noise, confounding, and lagged effects\(Wang et al\.,[2024](https://arxiv.org/html/2606.11251#bib.bib38);Herdeanu et al\.,[2025](https://arxiv.org/html/2606.11251#bib.bib12)\)\. MF\-Net does not claim to solve general causal discovery from observations\. Instead, it contributes a structure\-readable rollout framework: when the observed system supports a stable and load\-bearing relation structure,DDcan be compared with known interactions or coupling support; when the data are partially observed, externally driven, or dominated by shared latent dynamics,DDis interpreted more weakly as functional predictive coupling\.

### From stable structure to realized motion

The important separation in MF\-Net is between a persistent directed channel and its current expression during rollout\. The matrixDDstores the relation component that is reused across time steps\. The realized flowJ​\(t\)J\(t\)activates this relation under the current state, the field responseG​\(t\)G\(t\)collects the resulting target side effects, and the tendency variableη​\(t\)\\eta\(t\)converts the response into motion of the field state\. These quantities therefore sit at different levels of the transition:DDdescribes a stable coupling channel, whileJ​\(t\)J\(t\),G​\(t\)G\(t\), andη​\(t\)\\eta\(t\)describe how that channel is expressed at a particular moment\. This separation is what gives the equations their role in the model, and it is also why the same directed component can be inspected after training\.

### A factorized transition view

At a higher level, MF\-Net can be viewed as a structured factorization of the finite\-time prediction operator\. A generic forecaster learns a map from an observed history window to future observations\. MF\-Net instead factors this map through a current field state and a recurrent transition:

𝐳t−L:t↦Qt,Qt\+k\+1=Φθ,D​\(Qt\+k\),𝐳^t\+k=Rθ​\(Qt\+k\)\.\\mathbf\{z\}\_\{t\-L:t\}\\mapsto Q\_\{t\},\\qquad Q\_\{t\+k\+1\}=\\Phi\_\{\\theta,D\}\(Q\_\{t\+k\}\),\\qquad\\hat\{\\mathbf\{z\}\}\_\{t\+k\}=R\_\{\\theta\}\(Q\_\{t\+k\}\)\.The transitionΦθ,D\\Phi\_\{\\theta,D\}is expressed through the stable directed channel and its state\-dependent motion variables described above\. The specific parameterization ofJJ,GG, orη\\etacan be changed, but the modeling principle remains the same: future states are generated by repeatedly moving a structured field state, and predictions are read from the rolled states\. This view explains why the contribution of MF\-Net lies in the organization of the transition, rather than in any single intermediate formula\.

### Limiting cases of the MF\-Net transition

The MF\-Net transition also has useful limiting cases\. Here a limiting case means a controlled simplification of the transition: some internal variables are suppressed, some maps are made linear, or the structured mechanical path is absorbed into a more generic state update\. These limits do not imply that the corresponding models are special cases of the implemented code in every detail\. They clarify what assumptions MF\-Net adds beyond several familiar modeling views\.

The first limit is the residual or recurrent state\-update view\. With a fixed step size, the rollout can be written abstractly as

Qt\+1=Qt\+Δ​t​Fθ,D​\(Qt\)\.Q\_\{t\+1\}=Q\_\{t\}\+\\Delta t\\,F\_\{\\theta,D\}\(Q\_\{t\}\)\.Thus MF\-Net belongs to the recurrent state\-evolution family: future states are produced by repeatedly updating a current state\. This connects the model to the residual/Euler view of deep networks\(He et al\.,[2016](https://arxiv.org/html/2606.11251#bib.bib11)\)and to recurrent state\-space models\(Rangapuram et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib31)\)\. The difference is that the update fieldFθ,DF\_\{\\theta,D\}is structured\. It is not an unrestricted transition cell; it is expressed through a persistent directed matrix, realized flows, field responses, and motion tendencies\.

A second limit comes from local linearization\. For a smooth one\-step transitionzt\+1=Φ​\(zt\)z\_\{t\+1\}=\\Phi\(z\_\{t\}\), a first\-order Taylor expansion around the current state gives a local linear approximation of the form

Δ​zi​\(t\)≈ai​\(t\)\+∑jAi​j​\(t\)​zj​\(t\),\\Delta z\_\{i\}\(t\)\\approx a\_\{i\}\(t\)\+\\sum\_\{j\}A\_\{ij\}\(t\)z\_\{j\}\(t\),whereAi​j​\(t\)A\_\{ij\}\(t\)acts as a local effective coupling from variablejjto variableii\. If this local coefficient is fixed over time, the update becomes VAR\-like\(Lütkepohl,[2005](https://arxiv.org/html/2606.11251#bib.bib23)\):

zt\+1=c\+B​zt\.z\_\{t\+1\}=c\+Bz\_\{t\}\.MF\-Net approaches this limit when the hidden field variables are suppressed, the source strength is linear inzj​\(t\)z\_\{j\}\(t\), and the realized directed flow is reduced to

Jj→i​\(t\)=Dj,i​zj​\(t\)\.J\_\{j\\to i\}\(t\)=D\_\{j,i\}z\_\{j\}\(t\)\.The observed\-coordinate update then becomes

z˙i​\(t\)=ri\+αi​zi​\(t\)\+∑j≠iDj,i​zj​\(t\),\\dot\{z\}\_\{i\}\(t\)=r\_\{i\}\+\\alpha\_\{i\}z\_\{i\}\(t\)\+\\sum\_\{j\\neq i\}D\_\{j,i\}z\_\{j\}\(t\),up to the source\-target convention used forDD\. In this limit,DDplays the role of a stable interaction coefficient\. The full MF\-Net transition generalizes this local\-linear view by allowing the directed component to be expressed through state\-dependent realized flows, target\-side field responses, and motion tendencies\.

