ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


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Mukhin D. Н., Seleznev A. Ф., Gavrilov A. С., Feigin A. M. Optimal data-driven models of forced dynamical systems: General approach and examples from climate. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 4, pp. 571-602. DOI: 10.18500/0869-6632-2021-29-4-571-602

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
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Language: 
Russian
Article type: 
Review
UDC: 
530.182

Optimal data-driven models of forced dynamical systems: General approach and examples from climate

Autors: 
Mukhin Dmitry Николаевич, Institute of Applied Physics of the Russian Academy of Sciences
Seleznev Aleksei Фёдорович, Institute of Applied Physics of the Russian Academy of Sciences
Gavrilov Andrey Сергеевич, Institute of Applied Physics of the Russian Academy of Sciences
Feigin Aleksandr Markovich, Institute of Applied Physics of the Russian Academy of Sciences
Abstract: 

Purpose. Purpose of this article is to review recent results (over the past three years) obtained at the Institute of Applied Physics (IAP RAS) relating of applications of the method for constructing optimal empirical models to climatic systems. Methods. This method, developed by the authors of the article, includes the construction of reduced models of the system under study in the form of random dynamical systems. In combination with Bayesian optimization of the model structure, this method allows us to reconstruct statistically justified laws underlying the observed dynamics. Results. The article describes results of applying this method to modeling three climatic subsystems corresponding to different time scales: the Pleistocene climate characterized by glacial cycles, El Nino – Southern Oscillation in the modern climate – a phenomenon with a scale of the order of a year, and the climate of the tropical Pacific Ocean on a centennial scale. Conclusions. Based on the presented results, it can be concluded that the method used for constructing optimal models is a useful tool for verifying the mechanisms underlying the observed climatic variability, e.g., analyzing the response of the system to external signals.

Acknowledgments: 
The results described in the sections 1, 2.1 и 2.2 were supported by the Russian Science Foundation (grant 19-42-04121). The result from the section 2.3 was supported by the Russian Foundation for Basic Research (grant 19-02-00502)
Reference: 
  1. Bezruchko BP, Smirnov DA. Extracting Knowledge From Time Series: An Introduction to Nonlinear Empirical Modeling. In Springer Series in Synergetics. New York: Springer; 2010. DOI: 10.1007/978-3-642-12601-7
  2. Abarbanel HDI. Analysis of Observed Chaotic Data. New York: Springer; 1996. 272 p. DOI: 10.1007/978-1-4612-0763-4.
  3. Anishchenko VS, Astakhov V, Neiman A, Vadivasova T, Schimansky-Geier L. Nonlinear Dynamics of Chaotic and Stochastic Systems: Tutorial and Modern Developments. Springer Series in Synergetics. Berlin, Heidelberg: Springer-Verlag; 2007. 446 p. DOI: 10.1007/978-3-540-38168-6.
  4. Gouesbet G, Letellier C. Global vector-field reconstruction by using a multivariate polynomial L2 approximation on nets. Phys. Rev. E. 1994;49(6):4955–4972. DOI: 10.1103/PhysRevE.49.4955.
  5. Anishchenko V, Pavlov A, Janson N. Global reconstruction in the presence of a priori information. Chaos, Solitons & Fractals. 1998;9(8):1267–1278. DOI: 10.1016/S0960-0779(98)00061-7.
  6. Schelter B, Mader M, Mader W, Sommerlade L, Platt B, Lai YC, Grebogi C, Thiel M. Overarching framework for data-based modelling. EPL. 2014;105(3):30004. DOI: 10.1209/0295-5075/105/30004.
  7. Gorur-Shandilya S, Timme M. Inferring network topology from complex dynamics. New J. Phys. 2011;13:013004. DOI: 10.1088/1367-2630/13/1/013004.
  8. Wang WX, Yang R, Lai YC, Kovanis V, Grebogi C. Predicting catastrophes in nonlinear dynamical systems by compressive sensing. Phys. Rev. Lett. 2011;106(15):154101. DOI: 10.1103/PhysRevLett.106.154101.
