SINDyG: Sparse Identification of Nonlinear Dynamical Systems from Graph-Structured Data, with Applications to Stuart-Landau Oscillator Networks

Mohammad Amin Basiri; Sina Khanmohammadi

arxiv: 2409.04463 · v5 · submitted 2024-09-02 · 📡 eess.SY · cs.CE· cs.LG· cs.SY

SINDyG: Sparse Identification of Nonlinear Dynamical Systems from Graph-Structured Data, with Applications to Stuart-Landau Oscillator Networks

Mohammad Amin Basiri , Sina Khanmohammadi This is my paper

Pith reviewed 2026-05-23 21:07 UTC · model grok-4.3

classification 📡 eess.SY cs.CEcs.LGcs.SY

keywords sparse identificationdynamical systemsgraph-structured dataStuart-Landau oscillatorsnetwork dynamicssymbolic regressionSINDyneuronal modeling

0 comments

The pith

Incorporating known network graphs into sparse regression yields simpler and more accurate dynamical models for oscillator networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard sparse identification methods treat a networked system as a single undifferentiated whole and therefore miss small changes in emergent behavior caused by subsystem interactions. SINDyG modifies the regression by adding a penalty term that directly encodes the supplied graph structure, so coefficients are encouraged to respect actual node connections. When applied to populations of Stuart-Landau oscillators that model macroscopic neuronal oscillations, the resulting equations reproduce observed network dynamics with fewer terms and lower error than the unmodified SINDy procedure. The same graph-informed penalty can be dropped into other symbolic regression algorithms without changing their core optimization.

Core claim

By translating the known adjacency structure of the network into a structured sparsity penalty, SINDyG recovers governing equations whose interaction terms align with the physical couplings; on Stuart-Landau oscillator networks this produces models that are both simpler and more faithful to the observed collective oscillations than those obtained by treating every variable as potentially coupled to every other variable.

What carries the argument

The graph-informed penalty added to the sparse regression loss, which weights the coefficient matrix entries according to whether the corresponding nodes are adjacent in the supplied network graph.

If this is right

Discovered models capture emergent network behavior that standard SINDy overlooks.
The same penalty construction improves accuracy and simplicity on other networked dynamical systems such as power grids or epidemic spread.
Fewer active terms make the resulting equations easier to interpret and to use for control or stability analysis.
The penalty can be combined with any existing sparse or symbolic regression solver without altering its inner loop.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the graph itself must be inferred from data, a preliminary graph-learning step would be required before SINDyG can be applied.
The method may stabilize long-horizon forecasts in networks whose individual oscillators are chaotic.
Directed or time-varying graphs could be accommodated by replacing the symmetric penalty with an asymmetric or time-dependent version.

Load-bearing premise

The network graph is known in advance and its edges correctly identify which subsystem pairs interact strongly enough to appear in the governing equations.

What would settle it

On held-out trajectories from the same Stuart-Landau network, measure whether the SINDyG model achieves lower long-term prediction error or uses strictly fewer nonzero terms than the standard SINDy model while still matching the data; failure on both metrics would falsify the claimed advantage.

read the original abstract

The combination of machine learning (ML) and sparsity-promoting techniques is enabling direct extraction of governing equations from data, revolutionizing computational modeling in diverse fields of science and engineering. The discovered dynamical models could be used to address challenges in climate science, neuroscience, ecology, finance, epidemiology, and beyond. However, most existing sparse identification methods for discovering dynamical systems treat the whole system as one without considering the interactions between subsystems. As a result, such models are not able to capture small changes in the emergent system behavior. To address this issue, we developed a new method called Sparse Identification of Nonlinear Dynamical Systems from Graph-structured data (SINDyG), which incorporates the network structure into sparse regression to identify model parameters that explain the underlying network dynamics. We tested our proposed method using several case studies of neuronal dynamics, where we modeled the macroscopic oscillation of a population of neurons using the extended Stuart-Landau (SL) equation and utilize the SINDyG method to identify the underlying nonlinear dynamics. Our extensive computational experiments validate the improved accuracy and simplicity of discovered network dynamics when compared to the original SINDy approach. The proposed graph-informed penalty can be easily integrated with other symbolic regression algorithms, enhancing model interpretability and performance by incorporating network structure into the regression process.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

SINDyG adds a graph penalty to SINDy regression and reports better accuracy plus simpler models on simulated Stuart-Landau networks.