A third limit connects MF\-Net to latent neural dynamics\. If the directed relation channel and the mechanical variables are absorbed into an unrestricted vector field, the state equation becomes

Q˙=fθ​\(Q,t\),\\dot\{Q\}=f\_\{\\theta\}\(Q,t\),with predictions read fromQQ\. This is the modeling form used by Neural ODE and Latent ODE style models\(Chen et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib4);Rubanova et al\.,[2019](https://arxiv.org/html/2606.11251#bib.bib32)\)\. MF\-Net keeps the state\-evolution view of these models, but restricts the transition to pass through a directed mechanical path\. The latent state is therefore not only moved by a generic learned vector field; its motion is organized through a stable directed component and its current realization\.

A fourth limit connects MF\-Net to relational graph models such as NRI\(Kipf et al\.,[2018](https://arxiv.org/html/2606.11251#bib.bib17)\)\. If the mechanical tendency and field\-motion variables are removed, and the learned relation is used only as an adjacency\-like routing weight, the update becomes

hi​\(t\)=∑j≠iDj,i​mj​\(t\),qi​\(t\+1\)=gθ​\(qi​\(t\),hi​\(t\)\)\.h\_\{i\}\(t\)=\\sum\_\{j\\neq i\}D\_\{j,i\}m\_\{j\}\(t\),\\qquad q\_\{i\}\(t\+1\)=g\_\{\\theta\}\(q\_\{i\}\(t\),h\_\{i\}\(t\)\)\.This is a graph message\-passing transition\. MF\-Net retains the relational idea, but assigns the directed relation a different role\. The relation does not only route messages; it is converted into realized flow, field response, and motion tendency before the state is advanced\.

These limits clarify the position of MF\-Net\. Compared with VAR or local Taylor models, MF\-Net adds a latent field state and state\-dependent mechanical realization\. Compared with Latent ODE models, it keeps a readable directed component inside the transition instead of absorbing all dynamics into a generic vector field\. Compared with NRI\-like relational models, it uses the learned relation to drive field motion rather than only to define message passing\. The contribution is therefore the factorization of the transition itself: a stable directed structure is reused across rollout steps, expressed through current\-state effects, and converted into motion before prediction\.

### Structural Regularization and Rollout Generalization

![Refer to caption](https://arxiv.org/html/2606.11251v1/overfitting.png)Figure 6:Training dynamics and open\-loop rollout behavior on Lorenz–96 withN=40N=40\. The training objective decreases rapidly for high\-capacity baselines such as LSTM and LatentODE, but their validation rollout error remains high or deteriorates\. MF\-Net does not achieve the lowest training objective, yet maintains better validation behavior and lower open\-loop test error\. This supports the view that the relation\-to\-motion transition acts as an architectural regularizer in high\-dimensional chaotic rollout\.The training curves in Figure[6](https://arxiv.org/html/2606.11251#S4.F6)suggest that the advantage of MF\-Net on Lorenz–96 is not simply stronger fitting capacity\. High\-capacity baselines such as LSTM and LatentODE can reduce the training objective rapidly, but their validation rollout error remains high or deteriorates, indicating poor open\-loop generalization\. A generic latent transition has the form

Q˙=Fθ​\(Q\),\\dot\{Q\}=F\_\{\\theta\}\(Q\),so cross\-variable dependence can be absorbed into an unrestricted learned vector field\. MF\-Net restricts this transition by forcing cross\-variable effects through the relation\-to\-motion path

D→J​\(t\)→h​\(t\)→G​\(t\)→η​\(t\)→Q˙​\(t\)\.D\\rightarrow J\(t\)\\rightarrow h\(t\)\\rightarrow G\(t\)\\rightarrow\\eta\(t\)\\rightarrow\\dot\{Q\}\(t\)\.At the observed\-coordinate level, the direct cross\-variable sensitivity is gated by the stable relation law,

∂z˙i​\(t\)∂zj​\(t\)=Dj,i​sj′​\(zj​\(t\)\),j≠i,\\frac\{\\partial\\dot\{z\}\_\{i\}\(t\)\}\{\\partial z\_\{j\}\(t\)\}=D\_\{j,i\}s^\{\\prime\}\_\{j\}\(z\_\{j\}\(t\)\),\\qquad j\\neq i,with additional field\-mediated effects also passing through the same realized flow path\. Moreover, the sameDDis reused across rollout steps, so relation choices that fit one\-step fragments but destabilize multi\-step prediction are penalized by the multi\-horizon objective\. In this sense, MF\-Net is less free than an unrestricted recurrent or latent\-ODE transition, but this restriction acts as an architectural regularizer: it sacrifices some training\-set flexibility while improving validation and open\-loop rollout stability in coupled chaotic dynamics\.

### Interpreting the Learned Relation Structure

The limiting cases above also clarify the interpretation of the learned matrixDD\. In controlled systems, where the data\-generating structure is known,DDcan be evaluated against a true interaction matrix or a known coupling support\. In observational real data, the appropriate interpretation is weaker and more operational:DDis an effective directed predictive structure used by the MF\-Net transition\. This distinction is central to how we read the learned relation matrix across different datasets\.