  9. Baake E, Baake M, Bock HG, Briggs KM. Fitting ordinary differential equations to chaotic data. Phys. Rev. A. 1992;45(8):5524–5529. DOI: 10.1103/PhysRevA.45.5524.
  10. Bezruchko BP, Smirnov DA, Sysoev IV. Identification of chaotic systems with hidden variables (modified Bock’s algorithm). Chaos, Solitons & Fractals. 2006;29(1):82–90. DOI: 10.1016/j.chaos.2005.08.204.
  11. Gorodetskyi V, Osadchuk M. Analytic reconstruction of some dynamical systems. Phys. Lett. A. 2013;377(9):703–713. DOI: 10.1016/j.physleta.2012.12.043.
  12. Mukhin DN, Feigin AM, Loskutov EM, Molkov YI. Modified Bayesian approach for the reconstruction of dynamical systems from time series. Phys. Rev. E. 2006;73(3):036211. DOI: 10.1103/PhysRevE.73.036211.
  13. Molkov YI, Mukhin DN, Loskutov EM, Timushev RI, Feigin AM. Prognosis of qualitative system behavior by noisy, nonstationary, chaotic time series. Phys. Rev. E. 2011;84(3):036215. DOI: 10.1103/PhysRevE.84.036215.
  14. Molkov YI, Loskutov EM, Mukhin DN, Feigin AM. Random dynamical models from time series. Phys. Rev. E. 2012;85(3):036216. DOI: 10.1103/PhysRevE.85.036216.
  15. Bezruchko BP, Smirnov DA. Constructing nonautonomous differential equations from experimental time series. Phys. Rev. E. 2001;63(1):016207. DOI: 10.1103/PhysRevE.63.016207.
  16. Smirnov DA, Sysoev IV, Seleznev EP, Bezruchko BP. Reconstructing nonautonomous system models with discrete spectrum of external action. Tech. Phys. Lett. 2003;29(10):824–827. DOI: 10.1134/1.1623857.
  17. Ponomarenko VI, Prokhorov MD. Extracting information masked by the chaotic signal of a time-delay system. Phys. Rev. E. 2002;66(2):026215. DOI: 10.1103/PhysRevE.66.026215.
  18. Sysoev IV, Prokhorov MD, Ponomarenko VI, Bezruchko BP. Reconstruction of ensembles of coupled time-delay systems from time series. Phys. Rev. E. 2014;89(6):062911. DOI: 10.1103/Phys-RevE.89.062911.
  19. Sysoev IV, Ponomarenko VI, Kulminskiy DD, Prokhorov MD. Recovery of couplings and parameters of elements in networks of time-delay systems from time series. Phys. Rev. E. 2016;94(5):052207. DOI: 10.1103/PhysRevE.94.052207.
  20. Han X, Shen Z, Wang WX, Di Z. Robust reconstruction of complex networks from sparse data. Phys. Rev. Lett. 2015;114(2):028701. DOI: 10.1103/PhysRevLett.114.028701.
  21. Brunton SL, Proctor JL, Kutz JN. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences of the United States of America. 2016;113(15):3932–3937. DOI: 10.1073/pnas.1517384113.
  22. Mangan NM, Brunton SL, Proctor JL, Kutz JN. Inferring biological networks by sparse identification of nonlinear dynamics. IEEE Trans. Mol. Biol. Multi-Scale Commun. 2016;2(1):52–63. DOI: 10.1109/TMBMC.2016.2633265.
  23. Pikovsky A. Reconstruction of a neural network from a time series of firing rates. Phys. Rev. E. 2016;93(6):062313. DOI: 10.1103/PhysRevE.93.062313.
  24. Sysoev IV, Ponomarenko VI, Pikovsky A. Reconstruction of coupling architecture of neural field networks from vector time series. Communications in Nonlinear Science and Numerical Simulation. 2018;57:342–351. DOI: 10.1016/j.cnsns.2017.10.006.
  25. Feigin AM, Molkov YI, Mukhin DN, Loskutov EM. Investigation of nonlinear dynamical properties by the observed complex behaviour as a basis for construction of dynamical models of atmospheric photochemical systems. Faraday Discuss. 2002;120:105–123. DOI: 10.1039/b102985c.