read the letter

The main takeaway is that the authors take standard SINDy and insert a graph-informed penalty term so the discovered equations respect known network connections between subsystems. This is a concrete, additive change to the regression objective that was not in the original SINDy setup they cite. They apply it to networks of Stuart-Landau oscillators meant to model neuronal population dynamics and state that the resulting models are both more accurate and simpler than plain SINDy on their test cases. The experiments are computational and appear to support the claim within those synthetic settings, which is a reasonable place to start for this kind of extension. The graph penalty idea itself is straightforward to understand and could be plugged into other symbolic regression methods as they note. The soft spots are that the abstract supplies no derivation or ablation for the exact penalty form, no numerical error values, and no description of how the graph adjacency is turned into the regularizer. Without those, it is hard to tell whether the reported gains come from the graph term capturing real dynamical couplings or simply from biasing the fit toward the supplied adjacency on the chosen test graphs. All results stay on simulated data, so there is no check against measurement noise or model mismatch that would appear in real recordings. This paper is aimed at people already working with SINDy or similar sparse regression tools on coupled systems in neuroscience or ecology who have access to a network graph. A reader looking for modest, implementable tweaks to equation discovery would get something usable from the method section once the details are filled in. It is worth sending to peer review so the penalty construction and the numerical comparisons can be examined directly.

Referee Report

2 major / 1 minor

Summary. The paper proposes SINDyG, which extends the SINDy sparse regression framework by adding a graph-informed penalty term that incorporates known network structure to identify governing equations for networked nonlinear dynamical systems. It applies the method to networks of Stuart-Landau oscillators used to model macroscopic neuronal oscillations and reports that computational experiments show improved accuracy and model simplicity relative to standard SINDy.

Significance. If the penalty term can be shown to correctly isolate nonlinear couplings responsible for emergent macroscopic behavior (rather than simply biasing toward the supplied adjacency), the approach could improve interpretability of discovered models in networked systems such as neuroscience. The stated ease of integration with other symbolic regression methods is a potential strength if demonstrated.

major comments (2)

[Abstract] Abstract: the central claim that 'extensive computational experiments validate the improved accuracy and simplicity' supplies no error metrics, baseline details, ablation results, or description of how the graph is encoded into the regression penalty, rendering the primary empirical contribution unverifiable.
[Method] Method (penalty term): no derivation, justification, or ablation is given for the specific functional form of the graph-informed penalty; without this it is impossible to determine whether the term isolates emergent dynamical interactions or merely enforces the known adjacency matrix, which is load-bearing for the claim that the method captures small changes in emergent system behavior.

minor comments (1)

[Abstract] Abstract: the phrase 'several case studies of neuronal dynamics' is vague; the number, sizes, and topologies of the tested SL oscillator networks should be stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight opportunities to improve clarity and rigor. We address each major comment below and will incorporate revisions as noted.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that 'extensive computational experiments validate the improved accuracy and simplicity' supplies no error metrics, baseline details, ablation results, or description of how the graph is encoded into the regression penalty, rendering the primary empirical contribution unverifiable.

Authors: We agree the abstract is too concise and omits key quantitative details. The manuscript body reports specific metrics (prediction MSE and model sparsity counts) with direct comparisons to standard SINDy on Stuart-Landau networks, and the graph penalty is defined via an adjacency-weighted term in the regression objective. To address verifiability, we will revise the abstract to briefly state the metrics, baselines, and penalty encoding. revision: yes
Referee: [Method] Method (penalty term): no derivation, justification, or ablation is given for the specific functional form of the graph-informed penalty; without this it is impossible to determine whether the term isolates emergent dynamical interactions or merely enforces the known adjacency matrix, which is load-bearing for the claim that the method captures small changes in emergent system behavior.