#### Relation\-to\-motion role ofDD\.

The matrixDDis not a post\-hoc score attached to a completed predictor\. For a directed edgea→ba\\to b, MF\-Net forms the realized flow

Ja→b​\(t\)=Da,b​sa​\(t\)\.J\_\{a\\to b\}\(t\)=D\_\{a,b\}s\_\{a\}\(t\)\.Since∑j≠bJj→b​\(t\)\\sum\_\{j\\neq b\}J\_\{j\\to b\}\(t\)enters the observed\-coordinate vector field, we have the direct path

∂z˙b​\(t\)∂Da,b=sa​\(t\)\.\\frac\{\\partial\\dot\{z\}\_\{b\}\(t\)\}\{\\partial D\_\{a,b\}\}=s\_\{a\}\(t\)\.The same entry also acts through the field\-response path\. Because

hb​\(t\)=∑j≠bDj,b​sj​\(t\)​mj​\(t\),h\_\{b\}\(t\)=\\sum\_\{j\\neq b\}D\_\{j,b\}s\_\{j\}\(t\)m\_\{j\}\(t\),a perturbation ofDa,bD\_\{a,b\}gives

∂hb​\(t\)∂Da,b=sa​\(t\)​ma​\(t\)\.\\frac\{\\partial h\_\{b\}\(t\)\}\{\\partial D\_\{a,b\}\}=s\_\{a\}\(t\)m\_\{a\}\(t\)\.WithGb​\(t\)=Rθ​\(qb​\(t\)\)​hb​\(t\)G\_\{b\}\(t\)=R\_\{\\theta\}\(q\_\{b\}\(t\)\)h\_\{b\}\(t\)and

η˙b​\(t\)=λb​\(Gb​\(t\)−ηb​\(t\)\),\\dot\{\\eta\}\_\{b\}\(t\)=\\lambda\_\{b\}\(G\_\{b\}\(t\)\-\\eta\_\{b\}\(t\)\),this implies

∂η˙b​\(t\)∂Da,b=λb​Rθ​\(qb​\(t\)\)​sa​\(t\)​ma​\(t\)\.\\frac\{\\partial\\dot\{\\eta\}\_\{b\}\(t\)\}\{\\partial D\_\{a,b\}\}=\\lambda\_\{b\}R\_\{\\theta\}\(q\_\{b\}\(t\)\)s\_\{a\}\(t\)m\_\{a\}\(t\)\.Thus,DDenters both the direct observed\-flow term and the field\-mediated motion term:

D→J​\(t\)→z˙​\(t\),D→J​\(t\)→h​\(t\)→G​\(t\)→η​\(t\)→Q˙​\(t\)\.D\\rightarrow J\(t\)\\rightarrow\\dot\{z\}\(t\),\\qquad D\\rightarrow J\(t\)\\rightarrow h\(t\)\\rightarrow G\(t\)\\rightarrow\\eta\(t\)\\rightarrow\\dot\{Q\}\(t\)\.The structural readout is therefore part of the same computation that produces the forecast\.

#### Relation law versus local effective coupling\.

The learned matrixDDshould be distinguished from a local Jacobian\. At the vector\-field level,

z˙i​\(t\)=ri\+αi​zi​\(t\)\+∑j≠iDj,i​sj​\(zj​\(t\)\)\+ηiz​\(t\)\.\\dot\{z\}\_\{i\}\(t\)=r\_\{i\}\+\\alpha\_\{i\}z\_\{i\}\(t\)\+\\sum\_\{j\\neq i\}D\_\{j,i\}s\_\{j\}\(z\_\{j\}\(t\)\)\+\\eta\_\{i\}^\{z\}\(t\)\.Holding the other field\-state coordinates fixed, the direct local derivative is

∂z˙i​\(t\)∂zj​\(t\)=Dj,i​sj′​\(zj​\(t\)\),j≠i\.\\frac\{\\partial\\dot\{z\}\_\{i\}\(t\)\}\{\\partial z\_\{j\}\(t\)\}=D\_\{j,i\}s^\{\\prime\}\_\{j\}\(z\_\{j\}\(t\)\),\\qquad j\\neq i\.Thus,DDis not itself the instantaneous Jacobian; it is a stable relation law that contributes to the local effective coupling through the realized source strength\. More generally, the finite\-horizon effective coupling induced by MF\-Net can be written as

Aj→i\(h\)​\(t\)=∂z^i​\(t\+h\)∂zj​\(t\)\.A^\{\(h\)\}\_\{j\\to i\}\(t\)=\\frac\{\\partial\\hat\{z\}\_\{i\}\(t\+h\)\}\{\\partial z\_\{j\}\(t\)\}\.This quantity is state\- and horizon\-dependent\. It includes direct realized flow, field\-mediated response, tendency memory, hidden\-state motion, and indirect effects accumulated through rollout\. In simple identifiable systems,DDand the effective coupling can align closely; in partially observed or strongly driven systems, the same matrix should be read more cautiously as functional predictive coupling\.

#### Finite\-time mixing of direct and indirect effects\.

Even if the true system is linear in continuous time,

finite\-interval observations satisfy

x​\(t\+Δ​t\)=exp⁡\(A​Δ​t\)​x​\(t\),x\(t\+\\Delta t\)=\\exp\(A\\Delta t\)x\(t\),where

exp⁡\(A​Δ​t\)=I\+Δ​t​A\+Δ​t22​A2\+Δ​t36​A3\+⋯\.\\exp\(A\\Delta t\)=I\+\\Delta tA\+\\frac\{\\Delta t^\{2\}\}\{2\}A^\{2\}\+\\frac\{\\Delta t^\{3\}\}\{6\}A^\{3\}\+\\cdots\.The first\-order term contains the direct matrixAA, but higher\-order terms contain paths through intermediate variables\. For example,

\(A2\)i​j=∑kAi​k​Ak​j\(A^\{2\}\)\_\{ij\}=\\sum\_\{k\}A\_\{ik\}A\_\{kj\}contains the pathj→k→ij\\to k\\to i\. Therefore, a finite\-time predictive relation can mix direct and indirect effects even when the underlying continuous\-time interaction matrix is fixed\.