  26. Loskutov EM, Molkov YI, Mukhin DN, Feigin AM. Markov chain Monte Carlo method in Bayesian reconstruction of dynamical systems from noisy chaotic time series. Phys. Rev. E. 2008;77(6):066214. DOI: 10.1103/PhysRevE.77.066214.
  27. Mukhin D, Loskutov E, Mukhina A, Feigin A, Zaliapin I, Ghil M. Predicting critical transitions in ENSO models. Part I: Methodology and simple models with memory. Journal of Climate. 2015;28(5):1940–1961. DOI: 10.1175/JCLI-D-14-00239.1.
  28. Mukhin D, Kondrashov D, Loskutov E, Gavrilov A, Feigin A, Ghil M. Predicting critical transitions in ENSO models. Part II: Spatially dependent models. Journal of Climate. 2015;28(5):1962– 1976. DOI: 10.1175/JCLI-D-14-00240.1.
  29. Molkov YI, Mukhin DN, Loskutov EM, Feigin AM, Fidelin GA. Using the minimum description length principle for global reconstruction of dynamic systems from noisy time series. Phys. Rev. E. 2009;80(4):046207. DOI: 10.1103/PhysRevE.80.046207.
  30. Gavrilov A, Loskutov E, Mukhin D. Bayesian optimization of empirical model with state dependent stochastic forcing. Chaos, Solitons & Fractals. 2017;104:327–337. DOI: 10.1016/j.chaos.2017.08.032.
  31. Arnold L. Random Dynamical Systems. Springer Monographs in Mathematics. Berlin: SpringerVerlag; 1998. 586 p. DOI: 10.1007/978-3-662-12878-7.
  32. Gavrilov A, Seleznev A, Mukhin D, Loskutov E, Feigin A, Kurths J. Linear dynamical modes as new variables for data-driven ENSO forecast. Clim. Dyn. 2019;52(3–4):2199–2216. DOI: 10.1007/s00382-018-4255-7.
  33. Mukhin D, Gavrilov A, Loskutov E, Kurths J, Feigin A. Bayesian data analysis for revealing causes of the middle pleistocene transition. Sci. Rep. 2019;9(1):7328. DOI: 10.1038/s41598-019-43867-3.
  34. Seleznev A, Mukhin D, Gavrilov A, Loskutov E, Feigin A. Bayesian framework for simulation of dynamical systems from multidimensional data using recurrent neural network. Chaos. 2019;29(12):123115. DOI: 10.1063/1.5128372.
  35. Mukhin D, Gavrilov A, Seleznev A, Buyanova M. An atmospheric signal lowering the spring predictability barrier in statistical ENSO forecasts. Geophysical Research Letters. 2021;48(6): e2020GL091287. DOI: 10.1029/2020GL091287.
  36. Gildor H, Tziperman E. Sea ice as the glacial cycles’ Climate switch: role of seasonal and orbital forcing. Paleoceanography and Paleoclimatology. 2000;15(6):605–615. DOI: 10.1029/1999PA000461.
  37. Smirnov DA, Mokhov II. From Granger causality to long-term causality: Application to climatic data. Phys. Rev. E. 2009;80(1):016208. DOI: 10.1103/PhysRevE.80.016208.
  38. Mokhov II, Smirnov DA. Empirical estimates of the influence of natural and anthropogenic factors on the global surface temperature. Doklady Earth Sciences. 2009;427(1):798–803. DOI: 10.1134/S1028334X09050201.
  39. Mokhov II, Smirnov DA. Estimating the contributions of the Atlantic Multidecadal Oscillation and variations in the atmospheric concentration of greenhouse gases to surface air temperature trends from observations. Doklady Earth Sciences. 2018;480(1):602–606. DOI: 10.1134/S1028334X18050069.
  40. Mokhov II, Smirnov DA. Contribution of greenhouse gas radiative forcing and Atlantic Multidecadal Oscillation to surface air temperature trends. Russian Meteorology and Hydrology. 2018;43(9):557–564. DOI: 10.3103/S1068373918090017.
