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REVIEW 2 major objections 1 minor 52 references

Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy

T0 review · 2 major / 1 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read Weak form SINDy recovers effective ODE models from stochastic network data where mean-field approximations fail.

desk verdict WSINDy applied to coupled online-offline network data recovers the generating mean-field ODEs from its own trajectories, with secondary averaged-stochastic runs that do not yet test the failure-regime claim. read the letter →

arxiv 2605.30432 v2 pith:Y5PC2FMM submitted 2026-05-28 math.DS cs.LGcs.SInlin.AOphysics.soc-ph

Moyi Tian , Daniel A. Messenger , Vanja Dukic , Nancy Rodríguez , David M. Bortz This is my paper · ORCID
classification math.DScs.LGcs.SInlin.AOphysics.soc-ph
keywords weakformSINDynetworkdynamicsstochasticprocessesmean-fieldapproximationODEdiscoverysocialnetworksdata-drivenmodeling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that governing ODEs for processes on networks can be identified directly from data generated by stochastic interactions, using the weak form SINDy method applied to multiple trajectories under varying noise. This is tested on a coupled online-offline social activity model, showing that accuracy improves with additional trajectories mainly at high noise levels but plateaus quickly. A reader would care because the approach yields models that match the observed data better than traditional mean-field reductions when those reductions are inaccurate. The work demonstrates recovery of continuum equations from averaged stochastic realizations on networks.

What carries the argument

Weak Form Sparse Identification of Nonlinear Dynamics (WSINDy), which discovers governing ODEs from noisy data using a weak formulation that integrates against test functions and handles multiple initial conditions.

What would settle it

Apply the learned ODEs to predict long-term behavior on fully simulated individual-level stochastic network trajectories and check whether they match those trajectories more closely than the original mean-field equations do.

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Extended reading notes

Core claim

When traditional mean-field approximations fail, identifying continuum ODEs directly from stochastic processes yields efficient models that better match the data and provide deeper insight into the underlying dynamics. The method uses WSINDy on data from a mean-field approximation of a stochastic interaction process, testing recovery under different noise levels and numbers of trajectories.

Load-bearing premise

The test data are generated by a mean-field approximation of the true stochastic network process.

Editorial extensions

If this is right

  • Accuracy gains from extra trajectories are largest at high noise but require only a small number to capture most benefit.
  • Effective ODE models can be learned from averaged realizations of the stochastic process on networks.
  • The recovered models provide insight into dynamics beyond what failing mean-field approximations capture.
  • The approach works for systems coupling multiple activity types such as online and offline social behavior.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same workflow could extend to other network processes such as epidemic spread or opinion dynamics where mean-field closures are known to be imprecise.
  • Real-time parameter updates might become feasible if the method scales to streaming social media interaction counts.
  • Comparing learned models across different network topologies could reveal which structural features most affect the recovered equations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper proposes using Weak Form Sparse Identification of Nonlinear Dynamics (WSINDy) to discover effective continuum ODE models directly from network dynamics data with multiple initial conditions. Experiments assess recovery accuracy on data generated by a mean-field approximation of a stochastic interaction process, reporting that additional trajectories improve performance at high noise levels with diminishing returns, and that the approach can yield models that better match data than traditional mean-field approximations when the latter fail, including via averaged stochastic trajectories.

Significance. If validated in appropriate regimes, the work could provide a practical data-driven alternative for obtaining reduced-order models of social network processes, particularly in settings where standard mean-field closures are inaccurate, thereby enabling more efficient simulation and insight from limited trajectory data.

major comments (2)
  1. [Abstract] Abstract: The claim that WSINDy 'yields efficient models that better match the data' specifically 'when traditional mean-field approximations fail' is not supported by the described validation, which generates test data from the mean-field approximation itself and measures recovery of that same approximation rather than direct comparison against a failing mean-field baseline on stochastic trajectories.
  2. [Abstract] Abstract: No quantitative metrics (e.g., L2 error, R² values), error bars, or explicit description of the accuracy measure are provided for the reported improvements with additional trajectories at high noise, preventing assessment of effect size or statistical significance.
minor comments (1)
  1. [Abstract] The abstract mentions 'averaged stochastic data on networks' but does not specify the averaging procedure, number of realizations, or how this differs from the mean-field data generation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on the abstract. We agree that the claims require precise support from the experiments and will revise the abstract accordingly for clarity. Our responses to the major comments are below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that WSINDy 'yields efficient models that better match the data' specifically 'when traditional mean-field approximations fail' is not supported by the described validation, which generates test data from the mean-field approximation itself and measures recovery of that same approximation rather than direct comparison against a failing mean-field baseline on stochastic trajectories.