#### Scale and gauge\.

The raw scale ofDDis not fully identifiable by itself\. Since the realized flow depends on the product

Jj→i​\(t\)=Dj,i​sj​\(t\),J\_\{j\\to i\}\(t\)=D\_\{j,i\}s\_\{j\}\(t\),the transformation

Dj,i↦cj​Dj,i,sj​\(t\)↦1cj​sj​\(t\)D\_\{j,i\}\\mapsto c\_\{j\}D\_\{j,i\},\\qquad s\_\{j\}\(t\)\\mapsto\\frac\{1\}\{c\_\{j\}\}s\_\{j\}\(t\)leavesJj→i​\(t\)J\_\{j\\to i\}\(t\)unchanged\. Thus, the absolute magnitude of a single entry ofDDshould not be over\-interpreted\. Support, sign under a fixed convention, ranking, calibrated agreement, perturbation effects, and alignment with external temporal evidence are more meaningful structural readouts\. A similar basis ambiguity can occur inside the message\-response pathway, so internal message coordinates are not themselves treated as the structural object\.

#### Context and tendency states\.

The auxiliary states have specific dynamical roles\. The context state satisfies

χ˙i​\(t\)=−λχ​χi​\(t\),\\dot\{\\chi\}\_\{i\}\(t\)=\-\\lambda\_\{\\chi\}\\chi\_\{i\}\(t\),so

χi​\(t\+τ\)=e−λχ​τ​χi​\(t\)\.\\chi\_\{i\}\(t\+\\tau\)=e^\{\-\\lambda\_\{\\chi\}\\tau\}\\chi\_\{i\}\(t\)\.It is therefore a decaying history condition: it initializes the rollout but does not directly decode future observations\. The tendency state satisfies

η˙i​\(t\)=λi​\(Gi​\(t\)−ηi​\(t\)\),\\dot\{\\eta\}\_\{i\}\(t\)=\\lambda\_\{i\}\(G\_\{i\}\(t\)\-\\eta\_\{i\}\(t\)\),whose integral form is

ηi​\(t\+τ\)=e−λi​τ​ηi​\(t\)\+∫0τλi​e−λi​\(τ−s\)​Gi​\(t\+s\)​𝑑s\.\\eta\_\{i\}\(t\+\\tau\)=e^\{\-\\lambda\_\{i\}\\tau\}\\eta\_\{i\}\(t\)\+\\int\_\{0\}^\{\\tau\}\\lambda\_\{i\}e^\{\-\\lambda\_\{i\}\(\\tau\-s\)\}G\_\{i\}\(t\+s\)\\,ds\.Thus,ηi\\eta\_\{i\}is an exponential memory of past field responses\. Field responses are not used as instantaneous predictions; they are accumulated into motion tendencies that help move the field state forward\.

#### Evidence\-dependent interpretation\.

Partial observation creates additional ambiguity\. If observed variables are coupled to hidden variables, a model trained only on the observed trajectories may absorb hidden\-mediated effects into the effective relations among observed variables\. For this reason, we interpretDDat different levels in different settings: as a recovered interaction or coupling support in controlled benchmarks, as a functional predictive coupling signal in real neural recordings, and as a weak or non\-interpretable relation when the observed variables are dominated by missing external drivers\. The appropriate interpretation is therefore determined by the available evidence, not by the existence of a learned matrix alone\.

### Model scope and future directions

Overall, MF\-Net should be viewed as a mechanics\-inspired recurrent framework for organizing multivariate prediction through structured motion\. Its purpose is not to replace every specialized forecaster, since simple linear models can remain very strong on smooth signals\(Zeng et al\.,[2023](https://arxiv.org/html/2606.11251#bib.bib42)\)and tuned feature\-based methods can be highly effective on clean low\-dimensional systems\. Its value lies in combining useful rollout accuracy with a directed component that remains inside the transition and can still be inspected after training\. The present implementation is only one realization of this idea\. A first direction is to make the model fully continuous, replacing the fixed\-step rollout with a continuous field evolution in which directed effects are transmitted through the field along the trajectory\. A second direction is to distill the current equations into cleaner learned maps, for example from relation to realized flow, from flow to field response, and from response to motion, while preserving the same structure\-to\-motion principle\. Future work should also test the learned directed component under perturbation or intervention data and quantify uncertainty in the readout\. These extensions all keep the central view of MF\-Net unchanged: multivariate forecasting can be organized as a process in which stable directed structure shapes field motion, and field motion produces prediction\.

### Limitations

MF\-Net has several limitations\.

#### Relation interpretation\.

First, the learned relation matrix should not be interpreted as a universal interaction graph\. AlthoughDDis part of the rollout, finite\-time prediction can mix direct, indirect, and hidden\-mediated effects\. In addition, the realized flow depends on products such asDj,i​sj​\(t\)D\_\{j,i\}s\_\{j\}\(t\), and the message\-response pathway also has basis ambiguities\. Thus, raw magnitudes of the learned relation matrix are not fully identifiable; support, ranking, sign under a fixed convention, perturbation effects, and agreement with external evidence are more reliable than individual entry values\.

#### Architectural form\.

Second, the present architecture is an explicit realization of the relation\-to\-motion idea rather than a final minimal form\. We intentionally expose intermediate quantities such as realized flow, field response, and motion tendency so that the transition remains inspectable\. This makes the model less compact than a single unrestricted neural transition, and some components may be distilled into cleaner learned maps in future versions without changing the central principle\.

#### Complex spatiotemporal systems\.

Third, MF\-Net is not designed to solve all forms of highly complex spatiotemporal dynamics\. Systems governed by dense spatial fields, strong hyperchaos, or PDE\-like operators may require representations closer to neural operators or spatially continuous models\. This is a shared difficulty for many recurrent forecasting models, not a limitation unique to MF\-Net\.

#### Scope of useful structure\.

MF\-Net is most useful when cross\-variable structure is informative for prediction\. For nearly independent channels, strongly periodic signals, or low\-rank shared temporal dynamics, a relation\-structured model is not necessarily the best choice\. A simple amplitude\-scaled sinusoidal panel illustrates this limitation: all channels share the same latent periodic signal, so the task is essentially to learn a common oscillator rather than a directed relation law\. In such cases, latent\-dynamics, spectral, or classical periodic models may perform as well as or better than MF\-Net, and the learned relation matrix should be interpreted weakly or not interpreted at all\.