  41. Hornik K, Stinchcombe M, White H. Multilayer feedforward networks are universal approximators. Neural Networks. 1989;2(5):359–366. DOI: 10.1016/0893-6080(89)90020-8.
  42. Raymo ME, Nisancioglu KH. The 41 kyr world: Milankovitch’s other unsolved mystery. Paleoceanography and Paleoclimatology. 2003;18(1):1011. DOI: 10.1029/2002PA000791.
  43. Clark PU, Archer D, Pollard D, Blum JD, Rial JA, Brovkin V, Mix AC, Pisias NG, Roy M. The middle Pleistocene transition: characteristics, mechanisms, and implications for long-term changes in atmospheric pCO2. Quaternary Science Reviews. 2006;25(23–24):3150–3184. DOI: 10.1016/j.quascirev.2006.07.008.
  44. Maslin MA, Brierley CM. The role of orbital forcing in the Early Middle Pleistocene Transition. Quaternary International. 2015;389:47–55. DOI: 10.1016/j.quaint.2015.01.047.
  45. Elderfield H, Ferretti P, Greaves M, Crowhurst S, McCave IN, Hodell D, Piotrowski AM. Evolution of ocean temperature and ice volume through the mid-Pleistocene climate transition. Science. 2012;337(6095):704–709. DOI: 10.1126/science.1221294.
  46. Gildor H, Tziperman E. A sea ice climate switch mechanism for the 100-kyr glacial cycles. Journal of Geophysical Research: Oceans. 2001;106(C5):9117–9133. DOI: 10.1029/1999JC000120.
  47. Crucifix M. Oscillators and relaxation phenomena in Pleistocene climate theory. Phil. Trans. R. Soc. A. 2012;370(1962):1140–1165. DOI: 10.1098/rsta.2011.0315.
  48. Rial JA, Oh J, Reischmann E. Synchronization of the climate system to eccentricity forcing and the 100,000-year problem. Nature Geosci. 2013;6(4):289–293. DOI: 10.1038/ngeo1756.
  49. Ditlevsen PD. Bifurcation structure and noise-assisted transitions in the Pleistocene glacial cycles. Paleoceanography and Paleoclimatology. 2009;24(3):PA3204. DOI: 10.1029/2008PA001673.
  50. Huybers P. Pleistocene glacial variability as a chaotic response to obliquity forcing. Clim. Past. 2009;5(3):481–488. DOI: 10.5194/cp-5-481-2009.
  51. Benzi R, Parisi G, Sutera A, Vulpiani A. Stochastic resonance in climatic change. Tellus. 1982;34(1):10–15. DOI: 10.3402/tellusa.v34i1.10782.
  52. Lisiecki LE, Raymo ME. A Pliocene-Pleistocene stack of 57 globally distributed benthic δ 18O records. Paleoceanography and Paleoclimatology. 2005;20(1):PA1003. DOI: 10.1029/2004PA001071.
  53. Berger A, Loutre MF. Insolation values for the climate of the last 10 million years. Quaternary Science Reviews. 1991;10(4):297–317. DOI: 10.1016/0277-3791(91)90033-Q.
  54. Berger A, Li XS, Loutre MF. Modelling northern hemisphere ice volume over the last 3 Ma. Quaternary Science Reviews. 1999;18(1):1–11. DOI: 10.1016/S0277-3791(98)00033-X.
  55. Rial JA. Abrupt climate change: chaos and order at orbital and millennial scales. Global and Planetary Change. 2004;41(2):95–109. DOI: 10.1016/j.gloplacha.2003.10.004.
  56. Tziperman E, Gildor H. On the mid-Pleistocene transition to 100-kyr glacial cycles and the asymmetry between glaciation and deglaciation times. Paleoceanography and Paleoclimatology. 2003;18(1):1–1–1–8. DOI: 10.1029/2001pa000627.
  57. McManus JF, Oppo DW, Cullen JL. A 0.5-million-year record of millennial-scale climate variability in the North Atlantic. Science. 1999;283(5404):971–975. DOI: 10.1126/science.283.5404.971.