    Authors: The manuscript includes two distinct experimental regimes. The first uses data generated from the mean-field approximation to quantify recovery accuracy under noise and varying numbers of trajectories. The second uses averaged trajectories from the underlying stochastic network process, where we explicitly compare WSINDy-derived ODEs against the traditional mean-field closure and show improved data match when the mean-field is inaccurate. The abstract claim refers to this second regime. We will revise the abstract to explicitly distinguish the two regimes and tie the 'when mean-field fails' statement only to the stochastic-trajectory results, with a brief pointer to the relevant section. revision: yes

  2. Referee: [Abstract] Abstract: No quantitative metrics (e.g., L2 error, R² values), error bars, or explicit description of the accuracy measure are provided for the reported improvements with additional trajectories at high noise, preventing assessment of effect size or statistical significance.

    Authors: We accept that the abstract would benefit from concrete quantitative anchors. Detailed L2 recovery errors, standard deviations across noise realizations, and the precise accuracy measure (normalized L2 distance to the true coefficients) appear in Section 4 and Figures 3–5. We will add one or two representative quantitative statements to the abstract (e.g., “additional trajectories reduce median coefficient error by X% at noise level σ=0.3”) while remaining within length limits. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected; derivation self-contained

full rationale

The paper applies the established WSINDy method to recover governing ODEs from data generated externally by a mean-field model or averaged stochastic trajectories. No equations, fitted parameters, or self-citations are shown that reduce a claimed prediction or result to its own inputs by construction. The central task is identifiability testing on independently generated data, with no self-definitional loop or load-bearing self-citation chain in the derivation.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new postulated entities; full manuscript unavailable for audit.

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Cite this review

Pith. "Pith review of Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy." pith.science (2026). https://pith.science/paper/Y5PC2FMM

@misc{pith2026260530432,
  author       = {Pith},
  title        = {Pith review of: Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y5PC2FMM}},
  note         = {Machine review of arXiv:2605.30432}
}
read the original abstract

Social systems consist of networks of individuals who influence one another through social interactions. Studying how processes evolve on these networks can help us better understand patterns of social behavior. We study a system that couples online and offline social activity and investigate how to learn effective models directly from data using Weak Form Sparse Identification of Nonlinear Dynamics (WSINDy), a method for discovering governing equations. We assess learning performance using data generated by a mean-field approximation model of a stochastic interaction process on networks and test how accurately the system can be recovered under different noise levels. Our results show that using more trajectories improves accuracy when noise is high, but only a small number of additional trajectories is needed to gain most of the benefit, with little improvement beyond that. We also learn effective ODE models from averaged stochastic data on networks. When traditional mean-field approximations fail, identifying continuum ODEs directly from stochastic processes yields efficient models that better match the data and provide deeper insight into the underlying dynamics.

Figures

Figures reproduced from arXiv: 2605.30432 by the authors.