## Implementation and Reproducibility Notes

The implementation code is released on[GitHub](https://github.com/Cuixjnoob/MFNET/tree/main)\.

For the fixed\-horizon rollout experiments in this paper, we recommend fixed\-step Euler or Heun integration\. The latest reported MF\-Net runs use Heun integration, which was more stable than Euler at comparable step sizes while remaining easier to reproduce than adaptive solvers\. For a stateyty\_\{t\}and vector fieldff, the Heun update is

k1=f​\(yt\),y~=yt\+Δ​t​k1,k2=f​\(y~\),k\_\{1\}=f\(y\_\{t\}\),\\qquad\\tilde\{y\}=y\_\{t\}\+\\Delta t\\,k\_\{1\},\\qquad k\_\{2\}=f\(\\tilde\{y\}\),yt\+1=yt\+Δ​t2​\(k1\+k2\)\.y\_\{t\+1\}=y\_\{t\}\+\\frac\{\\Delta t\}\{2\}\(k\_\{1\}\+k\_\{2\}\)\.Adaptive solvers such as RK45 are not required for these sampled, fixed\-horizon experiments and may introduce implementation\-dependent internal step sizes\. Training uses a weighted multi\-horizon rollout loss over causal prediction origins,

ℒpred=∑h∈Hwh​ℓ​\(z^t\+h,zt\+h\),\\mathcal\{L\}\_\{\\mathrm\{pred\}\}=\\sum\_\{h\\in H\}w\_\{h\}\\,\\ell\(\\hat\{z\}\_\{t\+h\},z\_\{t\+h\}\),with small regularization terms onDD, the tendency state, and the recognition context\. Optimization used AdamW, gradient clipping, and validation\-based checkpoint selection\. Synthetic benchmarks should be trained until both rollout error and the learned relation matrix are stable; early stopping based only on short\-horizon prediction loss can stop before the structural readout has converged\. Reported multi\-seed results use independent random initializations and are summarized as mean and standard deviation unless otherwise stated\.

SinCos benchmark was generated as a deterministic multi\-channel periodic panel from four shared basis functions:sin⁡\(2​π​t/P\)\\sin\(2\\pi t/P\),cos⁡\(2​π​t/P\)\\cos\(2\\pi t/P\),sin⁡\(4​π​t/P\)\\sin\(4\\pi t/P\), andcos⁡\(4​π​t/P\)\\cos\(4\\pi t/P\)\. Each node was a linear mixtureyi​\(t\)=B​\(t\)​ciy\_\{i\}\(t\)=B\(t\)c\_\{i\}\. We usedT=900T=900,N=8N=8, periodP=64P=64, and no observation noise\. The first two channels were fixed assin\+cos\\sin\+\\cosandsin−cos\\sin\-\\cos; the remaining channels used fixed or seeded Gaussian mixture coefficients\. Each generated channel was variance\-normalized, then the final model inputs were standardized using only the chronological training split\.

We present hyperparameters in Appendix[B](https://arxiv.org/html/2606.11251#A2)\.

The author used ChatGPT for coding assistance, grammar checking, and LaTeX\-related editing\.

## Acknowledgements

I am grateful to Mr\. Jun Zhao of Suzhou High School SIP for supporting this project\. He trusted my ability when the work was still uncertain and unfinished, and his recognition and encouragement gave me the confidence to continue\. He saw my potential before it was visible\.

I thank my parents for their understanding and support, which gave me the time and space to pursue a project more demanding than ordinary coursework\. I am also grateful to Chenkai Liu, Tianyou Zhang, Zhenghong Zou, Yifan Situ, Yizheng Yan, and Yanhong Pan for their encouragement\.

Completing this work through a period of mental struggle, self\-doubt, and being underestimated by people around me has made it especially meaningful to me\.