  58. Schulz M, Berger WH, Sarnthein M, Grootes PM. Amplitude variations of 1470-year climate oscillations during the last 100,000 years linked to fluctuations of continental ice mass. Geophysical Research Letters. 1999;26(22):3385–3388. DOI: 10.1029/1999GL006069.
  59. Burgers G, Jin FF, van Oldenborgh GJ. The simplest ENSO recharge oscillator. Geophysical Research Letters. 2005;32(13):L13706. DOI: 10.1029/2005GL022951.
  60. Jin FF. An equatorial ocean recharge paradigm for ENSO. Part I: Conceptual model. Journal of the Atmospheric Sciences. 1997;54(7):811–829. DOI: 10.1175/1520-0469(1997)054<0811:AEORPF>2.0.CO;2.
  61. McPhaden MJ. Tropical Pacific Ocean heat content variations and ENSO persistence barriers. Geophysical Research Letters. 2003;30(9):1480. DOI: 10.1029/2003GL016872.
  62. Timmermann A, An SI, Kug JS, Jin FF, Cai W, Capotondi A, Cobb KM, Lengaigne M, McPhaden MJ, Stuecker MF, Stein K, Wittenberg AT, Yun KS, Bayr T, Chen HC, Chikamoto Y, Dewitte B, Dommenget D, Grothe P, Guilyardi E, Ham YG, Hayashi M, Ineson S, Kang D, Kim S, Kim W, Lee JY, Li T, Luo JJ, McGregor S, Planton Y, Power S, Rashid H, Ren HL, Santoso A, Takahashi K, Todd A, Wang G, Wang G, Xie R, Yang WH, Yeh SW, Yoon J, Zeller E, Zhang X. El Nino–Southern Oscillation complexity. Nature. 2018;559(7715):535–545. DOI: 10.1038/s41586-018-0252-6.
  63. Kondrashov D, Kravtsov S, Robertson AW, Ghil M. A hierarchy of data-based ENSO models. Journal of Climate. 2005;18(21):4425–4444. DOI: 10.1175/JCLI3567.1.
  64. Tippett MK, L’Heureux ML. Low-dimensional representations of Nino 3.4 evolution and the spring persistence barrier. npj Clim. Atmos. Sci. 2020;3(1):24. DOI: 10.1038/s41612-020-0128-y.
  65. Vimont DJ, Wallace JM, Battisti DS. The seasonal footprinting mechanism in the pacific: Implications for ENSO. Journal of Climate. 2003;16(16):2668–2675. DOI: 10.1175/1520-0442(2003)016<2668:TSFMIT>2.0.CO;2.
  66. Yu JY, Fang SW. The distinct contributions of the seasonal footprinting and charged-discharged mechanisms to ENSO complexity. Geophysical Research Letters. 2018;45(13):6611–6618. DOI: 10.1029/2018GL077664.
  67. Vimont DJ, Alexander M, Fontaine A. Midlatitude excitation of tropical variability in the pacific: The role of thermodynamic coupling and seasonality. Journal of Climate. 2009;22(3):518–534. DOI: 10.1175/2008JCLI2220.1.
  68. Fang XH, Mu M. Both air-sea components are crucial for El Nino forecast from boreal spring. Sci. Rep. 2018;8(1):10501. DOI: 10.1038/s41598-018-28964-z.
  69. Mokhov II, Smirnov DA. The trivariate seasonal analysis of couplings between El Nino, North Atlantic Oscillation, and Indian monsoon. Russian Meteorology and Hydrology. 2016;41(11– 12):798–807. DOI: 10.3103/S106837391611008X.
  70. Mokhov II, Smirnov DA. Estimates of the mutual influence of variations in the sea surface temperature in tropical latitudes of the Pacific, Atlantic, and Indian Oceans from long-period data series. Izvestiya, Atmospheric and Oceanic Physics. 2017;53(6):613–623. DOI: 10.1134/S0001433817060081.
