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Data-Driven Discovery of High-Dimensional Dynamical Systems with Sparse Interpretable Neural Networks

Siyuan Xing1,*, Qingyu Han2, Efstathios G. Charalampidis3,4, and Ying-Cheng Lai5

  • *Contact author: sixing@calpoly.edu
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PRX Intelligence 1, 013024 – Published 29 September, 2026

DOI: https://doi.org/10.1103/pdkf-98zb

Abstract

Existing approaches to explicit data-driven discovery of nonlinear dynamical systems face a curse of dimensionality because candidate libraries grow combinatorially with system dimension. This challenge is particularly severe for high-dimensional systems arising from spatially discretized fields and large complex networks. We develop sparse regression embedded interpretable network (SREINet), a machine-learning framework that embeds sparse regression in an interpretable neural network and uses a sparsity-promoting periodic pruning scheme. Across six paradigmatic systems, SREINet accurately identifies velocity fields with more than 100 dimensions and extrapolates coherent structures from untrained data. Head-to-head benchmarks against representative equation-discovery and predictive methods on clean and noisy systems show that SREINet combines scalable computation with accurate, parsimonious equation recovery, particularly in high-dimensional settings. We further validate the framework using empirical data from a triple-pendulum experiment. These results demonstrate a practical route toward interpretable model discovery for large-scale nonlinear systems and suggest that the framework can be extended to systems with thousands of dimensions.

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References (71)