## References

- Atanas et al\. \(2023\)Atanas, A\.A\., Kim, J\., Wang, Z\., Bueno, E\., Becker, M\., Kang, D\., Park, J\., Kramer, T\.S\., Wan, F\.K\., Baskoylu, S\., Dag, U\., Kalogeropoulou, E\., Gomes, M\.A\., Estrem, C\., Cohen, N\., Mansinghka, V\.K\., Flavell, S\.W\., 2023\.Brain\-wide representations of behavior spanning multiple timescales and states in C\. elegans\.Cell 186, 4134–4151\.e31\.doi:[10\.1016/j\.cell\.2023\.07\.035](http://dx.doi.org/10.1016/j.cell.2023.07.035)\.
- Blasius et al\. \(2020\)Blasius, B\., Rudolf, L\., Weithoff, G\., Gaedke, U\., Fussmann, G\.F\., 2020\.Long\-term cyclic persistence in an experimental predator–prey system\.Nature 577, 226–230\.doi:[10\.1038/s41586\-019\-1857\-0](http://dx.doi.org/10.1038/s41586-019-1857-0)\.
- Brunton et al\. \(2016\)Brunton, S\.L\., Proctor, J\.L\., Kutz, J\.N\., 2016\.Discovering governing equations from data by sparse identification of nonlinear dynamical systems\.Proceedings of the National Academy of Sciences 113, 3932–3937\.doi:[10\.1073/pnas\.1517384113](http://dx.doi.org/10.1073/pnas.1517384113)\.
- Chen et al\. \(2018\)Chen, R\.T\.Q\., Rubanova, Y\., Bettencourt, J\., Duvenaud, D\.K\., 2018\.Neural ordinary differential equations, in: Advances in Neural Information Processing Systems, pp\. 6571–6583\.
- Cranmer et al\. \(2020\)Cranmer, M\., Greydanus, S\., Hoyer, S\., Battaglia, P\., Spergel, D\., Ho, S\., 2020\.Lagrangian neural networks\.arXiv preprint arXiv:2003\.04630 doi:[10\.48550/arXiv\.2003\.04630](http://dx.doi.org/10.48550/arXiv.2003.04630),[arXiv:2003\.04630](http://arxiv.org/abs/2003.04630)\.
- Dupont et al\. \(2019\)Dupont, E\., Doucet, A\., Teh, Y\.W\., 2019\.Augmented neural ODEs\.arXiv preprint arXiv:1904\.01681 doi:[10\.48550/arXiv\.1904\.01681](http://dx.doi.org/10.48550/arXiv.1904.01681),[arXiv:1904\.01681](http://arxiv.org/abs/1904.01681)\. published in NeurIPS 2019\.
- FitzHugh \(1961\)FitzHugh, R\., 1961\.Impulses and physiological states in theoretical models of nerve membrane\.Biophysical Journal 1, 445–466\.doi:[10\.1016/S0006\-3495\(61\)86902\-6](http://dx.doi.org/10.1016/S0006-3495(61)86902-6)\.
- Gauthier et al\. \(2021\)Gauthier, D\.J\., Bollt, E\., Griffith, A\., Barbosa, W\.A\.S\., 2021\.Next generation reservoir computing\.Nature Communications 12, 5564\.doi:[10\.1038/s41467\-021\-25801\-2](http://dx.doi.org/10.1038/s41467-021-25801-2)\.
- Granger \(1969\)Granger, C\.W\.J\., 1969\.Investigating causal relations by econometric models and cross\-spectral methods\.Econometrica 37, 424–438\.doi:[10\.2307/1912791](http://dx.doi.org/10.2307/1912791)\.
- Greydanus et al\. \(2019\)Greydanus, S\., Dzamba, M\., Yosinski, J\., 2019\.Hamiltonian neural networks\.arXiv preprint arXiv:1906\.01563 doi:[10\.48550/arXiv\.1906\.01563](http://dx.doi.org/10.48550/arXiv.1906.01563),[arXiv:1906\.01563](http://arxiv.org/abs/1906.01563)\. published in NeurIPS 2019\.
- He et al\. \(2016\)He, K\., Zhang, X\., Ren, S\., Sun, J\., 2016\.Deep residual learning for image recognition, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp\. 770–778\.doi:[10\.1109/CVPR\.2016\.90](http://dx.doi.org/10.1109/CVPR.2016.90)\.
- Herdeanu et al\. \(2025\)Herdeanu, B\., Nathaniel, J\., Roesch, C\., Buch, J\., Ramien, G\., Haux, J\., Gentine, P\., 2025\.Causaldynamics: A large\-scale benchmark for structural discovery of dynamical causal models\.arXiv preprint arXiv:2505\.16620 doi:[10\.48550/arXiv\.2505\.16620](http://dx.doi.org/10.48550/arXiv.2505.16620),[arXiv:2505\.16620](http://arxiv.org/abs/2505.16620)\.
- Hochreiter and Schmidhuber \(1997\)Hochreiter, S\., Schmidhuber, J\., 1997\.Long short\-term memory\.Neural Computation 9, 1735–1780\.doi:[10\.1162/neco\.1997\.9\.8\.1735](http://dx.doi.org/10.1162/neco.1997.9.8.1735)\.
- Jin et al\. \(2024\)Jin, M\., Koh, H\.Y\., Wen, Q\., Zambon, D\., Alippi, C\., Webb, G\.I\., King, I\., Pan, S\., 2024\.A survey on graph neural networks for time series: Forecasting, classification, imputation, and anomaly detection\.IEEE Transactions on Pattern Analysis and Machine Intelligence 46, 10466–10485\.doi:[10\.1109/TPAMI\.2024\.3443141](http://dx.doi.org/10.1109/TPAMI.2024.3443141)\.
- Kang et al\. \(2024\)Kang, B\., Saha, P\., Sharma, S\., Chakraborty, B\., Mukhopadhyay, S\., 2024\.Online relational inference for evolving multi\-agent interacting systems\.arXiv preprint arXiv:2411\.01442 doi:[10\.48550/arXiv\.2411\.01442](http://dx.doi.org/10.48550/arXiv.2411.01442),[arXiv:2411\.01442](http://arxiv.org/abs/2411.01442)\.
- Kidger et al\. \(2020\)Kidger, P\., Morrill, J\., Foster, J\., Lyons, T\., 2020\.Neural controlled differential equations for irregular time series, in: Advances in Neural Information Processing Systems, pp\. 6696–6707\.
- Kipf et al\. \(2018\)Kipf, T\.N\., Fetaya, E\., Wang, K\.C\., Welling, M\., Zemel, R\.S\., 2018\.Neural relational inference for interacting systems, in: Proceedings of the 35th International Conference on Machine Learning, PMLR\. pp\. 2688–2697\.
- Liu et al\. \(2024\)Liu, Y\., Hu, T\., Zhang, H\., Wu, H\., Wang, S\., Ma, L\., Long, M\., 2024\.iTransformer: Inverted transformers are effective for time series forecasting\.arXiv preprint arXiv:2310\.06625 doi:[10\.48550/arXiv\.2310\.06625](http://dx.doi.org/10.48550/arXiv.2310.06625),[arXiv:2310\.06625](http://arxiv.org/abs/2310.06625)\. published at ICLR 2024\.
- Liu et al\. \(2025\)Liu, Z\., Wang, X\., Wang, B\., Huang, Z\., Yang, C\., Jin, W\., 2025\.Graph ODEs and beyond: A comprehensive survey on integrating differential equations with graph neural networks\.arXiv preprint arXiv:2503\.23167 doi:[10\.48550/arXiv\.2503\.23167](http://dx.doi.org/10.48550/arXiv.2503.23167),[arXiv:2503\.23167](http://arxiv.org/abs/2503.23167)\.
- Lorenz \(1996\)Lorenz, E\.N\., 1996\.Predictability: A problem partly solved, in: Proceedings of the Seminar on Predictability, European Centre for Medium\-Range Weather Forecasts, Reading, UK\. pp\. 1–18\.
- Lotka \(1925\)Lotka, A\.J\., 1925\.Elements of Physical Biology\.Williams & Wilkins, Baltimore\.
- Luo and Wang \(2024\)Luo, D\., Wang, X\., 2024\.ModernTCN: A modern pure convolution structure for general time series analysis, in: International Conference on Learning Representations\.URL:[https://openreview\.net/forum?id=vpJMJerXHU](https://openreview.net/forum?id=vpJMJerXHU)\.
- Lütkepohl \(2005\)Lütkepohl, H\., 2005\.New Introduction to Multiple Time Series Analysis\.Springer, Berlin, Heidelberg\.doi:[10\.1007/978\-3\-540\-27752\-1](http://dx.doi.org/10.1007/978-3-540-27752-1)\.