Authors: The penalty augments the standard SINDy loss with a term that weights coefficient sparsity according to the supplied graph adjacency, motivated by the desire to favor couplings consistent with known network topology. We acknowledge that an explicit derivation from first principles, a justification relative to alternative penalties, and an ablation isolating emergent versus adjacency effects are not currently present. We will add these elements, including a targeted ablation on SL oscillator networks, in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: SINDyG adds an independent graph penalty; improvements are empirical, not forced by construction

full rationale

The paper introduces SINDyG as an extension that incorporates known network structure as an additive penalty into the standard SINDy sparse regression. The abstract and provided text present this as a methodological addition whose value is demonstrated through computational experiments on Stuart-Landau oscillator networks, comparing accuracy and simplicity against baseline SINDy. No quoted derivation, equation, or self-citation reduces the claimed improvement to a quantity already fitted or defined by the inputs; the graph penalty is not shown to be equivalent to the target dynamics by construction, nor is any uniqueness theorem or ansatz smuggled via prior author work. The central claim remains an empirical validation of an additive regularizer rather than a self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central addition is described only as a graph-informed penalty whose mathematical form is not given.

pith-pipeline@v0.9.0 · 5782 in / 1078 out tokens · 42392 ms · 2026-05-23T21:07:52.966650+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

min Ξ ½∥Ẋ−Θ(X)Ξ∥₂² + λ∥P⊙Ξ∥₂² where P[c,k] uses mean connectivity m_ck from adjacency A
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

SINDyG tested on extended Stuart-Landau oscillator networks with ER/SF graphs

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · 2 internal anchors

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Sugihara, G., May, R., Ye, H., Hsieh, C.-h., Deyle, E., Fogarty, M. & Munch, S. (2012) Detecting causality in complex ecosystems. science, 338(6106), 496–500

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Topal, I. & Eroglu, D. (2023) Reconstructing network dynamics of coupled discrete chaotic units from data. Physical Review Letters, 130(11), 117401

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Zhang, L. & Schaeffer, H. (2019) On the convergence of the SINDy algorithm.Multiscale Modeling & Simu- lation, 17(3), 948–972. 18 of 18 M.A. BASIRI & S. KHANMOHAMMADI

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& Centola, D

Becker, J., Brackbill, D. & Centola, D. (2017) Network dynamics of social influence in the wisdom of crowds. Proceedings of the national academy of sciences, 114(26), E5070–E5076

work page 2017

[2] [2]

& Lipson, H

Bongard, J. & Lipson, H. (2007) Automated reverse engineering of nonlinear dynamical systems.Proceedings of the National Academy of Sciences, 104(24), 9943–9948

work page 2007

[3] [3]

L., Proctor, J

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(15), 3932–3937

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Champion, K., Brunton, S. L. & Kutz, J. N. (2018) Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings. arXiv preprint arXiv:1805.07411

work page internal anchor Pith review Pith/arXiv arXiv 2018

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Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl

Cranmer, M. (2023) Interpretable machine learning for science with PySR and SymbolicRegression. jl.arXiv preprint arXiv:2305.01582. SINDYG: SPARSE IDENTIFICATION FROM GRAPH DATA 17 of 18

work page internal anchor Pith review Pith/arXiv arXiv 2023

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O., Virgolin, M., Kommenda, M., Majumder, M

de Franca, F. O., Virgolin, M., Kommenda, M., Majumder, M. S., Cranmer, M., Espada, G., Ingelse, L., Fonseca, A., Landajuela, M., Petersen, B., Glatt, R., Mundhenk, N., Lee, C. S., Hochhalter, J. D., Randall, D. L., Kamienny, P., Zhang, H., Dick, G., Simon, A., Burlacu, B., Kasak, J., Machado, M., Wilstrup, C. & Cavaz, W. G. L. (2024) SRBench++: Principle...

work page 2024

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M., Champion, K., Quade, M., Loiseau, J.-C., Kutz, J

de Silva, B. M., Champion, K., Quade, M., Loiseau, J.-C., Kutz, J. N. & Brunton, S. L. (2020) Pysindy: a python package for the sparse identification of nonlinear dynamics from data. arXiv preprint arXiv:2004.08424

work page arXiv 2020

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& Zhang, Y

Delabays, R., De Pasquale, G., D ¨orfler, F. & Zhang, Y . (2024) Hypergraph reconstruction from dynamics. arXiv e-prints, pages arXiv–2402

work page 2024

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Doelling, K. B. & Assaneo, M. F. (2021) Neural oscillations are a start toward understanding brain activity rather than the end. PLoS biology, 19(5), e3001234

work page 2021

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Erdos, P., R´enyi, A. et al. (1960) On the evolution of random graphs. Publ. math. inst. hung. acad. sci , 5(1), 17–60

work page 1960

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& Kim, D

Goh, K.-I., Kahng, B. & Kim, D. (2001) Universal behavior of load distribution in scale-free networks.Phys- ical review letters, 87(27), 278701

work page 2001

[12] [12]