Figure 1
Figure 1. Illustrations of the model: (a) The flow within and between the online and offline spaces; (b) Diagram of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. (a) Median TPR over 100 realizations with varying initial conditions in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Median equation error for (a) 𝐸 equation and (b) 𝑃 equation over 100 realizations with varying initial conditions in 𝐸(0). The same parameters are used for data generation. Significant improvement is observed when transitioning from 2 to 3 trajectories, with minor improvement beyond that. We conducted another set of experiments using generated data obtained by forward-solve with small fractions of offline activity a… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Median output error for (a) 𝐸 trajectory and (b) 𝑃 trajectory over 100 realizations with varying initial conditions in 𝐸(0). The same parameters are used for data generation. Improvements occur when the number of trajectories changes from 1 to 2 and from 2 to 3, but li…
Figure 5
Figure 5. Figure 5: Mean output error relative to training data in base-10 logarithmic scale over number of trajectories for [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Failure rate heatmap (a) and example line plots for different [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Mean output error for (a) 𝑈 trajectory, (b) 𝐸 trajectory, and (c) 𝑃 trajectory across average degree 𝑘 and number of trajectories on Erdos-R ˝ enyi random networks. Each grid shows an average over 1000 values, with ´ 100 realizations including a random network generati…
Figure 8
Figure 8. Figure 8: Comparison of (a) 𝑈, (b) 𝐸, and (c) 𝑃 trajectories among the sample-averaged stochastic data, the inferred solutions from the learned model based on data, and the solutions from the mean-field approximation model. The average degree is 𝑘 = 5 in this example, with 10 tr…
Figure 9
Figure 9. Figure 9: Counts of terms appearing across 100 learned models (panels 9a, 9c, and 9e) and box plots of their corresponding weights [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: (e, q)-identifiability summary for 100 realizations with varying initial conditions in 𝐸(0). (a) Heatmap of the average relative ℓ 2 error of the vectorized learned coefficient matrix, using the true coefficient matrix as the reference. (b) Analogous heatmap using the…
Figure 11
Figure 11. Figure 11: Cross-sectional distributions of (a) TPR and (b) parameter error over 100 independent noise realizations [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Cross-sectional distributions of equation error for (a) [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Cross-sectional distributions of output error for (a) [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Cross-sectional distributions of output error for (a) [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

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Works this paper leans on

52 extracted references · 2 canonical work pages

  1. [1]

    Traveling wave solutions in a model for social outbursts in a tension-inhibitive regime.Studies in Applied Mathematics, 147(2):650–674, 2021

    Marzieh Bakhshi, Anna Ghazaryan, Vahagn Manukian, and Nancy Rodriguez. Traveling wave solutions in a model for social outbursts in a tension-inhibitive regime.Studies in Applied Mathematics, 147(2):650–674, 2021

    2021

  2. [2]

    Mohammad Amin Basiri and Sina Khanmohammadi. SINDyG: Sparse identification of nonlinear dynamical systems from graph-structured data, with applications to Stuart–Landau oscillator networks.Journal of Complex Networks, 13(5):cnaf029, September 2025

    2025

  3. [3]

    Bastos, Dan Mercea, and Arthur Charpentier

    Marco T. Bastos, Dan Mercea, and Arthur Charpentier. Tents, tweets, and events: The interplay between ongoing protests and social media.Journal of Communication, 65(2):320–350, March 2015

    2015

  4. [4]

    Berestycki and J.-P

    H. Berestycki and J.-P. Nadal. Self-organised critical hot spots of criminal activity.European Journal of Applied Mathematics, 21(4–5):371–399, 2010

    2010

  5. [5]

    A model of riots dynamics: Shocks, diffusion and thresholds.Networks and Heterogeneous Media, 10(3):443–475, July 2015

    Henri Berestycki, Jean-Pierre Nadal, and Nancy Rod ´ıguez. A model of riots dynamics: Shocks, diffusion and thresholds.Networks and Heterogeneous Media, 10(3):443–475, July 2015

    2015

  6. [6]

    Gordon, Sebastian Roch ´e, Nancy Rodriguez, and Jean-Pierre Nadal

    Laurent Bonnasse-Gahot, Henri Berestycki, Marie-Aude Depuiset, Mirta B. Gordon, Sebastian Roch ´e, Nancy Rodriguez, and Jean-Pierre Nadal. Epidemiological modelling of the 2005 French riots: A spreading wave and the role of contagion.Scientific Reports, 8(1):107, January 2018

    2005

  7. [7]

    Bortz, Daniel A

    David M. Bortz, Daniel A. Messenger, and Vanja Dukic. Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics.Bulletin of Mathematical Biology, 85(110), 2023

    2023

  8. [8]