- Nagumo et al\. \(1962\)Nagumo, J\., Arimoto, S\., Yoshizawa, S\., 1962\.An active pulse transmission line simulating nerve axon\.Proceedings of the IRE 50, 2061–2070\.doi:[10\.1109/JRPROC\.1962\.288235](http://dx.doi.org/10.1109/JRPROC.1962.288235)\.
- Nie et al\. \(2023\)Nie, Y\., Nguyen, N\.H\., Sinthong, P\., Kalagnanam, J\., 2023\.A time series is worth 64 words: Long\-term forecasting with transformers\.arXiv preprint arXiv:2211\.14730 doi:[10\.48550/arXiv\.2211\.14730](http://dx.doi.org/10.48550/arXiv.2211.14730),[arXiv:2211\.14730](http://arxiv.org/abs/2211.14730)\. published as an ICLR 2023 conference paper\.
- Norcliffe et al\. \(2020\)Norcliffe, A\., Bodnar, C\., Day, B\., Simidjievski, N\., Liò, P\., 2020\.On second order behaviour in augmented neural ODEs\.arXiv preprint arXiv:2006\.07220 doi:[10\.48550/arXiv\.2006\.07220](http://dx.doi.org/10.48550/arXiv.2006.07220),[arXiv:2006\.07220](http://arxiv.org/abs/2006.07220)\. published in NeurIPS 2020\.
- Oh et al\. \(2025\)Oh, Y\., Kam, S\., Lee, J\., Lim, D\.Y\., Kim, S\., Bui, A\.A\.T\., 2025\.Comprehensive review of neural differential equations for time series analysis, in: Proceedings of the Thirty\-Fourth International Joint Conference on Artificial Intelligence, pp\. 10621–10631\.doi:[10\.24963/ijcai\.2025/1179](http://dx.doi.org/10.24963/ijcai.2025/1179)\.
- Oreshkin et al\. \(2020\)Oreshkin, B\.N\., Carpov, D\., Chapados, N\., Bengio, Y\., 2020\.N\-BEATS: Neural basis expansion analysis for interpretable time series forecasting\.arXiv preprint arXiv:1905\.10437 doi:[10\.48550/arXiv\.1905\.10437](http://dx.doi.org/10.48550/arXiv.1905.10437),[arXiv:1905\.10437](http://arxiv.org/abs/1905.10437)\. published at ICLR 2020\.
- Pan et al\. \(2024\)Pan, L\., Shi, C\., Dokmanić, I\., 2024\.A graph dynamics prior for relational inference, in: Proceedings of the AAAI Conference on Artificial Intelligence, pp\. 14508–14516\.doi:[10\.1609/aaai\.v38i13\.29366](http://dx.doi.org/10.1609/aaai.v38i13.29366)\.
- Poli et al\. \(2019\)Poli, M\., Massaroli, S\., Park, J\., Yamashita, A\., Asama, H\., Park, J\., 2019\.Graph neural ordinary differential equations\.arXiv preprint arXiv:1911\.07532 doi:[10\.48550/arXiv\.1911\.07532](http://dx.doi.org/10.48550/arXiv.1911.07532),[arXiv:1911\.07532](http://arxiv.org/abs/1911.07532)\.
- Rangapuram et al\. \(2018\)Rangapuram, S\.S\., Seeger, M\.W\., Gasthaus, J\., Stella, L\., Wang, Y\., Januschowski, T\., 2018\.Deep state space models for time series forecasting, in: Advances in Neural Information Processing Systems\.
- Rubanova et al\. \(2019\)Rubanova, Y\., Chen, R\.T\.Q\., Duvenaud, D\.K\., 2019\.Latent ordinary differential equations for irregularly\-sampled time series\.arXiv preprint arXiv:1907\.03907 doi:[10\.48550/arXiv\.1907\.03907](http://dx.doi.org/10.48550/arXiv.1907.03907),[arXiv:1907\.03907](http://arxiv.org/abs/1907.03907)\.
- Rudin \(2019\)Rudin, C\., 2019\.Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead\.Nature Machine Intelligence 1, 206–215\.doi:[10\.1038/s42256\-019\-0048\-x](http://dx.doi.org/10.1038/s42256-019-0048-x)\.
- Sakaguchi and Kuramoto \(1986\)Sakaguchi, H\., Kuramoto, Y\., 1986\.A soluble active rotator model showing phase transitions via mutual entrainment\.Progress of Theoretical Physics 76, 576–581\.doi:[10\.1143/PTP\.76\.576](http://dx.doi.org/10.1143/PTP.76.576)\.
- Sugihara et al\. \(2012\)Sugihara, G\., May, R\., Ye, H\., Hsieh, C\.h\., Deyle, E\., Fogarty, M\., Munch, S\., 2012\.Detecting causality in complex ecosystems\.Science 338, 496–500\.doi:[10\.1126/science\.1227079](http://dx.doi.org/10.1126/science.1227079)\.
- Tank et al\. \(2022\)Tank, A\., Covert, I\., Foti, N\., Shojaie, A\., Fox, E\.B\., 2022\.Neural Granger causality\.IEEE Transactions on Pattern Analysis and Machine Intelligence 44, 4267–4279\.doi:[10\.1109/TPAMI\.2021\.3065601](http://dx.doi.org/10.1109/TPAMI.2021.3065601)\.
- Volterra \(1926\)Volterra, V\., 1926\.Fluctuations in the abundance of a species considered mathematically\.Nature 118, 558–560\.doi:[10\.1038/118558a0](http://dx.doi.org/10.1038/118558a0)\.
- Wang et al\. \(2024\)Wang, B\., Jennings, J\., Gong, W\., 2024\.Neural structure learning with stochastic differential equations\.arXiv preprint arXiv:2311\.03309 doi:[10\.48550/arXiv\.2311\.03309](http://dx.doi.org/10.48550/arXiv.2311.03309),[arXiv:2311\.03309](http://arxiv.org/abs/2311.03309)\. published at ICLR 2024\.
- WormWideWeb \(2023\)WormWideWeb, 2023\.Wormwideweb neural activity datasets\.URL:[https://wormwideweb\.org/activity/dataset/](https://wormwideweb.org/activity/dataset/)\. neural activity datasets associated with Atanas and Kim et al\. \(2023\)\.
- Wu et al\. \(2020\)Wu, Z\., Pan, S\., Long, G\., Jiang, J\., Chang, X\., Zhang, C\., 2020\.Connecting the dots: Multivariate time series forecasting with graph neural networks, in: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp\. 753–763\.doi:[10\.1145/3394486\.3403118](http://dx.doi.org/10.1145/3394486.3403118)\.
- Wu et al\. \(2019\)Wu, Z\., Pan, S\., Long, G\., Jiang, J\., Zhang, C\., 2019\.Graph WaveNet for deep spatial\-temporal graph modeling, in: Proceedings of the Twenty\-Eighth International Joint Conference on Artificial Intelligence, pp\. 1907–1913\.doi:[10\.24963/ijcai\.2019/264](http://dx.doi.org/10.24963/ijcai.2019/264)\.
- Zeng et al\. \(2023\)Zeng, A\., Chen, M\., Zhang, L\., Xu, Q\., 2023\.Are transformers effective for time series forecasting?, in: Proceedings of the AAAI Conference on Artificial Intelligence, pp\. 11121–11128\.doi:[10\.1609/aaai\.v37i9\.26317](http://dx.doi.org/10.1609/aaai.v37i9.26317)\.
- Zheng et al\. \(2024\)Zheng, Y\., Yi, L\., Wei, Z\., 2024\.A survey of dynamic graph neural networks\.arXiv preprint arXiv:2404\.18211 doi:[10\.48550/arXiv\.2404\.18211](http://dx.doi.org/10.48550/arXiv.2404.18211),[arXiv:2404\.18211](http://arxiv.org/abs/2404.18211)\.
- Zhong et al\. \(2020\)Zhong, Y\.D\., Dey, B\., Chakraborty, A\., 2020\.Symplectic ODE\-net: Learning hamiltonian dynamics with control, in: International Conference on Learning Representations\.URL:[https://openreview\.net/forum?id=ryxmb1rKDS](https://openreview.net/forum?id=ryxmb1rKDS)\.
- Zhou et al\. \(2024\)Zhou, J\., Lu, X\., Xiao, Y\., Tang, J\., Su, J\., Li, Y\., Liu, J\., Lyu, J\., Ma, Y\., Dou, D\., 2024\.SDWPF: A dataset for spatial dynamic wind power forecasting over a large turbine array\.Scientific Data 11, 649\.doi:[10\.1038/s41597\-024\-03427\-5](http://dx.doi.org/10.1038/s41597-024-03427-5)\.