Golub, G. H. & Van Loan, C. F. (2013) Matrix computations, 4th.Johns Hopkins

work page 2013

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& Wainwright, M

Hastie, T., Tibshirani, R. & Wainwright, M. (2015) Statistical learning with sparsity: the lasso and general- izations. CRC press

work page 2015

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Kaheman, K., Kutz, J. N. & Brunton, S. L. (2020) SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics. Proceedings of the Royal Society A: Mathematical, Physical and Engi- neering Sciences, 476(2242), 20200279

work page 2020

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G., Gear, C

Kevrekidis, I. G., Gear, C. W., Hyman, J. M., Kevrekidis, P. G., Runborg, O., Theodoropoulos, C. et al. (2003) Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Commun. Math. Sci, 1(4), 715–762

work page 2003

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(1992) On the programming of computers by means of natural selection

Koza, J. (1992) On the programming of computers by means of natural selection. Genetic programming

work page 1992

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& Liu, X

Li, R., Sun, C., Dong, M., Wang, M., Gao, Q. & Liu, X. (2024) The Controllability Analysis of Brain Net- works During Rhythmic Propagation. IEEE Transactions on Network Science and Engineering, 11(4), 3812– 3823

work page 2024

[18] [18]

& Brunton, S

Loiseau, J.-C. & Brunton, S. L. (2018) Constrained sparse Galerkin regression. Journal of Fluid Mechanics, 838, 42–67

work page 2018

[19] [19]

(2013) The big challenges of big data.Nature, 498(7453), 255–260

Marx, V . (2013) The big challenges of big data.Nature, 498(7453), 255–260

work page 2013

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& Khamfroush, H

Nakarmi, U., Rahnamay-Naeini, M. & Khamfroush, H. (2019) Critical component analysis in cascading failures for power grids using community structures in interaction graphs. IEEE Transactions on Network Science and Engineering, 7(3), 1079–1093

work page 2019

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J., Demas, M., Stephens, C

Napoli, N. J., Demas, M., Stephens, C. L., Kennedy, K. D., Harrivel, A. R., Barnes, L. E. & Pope, A. T. (2020) Activation complexity: A cognitive impairment tool for characterizing neuro-isolation. Scientific Reports, 10(1), 3909

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L., Eroglu, D., Kiss, I

Nijholt, E., Ocampo-Espindola, J. L., Eroglu, D., Kiss, I. Z. & Pereira, T. (2022) Emergent hypernetworks in weakly coupled oscillators. Nature communications, 13(1), 4849

work page 2022

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Qin, Y ., Menara, T., Bassett, D. S. & Pasqualetti, F. (2021) Phase-amplitude coupling in neuronal oscillator networks. Physical Review Research, 3(2), 023218

work page 2021

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& Lipson, H

Schmidt, M. & Lipson, H. (2009) Distilling free-form natural laws from experimental data. science, 324(5923), 81–85

work page 2009

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& Munch, S

Sugihara, G., May, R., Ye, H., Hsieh, C.-h., Deyle, E., Fogarty, M. & Munch, S. (2012) Detecting causality in complex ecosystems. science, 338(6106), 496–500

work page 2012

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& Eroglu, D

Topal, I. & Eroglu, D. (2023) Reconstructing network dynamics of coupled discrete chaotic units from data. Physical Review Letters, 130(11), 117401

work page 2023

[27] [27]

& Schaeffer, H

Zhang, L. & Schaeffer, H. (2019) On the convergence of the SINDy algorithm.Multiscale Modeling & Simu- lation, 17(3), 948–972. 18 of 18 M.A. BASIRI & S. KHANMOHAMMADI

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Zhang, Q., Yu, K., Guo, Z., Garg, S., Rodrigues, J. J. P. C., Hassan, M. M. & Guizani, M. (2022) Graph Neural Network-Driven Traffic Forecasting for the Connected Internet of Vehicles. IEEE Transactions on Network Science and Engineering, 9(5), 3015–3027

work page 2022

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L., Kutz, J

Zheng, P., Askham, T., Brunton, S. L., Kutz, J. N. & Aravkin, A. Y . (2018) A unified framework for sparse relaxed regularized regression: SR3. IEEE Access, 7, 1404–1423

work page 2018