    Bortz, Daniel A

    David M. Bortz, Daniel A. Messenger, and April Tran. Chapter 2 - Weak form-based data-driven modeling: Computationally efficient and noise robust equation learning and parameter inference. In Siddhartha Mishra and Alex Townsend, editors,Numerical Analysis Meets Machine Learning, volume 25 ofHandbook of Numerical Analysis, pages 53–82. Elsevier, 2024

    2024

Show all 52 references
  1. [9]

    Brunton, Joshua L

    Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems.Proceedings of the National Academy of Sciences, 113(15):3932–3937, April 2016

    2016

  2. [10]

    Burbeck, Walter J

    Stephen L. Burbeck, Walter J. Raine, and M. J. Abudu Stark. The dynamics of riot growth: An epidemio- logical approach.The Journal of Mathematical Sociology, 6(1):1–22, 1978

    1978

  3. [11]

    Bortz, and Vanja Dukic

    Abhi Chawla, David M. Bortz, and Vanja Dukic. Bias and Coverage Properties of the WENDy-IRLS Al- gorithm. In Bharath Sriraman, editor,Handbook of Visual, Experimental and Computational Mathematics: Bridges through Data, pages 1–108. Springer Nature Switzerland, Cham, October 2025. 21

    2025

  4. [12]

    Darling and J.R

    R.W.R. Darling and J.R. Norris. Differential equation approximations for Markov chains.Probability Surveys, 5(none):37–79, 2008

    2008

  5. [13]

    Davies, Hannah M

    Toby P. Davies, Hannah M. Fry, Alan G. Wilson, and Steven R. Bishop. A mathematical model of the London riots and their policing.Scientific Reports, 3(1):1303, February 2013

    2013

  6. [14]

    Number of internet and social media users worldwide as of October 2025

    Statista Research Department. Number of internet and social media users worldwide as of October 2025. Technical report, Statista, December 2025

    2025

  7. [15]

    Ken T. D. Eames and Matt J. Keeling. Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases.Proceedings of the National Academy of Sciences, 99(20):13330–13335, October 2002

    2002

  8. [16]

    Ethier and Thomas G

    Stewart N. Ethier and Thomas G. Kurtz.Markov Processes – Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986

    1986

  9. [17]

    Gadi Fibich. Bass-SIR model for diffusion of new products in social networks.Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 94(3):032305, September 2016

    2016

  10. [18]

    Autonomous inference of complex network dynamics from incomplete and noisy data.Nature Computational Science, 2(3):160–168, March 2022

    Ting-Ting Gao and Gang Yan. Autonomous inference of complex network dynamics from incomplete and noisy data.Nature Computational Science, 2(3):160–168, March 2022

    2022

  11. [19]

    Bortz, and Youngsoo Choi

    Xiaolong He, April Tran, David M. Bortz, and Youngsoo Choi. Physics-informed active learning with simultaneous weak-form latent space dynamics identification.International Journal for Numerical Methods in Engineering, 126(1):e7634, January 2025

    2025

  12. [20]

    Nora Heitzman-Breen, Vanja Dukic, and David M. Bortz. A practical identifiability criterion leveraging weak-form parameter estimation, 2025

    2025

  13. [21]

    Digital 2026 Global Overview Report

    Simon Kemp. Digital 2026 Global Overview Report. Technical report, DataReportal, October 2025

    2026

  14. [22]

    Social media and protests: An examination of Twitter images of the 2011 Egyptian revolution.New Media & Society, 18(9):1973–1992, 2016

    Tamara Kharroub and Ozen Bas. Social media and protests: An examination of Twitter images of the 2011 Egyptian revolution.New Media & Society, 18(9):1973–1992, 2016

    2011

  15. [23]

    Kiss, Joel C

    Istv ´an Z. Kiss, Joel C. Miller, and P ´eter L. Simon.Mathematics of Epidemics on Networks: From Ex- act to Approximate Models, volume 46 ofInterdisciplinary Applied Mathematics. Springer International Publishing, 2017

    2017

  16. [24]

    Lagergren, John T

    John H. Lagergren, John T. Nardini, G. Michael Lavigne, Erica M. Rutter, and Kevin B. Flores. Learning partial differential equations for biological transport models from noisy spatio-temporal data.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Scie...