  1. P. Davidson, Turbulence: An Introduction for Scientists and Engineers (Oxford University Press, Oxford, UK, 2004).
  2. P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment (Springer, Berlin, Heidelberg, 2008), Vol. 45.
  3. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, UK, 2001).
  4. J. V. José and E. J. Saletan, Classical Dynamics: A Contemporary Approach (Cambridge University Press, Cambridge, UK, 1998).
  5. J. P. Crutchfield and B. McNamara, Equations of motion from a data series, Complex Syst. 1, 417 (1987).
  6. T. Sauer, Reconstruction of dynamical systems from interspike intervals, Phys. Rev. Lett. 72, 3811 (1994).
  7. U. Parlitz, Estimating model parameters from time series by autosynchronization, Phys. Rev. Lett. 76, 1232 (1996).
  8. C. Yao and E. M. Bollt, Modeling and nonlinear parameter estimation with Kronecker product representation for coupled oscillators and spatiotemporal systems, Physica D 227, 78 (2007).
  9. J. Bongard and H. Lipson, Automated reverse engineering of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA 104, 9943 (2007).
  10. M. Schmidt and H. Lipson, Distilling free-form natural laws from experimental data, Science 324, 81 (2009).
  11. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, MA, USA, 1989).
  12. S.-M. Udrescu and M. Tegmark, AI Feynman: A physics-inspired method for symbolic regression, Sci. Adv. 6, eaay2631 (2020).
  13. B. K. Petersen, M. Landajuela, T. N. Mundhenk, C. P. Santiago, S. K. Kim, and J. T. Kim, in Proceedings of the International Conference on Learning Representations (OpenReview.net, Virtual Event, Austria, 2021).
  14. W.-X. Wang, R. Yang, Y.-C. Lai, V. Kovanis, and C. Grebogi, Predicting catastrophes in nonlinear dynamical systems by compressive sensing, Phys. Rev. Lett. 106, 154101 (2011).
  15. Y.-C. Lai, Finding nonlinear system equations and complex network structures from data: A sparse optimization approach, Chaos 31, 082101 (2021).
  16. R. G. Baraniuk, Compressive sensing, IEEE Signal Process. Mag. 24, 118 (2007).
  17. S. L. Brunton, J. L. Proctor, and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA 113, 3932 (2016).
  18. K. Ikeda, Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Opt. Commun. 30, 257 (1979).
  19. P. Gelß, S. Klus, J. Eisert, and C. Schütte, Multidimensional approximation of nonlinear dynamical systems, J. Comput. Nonlinear Dyn. 14, 061006 (2019).
  20. R. González-García, R. Rico-Martínez, and I. Kevrekidis, Identification of distributed parameter systems: A neural net based approach, Comp. Chem. Eng. 22, S965 (1998).
  21. M. Reichstein, G. Camps-Valls, B. Stevens, M. Jung, J. Denzler, N. Carvalhais, and Prabhat, Deep learning and process understanding for data-driven Earth system science, Nature (London) 566, 195 (2019).
  22. D. Durstewitz, G. Koppe, and M. I. Thurm, Reconstructing computational system dynamics from neural data with recurrent neural networks, Nat. Rev. Neurosci. 24, 693 (2023).
  23. R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, in Advances in Neural Information Processing Systems (NeurIPS) (Curran Associates, Inc., Red Hook, NY, 2018), pp. 6571–6583.
  24. H. Jaeger, The “Echo State” Approach to Analysing and Training Recurrent Neural Networks, GMD Report No. 148 (GMD Forschungszentrum Informationstechnik, Sankt Augustin, Germany, 2001), doi: 10.24406/publica-fhg-291111.
  25. H. Jaeger and H. Haas, Harnessing nonlinearity: Predicting chaotic systems and saving energy in wireless communication, Science 304, 78 (2004).
  26. L.-W. Kong, Y. Weng, B. Glaz, M. Haile, and Y.-C. Lai, Reservoir computing as digital twins for nonlinear dynamical systems, Chaos 33, 033111 (2023).
  27. K. Champion, B. Lusch, J. N. Kutz, and S. L. Brunton, Data-driven discovery of coordinates and governing equations, Proc. Natl. Acad. Sci. USA 116, 22445 (2019).
  28. M. Cranmer, A. Sanchez Gonzalez, P. Battaglia, R. Xu, K. Cranmer, D. Spergel, and S. Ho, in Proceedings of the 34th Conference on Neural Information Processing Systems (Vancouver, Canada, 2020), pp. 17429–17442.
  29. I. Mezić, The Koopman Operator in Systems and Control: Concepts, Methodologies, and Applications, Lecture Notes in Control and Information Sciences (Springer, Cham, Switzerland, 2020), Vol. 484.
  30. M. O. Williams, C. W. Rowley, and I. G. Kevrekidis, A kernel-based method for data-driven Koopman spectral analysis, J. Comp. Dyn. 2, 247 (2015).
  31. S. Sahoo, C. Lampert, and G. Martius, in International Conference on Machine Learning (PMLR, 2018), pp. 4442–4450.
  32. H. F. Tsoi, V. Loncar, S. Dasu, and P. Harris, SymbolNet: Neural symbolic regression with adaptive dynamic pruning for compression, Mach. Learn.: Sci. Technol. 6, 015021 (2025).
  33. Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljačić, T. Y. Hou, and M. Tegmark, KAN: Kolmogorov-Arnold networks, in Proceedings of the Thirteenth International Conference on Learning Representations (ICLR 2025) (OpenReview.net, 2025).
  34. Z. Liu, P. Ma, Y. Wang, W. Matusik, and M. Tegmark, KAN 2.0: Kolmogorov-Arnold networks meet science, arXiv:2408.10205.
  35. A. N. Kolmogorov, On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR 114, 953 (1957).
  36. A. B. Givental, B. A. Khesin, J. E. Marsden, A. N. Varchenko, V. A. Vassiliev, O. Y. Viro, and V. M. Zakalyukin, On the representation of functions of several variables as a superposition of functions of a smaller number of variables, in Collected Works: Representations of Functions, Celestial Mechanics and KAM Theory, 1957–1965 (Springer, Berlin, 2009), pp. 25–46.