## Appendix AAppendix A: Lorenz–96 Hyperparameter Sensitivity

We ran a one\-factor\-at\-a\-time hyperparameter sensitivity screen on the 40\-dimensional Lorenz–96 benchmark to check whether MF\-Net depends on a narrow configuration\. This screen is intended as a robustness diagnostic rather than as the main benchmark result\. All runs used seed 0, MF\-Net\-Heun, 220 epochs, and 4 steps per epoch\. The baseline configuration was history lengthL=32L=32,udim=χdim=6u\_\{\\dim\}=\\chi\_\{\\dim\}=6, field modes=8=8, hidden width=80=80, and learning rate7×10−47\\times 10^\{\-4\}\. For each factor, only one setting was changed at a time\.

Table 7:One\-factor hyperparameter sensitivity screen on the 40\-dimensional Lorenz–96 benchmark\. All runs use seed 0, MF\-Net\-Heun, 220 epochs, and 4 steps per epoch\. The baseline configuration isL=32L=32,udim=χdim=6u\_\{\\dim\}=\\chi\_\{\\dim\}=6, field modes=8=8, hidden width=80=80, and learning rate7×10−47\\times 10^\{\-4\}\.Overall, the sensitivity screen suggests that the Lorenz–96 structural recovery is not tied to a single fragile hyperparameter setting\. Moderate internal dimensions, field modes, and hidden widths preserve strong structural readout\. Very long histories degrade Precision@KK, suggesting that increasing the history window does not simply improve structure recovery through a longer history\-to\-future shortcut\. The low learning\-rate setting underperforms under the fixed training budget, consistent with under\-optimization\.

## Appendix BAppendix B: Main Hyperparameter Settings

Table[8](https://arxiv.org/html/2606.11251#A2.T8)summarizes the main MF\-Net configurations used in the reported experiments\. The table is intended as a compact reproducibility reference rather than a full internal experiment log\.

Table 8:Main MF\-Net hyperparameter settings\. All experiments use chronological splits, train\-only normalization, AdamW optimization, gradient clipping, and validation\-based checkpoint selection unless otherwise stated\.