    2020

  17. [25]

    Learning interaction kernels in mean-field equations of first-order systems of interacting particles.SIAM Journal on Scientific Computing, 44(1):A260–A285, 2022

    Quanjun Lang and Fei Lu. Learning interaction kernels in mean-field equations of first-order systems of interacting particles.SIAM Journal on Scientific Computing, 44(1):A260–A285, 2022

    2022

  18. [26]

    Donglin Liu and Alexandros Sopasakis. Enhancing sparse identification of nonlinear dynamics with Earth-Mover distance and group similarity.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(3):033139, March 2025

    2025

  19. [27]

    Rainey Lyons, Vanja Dukic, and David M. Bortz. Learning structured population models from data with WSINDy.PLOS Computational Biology, 21(12):e1013742, December 2025. 22

    2025

  20. [28]

    Learning interpretable closures for thermal radiation transport in optically-thin media using WSINDy, 2025

    Daniel Messenger, Ben Southworth, Hans Hammer, and Luis Chacon. Learning interpretable closures for thermal radiation transport in optically-thin media using WSINDy, 2025

    2025

  21. [29]

    Messenger and David M

    Daniel A. Messenger and David M. Bortz. Weak SINDy for partial differential equations.Journal of Computational Physics, 443:110525, 2021

    2021

  22. [30]

    Messenger and David M

    Daniel A. Messenger and David M. Bortz. Weak sindy: Galerkin-based data-driven model selection. Multiscale Modeling & Simulation, 19(3):1474–1497, 2021

    2021

  23. [31]

    Messenger and David M

    Daniel A. Messenger and David M. Bortz. Learning mean-field equations from particle data using WSINDy. Physica D: Nonlinear Phenomena, 439:133406, November 2022

    2022

  24. [32]

    Asymptotic consistency of the WSINDy algorithm in the limit of continuum data.IMA Journal of Numerical Analysis, page drae086, December 2024

    Daniel A Messenger and David M Bortz. Asymptotic consistency of the WSINDy algorithm in the limit of continuum data.IMA Journal of Numerical Analysis, page drae086, December 2024

    2024

  25. [33]

    Messenger, Joshua W

    Daniel A. Messenger, Joshua W. Burby, and David M. Bortz. Coarse-Graining Hamiltonian Systems Using WSINDy.Scientific Reports, 14(14457):1–24, June 2024

    2024

  26. [34]

    Messenger, Emiliano Dall’ Anese, and David M

    Daniel A. Messenger, Emiliano Dall’ Anese, and David M. Bortz. Online Weak-form Sparse Identification of Partial Differential Equations. InProceedings of The Third Mathematical and Scientific Machine Learning Conference, volume 190 ofProceedings of Machine Learning Research, p...

    2022

  27. [35]

    Messenger, April Tran, Vanja Dukic, and David M

    Daniel A. Messenger, April Tran, Vanja Dukic, and David M. Bortz. The Weak Form Is Stronger Than You Think.SIAM News, 57(8), October 2024

    2024

  28. [36]

    Elderd, David M

    Seth Minor, Bret D. Elderd, David M. Bortz, and Vanja Dukic. Weak Form Learning for Mean-Field Partial Differential Equations: An Application to Insect Movement.arXiv:2510.07786, October 2025

    2025

  29. [37]

    Messenger, Vanja Dukic, and David M

    Seth Minor, Daniel A. Messenger, Vanja Dukic, and David M. Bortz. Learning Physically Interpretable Atmospheric Models from Data with WSINDy.Journal of Geophysical Research: Machine Learning and Computation, 2(3):e2025JH000602, September 2025

    2025

  30. [38]

    Hawkes binomial topic model with applications to coupled conflict-Twitter data.The Annals of Applied Statistics, 14(4):1984–2002, 2020

    George Mohler, Erin McGrath, Cody Buntain, and Gary LaFree. Hawkes binomial topic model with applications to coupled conflict-Twitter data.The Annals of Applied Statistics, 14(4):1984–2002, 2020

    1984

  31. [39]

    Learning opinion dynamics from social traces

    Corrado Monti, Gianmarco De Francisci Morales, and Francesco Bonchi. Learning opinion dynamics from social traces. InProceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, Kdd ’20, pages 764–773, New York, NY, USA, 2020. Association f...