  37. S. Panahi, M. Moradi, E. M. Bollt, and Y.-C. Lai, Data-driven model discovery with Kolmogorov-Arnold networks, Phys. Rev. Res. 7, 023037 (2025).
  38. S. Xing, Q. Han, and E. G. Charalampidis, CombOpNet: A neural-network accelerator for SINDy, J. Vib. Test. Sys. Dyn. 9, 1 (2025).
  39. See Supplemental Material at http://link.aps.org/supplemental/10.1103/pdkf-98zb for a mathematical proof, computational-complexity analysis, algorithmic details, data-generation and training procedures, and additional numerical and experimental results, which includes Ref. [40].
  40. A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, Automatic differentiation in machine learning: A survey, J. Mach. Learn. Res. 18, 1 (2018).
  41. R. Tibshirani, Regression shrinkage and selection via the lasso, J. R. Stat. Soc. Ser. B (Method.) 58, 267 (1996).
  42. A. E. Hoerl and R. W. Kennard, Ridge regression: Biased estimation for nonorthogonal problems, Technometrics 12, 55 (1970).
  43. M. Zhu and S. Gupta, in Proceedings of the 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30–May 3, 2018, Workshop Track Proceedings (OpenReview.net, 2018).
  44. M. Zhang, S. Kim, P. Y. Lu, and M. Soljačić, Deep learning and symbolic regression for discovering parametric equations, IEEE Trans. Neural Networks Learn. Syst. 35, 16775 (2024).
  45. C. M. Bishop and H. Bishop, Deep Learning (Springer Nature, Cham, Switzerland, 2024).
  46. E. N. Lorenz, Predictability—A problem partly solved, in Seminar on Predictability (European Centre for Medium-Range Weather Forecasts (ECMWF), Reading, UK, 1995), Vol. 1, pp. 1–18.
  47. P. G. Kevrekidis and J. Cuevas-Maraver, A Dynamical Perspective on the ϕ4 Model: Past, Present and Future (Springer, Cham, Switzerland, 2019).
  48. P. Kevrekidis, The Discrete Nonlinear Schrödinger Equation, 1st ed. (Springer-Verlag, Heidelberg, 2009).
  49. M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations, J. Math. Phys. 16, 598 (1975).
  50. B. de Silva, K. Champion, M. Quade, J.-C. Loiseau, J. Kutz, and S. Brunton, PySINDy: A Python package for the sparse identification of nonlinear dynamical systems from data, JOSS 5, 2104 (2020).
  51. N. Manton and P. Sutcliffe, Topological Solitons (Cambridge University Press, Cambridge, UK, 2004).
  52. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, London, 2003).
  53. M. A. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, UK, 1991).
  54. C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue Waves in the Ocean, Advances in Geophysical and Environmental Mechanics and Mathematics (Springer-Verlag, Berlin, Heidelberg, 2009).
  55. J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. B 221, 87 (1984).
  56. J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77, 137 (2005).
  57. K. Kaheman, U. Fasel, J. J. Bramburger, B. Strom, J. N. Kutz, and S. L. Brunton, The experimental multi-arm pendulum on a cart: A benchmark system for chaos, learning, and control, HardwareX 15, e00465 (2023).
  58. A. Savitzky and M. J. E. Golay, Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem. 36, 1627 (1964).
  59. C. Hoffmann, E. Charalampidis, D. Frantzeskakis, and P. Kevrekidis, Peregrine solitons and gradient catastrophes in discrete nonlinear Schrödinger systems, Phys. Lett. A 382, 3064 (2018).
  60. W. Zhu, W. Khademi, E. G. Charalampidis, and P. G. Kevrekidis, Neural networks enforcing physical symmetries in nonlinear dynamical lattices: The case example of the Ablowitz–Ladik model, Physica D 434, 133264 (2022).
  61. B. Prokop and L. Gelens, From biological data to oscillator models using SINDy, iScience 27, 109316 (2024).
  62. E. Fermi, J. Pasta, and S. Ulam, Studies of Nonlinear Problems, I, Technical Report LA-1940 (Los Alamos Scientific Laboratory, Los Alamos, NM, 1955).
  63. I. V. Oseledets, Tensor-train decomposition, SIAM J. Sci. Comput. 33, 2295 (2011).
  64. M. J. Panaggio, M. V. Ciocanel, L. Lazarus, C. M. Topaz, and B. Xu, Model reconstruction from temporal data for coupled oscillator networks, Chaos 29, 103116 (2019).
  65. A. Banerjee, J. D. Hart, R. Roy, and E. Ott, Machine learning link inference of noisy delay-coupled networks with optoelectronic experimental tests, Phys. Rev. X 11, 031014 (2021).
  66. K. Srinivasan, N. Coble, J. Hamlin, T. Antonsen, E. Ott, and M. Girvan, Parallel machine learning for forecasting the dynamics of complex networks, Phys. Rev. Lett. 128, 164101 (2022).
  67. N. M. Mangan, S. L. Brunton, J. L. Proctor, and J. N. Kutz, Inferring biological networks by sparse identification of nonlinear dynamics, IEEE Trans. Mol. Biol. Multi-Scale Commun. 2, 52 (2016).
  68. K. Kaheman, J. N. Kutz, and S. L. Brunton, SINDy-PI: A robust algorithm for parallel implicit sparse identification of nonlinear dynamics, Proc. R. Soc. A 476, 20200279 (2020).
  69. S. Xing and Q. Han, SREINet: Sparse regression embedded interpretable network, version v1.0.0 [software], Zenodo, 2026, https://doi.org/10.5281/zenodo.22048994.
  70. https://github.com/siyuan-xing/SREINet.
  71. S. Xing, SREINet: Data and figure source files for data-driven discovery of high-dimensional dynamical systems with sparse interpretable neural networks, version v2 [dataset], Zenodo, 2026, https://doi.org/10.5281/zenodo.22058158.

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PRX Intelligence 1, 013024– Published 29 September, 2026
  • Received 6 March 2026
  • Accepted 19 August 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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