  71. Barnston AG, Tippett MK, L’Heureux ML, Li S, DeWitt DG. Skill of real-time seasonal ENSO model predictions during 2002–11: Is our capability increasing? Bulletin of the American Meteorological Society. 2012;93(5):631–651. DOI: 10.1175/BAMS-D-11-00111.1.
  72. Bond G, Kromer B, Beer J, Muscheler R, Evans MN, Showers W, Hoffmann S, Lotti-Bond R, Hajdas I, Bonani G. Persistent solar influence on North Atlantic climate during the Holocene. Science. 2001;294(5549):2130–2136. DOI: 10.1126/science.1065680.
  73. Emile-Geay J, Cane M, Seager R, Kaplan A, Almasi P. El Nino as a mediator of the solar influence on climate. Paleoceanography and Paleoclimatology. 2007;22(3):PA3210. DOI: 10.1029/2006PA001304.
  74. Shindell D, Rind D, Balachandran N, Lean J, Lonergan P. Solar cycle variability, ozone, and climate. Science. 1999;284(5412):305–308. DOI: 10.1126/science.284.5412.305.
  75. Shindell DT, Schmidt GA, Mann ME, Rind D, Waple A. Solar forcing of regional climate change during the Maunder Minimum. Science. 2001;294(5549):2149–2152. DOI: 10.1126/science.1064363.
  76. Zebiak SE. Oceanic heat content variability and El Nino cycles. Journal of Physical Oceanography. 1989;19(4):475–486. DOI: 10.1175/1520-0485(1989)019<0475:OHCVAE>2.0.CO;2.
  77. Bjerknes J. Atmospheric teleconnections from the equatorial Pacific. Monthly Weather Review. 1969;97(3):163–172. DOI: 10.1175/1520-0493(1969)097<0163:ATFTEP>2.3.CO;2.
  78. Yeh SW, Cai W, Min SK, McPhaden MJ, Dommenget D, Dewitte B, Collins M, Ashok K, An SI, Yim BY, Kug JS. ENSO atmospheric teleconnections and their response to greenhouse gas forcing. Reviews of Geophysics. 2018;56(1):185–206. DOI: 10.1002/2017RG000568.
  79. Mukhin D, Gavrilov A, Loskutov E, Feigin A, Kurths J. Nonlinear reconstruction of global climate leading modes on decadal scales. Clim. Dyn. 2018;51(5–6):2301–2310. DOI: 10.1007/s00382- 017-4013-2.
  80. Emile-Geay J, Cobb KM, Mann ME, Wittenberg AT. Estimating central equatorial Pacific SST variability over the past millennium. Part II: Reconstructions and implications. Journal of Climate. 2013;26(7):2329–2352. DOI: 10.1175/JCLI-D-11-00511.1.
  81. Emile-Geay J, Cobb KM, Mann ME, Wittenberg AT. Estimating central equatorial Pacific SST variability over the past millennium. Part I: Methodology and validation. Journal of Climate. 2013;26(7):2302–2328. DOI: 10.1175/JCLI-D-11-00510.1.
  82. Steinhilber F, Beer J, Frohlich C. Total solar irradiance during the Holocene. Geophysical Research Letters. 2009;36(19):L19704. DOI: 10.1029/2009GL040142.
  83. Feigin AM, Gavrilov AS, Loskutov EM, Mukhin DN, Seleznev AF. Nonlinear dynamical modes: A method for empirical reconstruction of complex systems. In: Nonlinear Waves’ 2018. Nizhny Novgorod: IAP RAS; 2019. P. 191–217 (in Russian).
  84. Mukhin D, Gavrilov A, Feigin A, Loskutov E, Kurths J. Principal nonlinear dynamical modes of climate variability. Sci. Rep. 2015;5(1):15510. DOI: 10.1038/srep15510.
  85. Gavrilov A, Mukhin D, Loskutov E, Volodin E, Feigin A, Kurths J. Method for reconstructing nonlinear modes with adaptive structure from multidimensional data. Chaos. 2016;26(12):123101. DOI: 10.1063/1.4968852.
Received: 
31.05.2021
Accepted: 
29.06.2021
Published: 
30.07.2021