    2020

  32. [40]

    M. E. J. Newman. The structure and function of complex networks.SIAM Review, 45(2):167–256, 2003

    2003

  33. [41]

    Vixie, Arghavan Talebanpour, and Yunfeng Hu

    Hossein Noorazar, Kevin R. Vixie, Arghavan Talebanpour, and Yunfeng Hu. From classical to modern opinion dynamics.International Journal of Modern Physics C, 31(07):2050101, July 2020

    2020

  34. [42]

    Epidemic dynamics and endemic states in complex networks.Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 63(6):066117, May 2001

    Romualdo Pastor-Satorras and Alessandro Vespignani. Epidemic dynamics and endemic states in complex networks.Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 63(6):066117, May 2001

    2001

  35. [43]

    Lindstrom, Christian Parkinson, Chuntian Wang, Andrea L

    Kaiyan Peng, Zheng Lu, Vanessa Lin, Michael R. Lindstrom, Christian Parkinson, Chuntian Wang, Andrea L. Bertozzi, and Mason A. Porter. A multilayer network model of the coevolution of the spread of a disease and competing opinions.Mathematical Models and Methods in Applied Sci...

    2021

  36. [44]

    Frontiers in Applied Dynamical Systems: Reviews and Tutorials

    Mason Porter and James Gleeson.Dynamical Systems on Networks: A Tutorial. Frontiers in Applied Dynamical Systems: Reviews and Tutorials. Springer Cham, 1 edition, 2016. 23

    2016

  37. [45]

    Predicting network dynamics without requiring the knowledge of the interaction graph.Proceedings of the National Academy of Sciences, 119(44):e2205517119, November 2022

    Bastian Prasse and Piet Van Mieghem. Predicting network dynamics without requiring the knowledge of the interaction graph.Proceedings of the National Academy of Sciences, 119(44):e2205517119, November 2022

    2022

  38. [46]

    Proctor, Steven L

    Joshua L. Proctor, Steven L. Brunton, and J. Nathan Kutz. Generalizing Koopman Theory to Allow for Inputs and Control.SIAM Journal on Applied Dynamical Systems, 17(1):909–930, January 2018

    2018

  39. [47]

    Jeffrey Brantingham, and Nancy Rodr´ıguez

    Moyi Tian, P. Jeffrey Brantingham, and Nancy Rodr´ıguez. Modelling the spillover from online engagement to offline protest: Stochastic dynamics and mean-field approximations on networks.Journal of Complex Networks, 2026

    2026

  40. [48]

    Jeffrey Brantingham, and Nancy Rodr´ıguez

    Moyi Tian, George Mohler, P. Jeffrey Brantingham, and Nancy Rodr´ıguez. Learning dynamics from online- offline systems of LLM agents, 2026

    2026

  41. [49]

    April Tran and David M. Bortz. Weak Form Scientific Machine Learning: Test Function Construction for System Identification.arXiv:2507.03206, July 2025

    2025

  42. [50]

    Messenger, Youngsoo Choi, and David M

    April Tran, Xiaolong He, Daniel A. Messenger, Youngsoo Choi, and David M. Bortz. Weak-Form Latent Space Dynamics Identification.Computer Methods in Applied Mechanics and Engineering, 427:116998, July 2024

    2024

  43. [51]

    Messenger, David M

    Gina Vasey, Daniel A. Messenger, David M. Bortz, Andrew Christlieb, and Brian O’Shea. Influence of initial conditions on data-driven model identification and information entropy for ideal mhd problems.Journal of Computational Physics, 524:113719, March 2025

    2025

  44. [52]

    Reuben R. W. Wang and Daniel A. Messenger. Physics-guided weak-form discovery of reduced-order models for trapped ultracold hydrodynamics.Physical Review A, 111(3):033311, March 2025. 24

    2025

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Reviewed June 29, 2026 · model on record in the stance chip above.