Data-driven early warning of Gaussian white noise-induced critical transitions

doi:10.1007/s10483-026-3346-9

摘要/Abstract

Abstract:

Many complex systems are frequently subject to the influence of uncertain disturbances, which can exert a profound effect on the critical transitions (CTs), potentially resulting in catastrophic consequences. Consequently, it is of uttermost importance to provide warnings for noise-induced CTs in various applications. Although capturing certain generic symptoms of transition behaviors from observational and simulated data poses a challenging problem, this work attempts to extract information regarding CTs from simulated data of a Gaussian white noise-induced tri-stable system. Using the extended dynamic mode decomposition (EDMD) algorithm, we initially obtain finite-dimensional approximations of both the stochastic Koopman operator and the generator. Subsequently, the drift parameters and the noise intensity within the system are identified from the simulated data. Utilizing the identified system, the parameter-dependent basin of the unsafe regime (PDBUR) is quantified, enabling data-driven early warning of Gaussian white noise-induced CTs. Finally, an error analysis is carried out to verify the effectiveness of the data-driven results. Our findings may serve as a paradigm for understanding and predicting noise-induced CTs in complex systems based on data.

Key words: Gaussian white noise, critical transition (CT), extended dynamic mode decomposition (EDMD), parameter-dependent basin of the unsafe regime (PDBUR)

中图分类号:

O211.63

. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(2): 389-400.

Ruifang WANG, Minhe JIA, Xuanqi FAN, Jinzhong MA, Yong XU. Data-driven early warning of Gaussian white noise-induced critical transitions[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(2): 389-400.

图/表 6

Basis	Relative error of drift coefficient/%				Relative error of diffusion coefficient $σ$ /%
	Data volume				Data volume
	$105$	$106$	$107$	$108$	$105$	$106$	$107$	$108$
1	28.83	4.73	3.43	2.53	7.26	0.78	3.12	2.12
$x$	2.04	1.33	0.38	0.73	0	0	0	0
$x 2$	6.78	1.1	8.86	4.88	0	0	0	0
$x 3$	5.10	1.74	0.25	0.29	0	0	0	0
$x 4$	12.83	10.4	6.4	2.5	0	0	0	0
$x 5$	9.93	2.14	1.28	0.37	0	0	0	0
$x 6$	0	0	0	0	0	0	0	0

Drift coefficient			Diffusion coefficient $σ$
Basis	True	Learned	Basis	True	Learned
1	$− 0.3$	$− 0.292 4$	1	0.5	0.489 4
$x$	$− 3.8$	$− 3.772 1$	$x$	0	0
$x 2$	0.5	0.475 6	$x 2$	0	0
$x 3$	4.6	4.586 5	$x 3$	0	0
$x 4$	$− 0.4$	$− 0.390 0$	$x 4$	0	0
$x 5$	$− 1.14$	$− 1.144 2$	$x 5$	0	0
$x 6$	0	0	$x 6$	0	0

$σ$	True	Predicted	Relative error/%
0.3	$(− 0.08, 1.5)$	$(− 0.09, 1.5)$	0.63
0.5	$(− 0.65, 1.5)$	$(− 0.64, 1.5)$	0.46

参考文献 39

[1]	SCHEFFER, M., CARPENTER, S., FOLEY, J. A., FOLKE, C., and WALKER, B. Catastrophic shifts in ecosystems. nature, 413(6856), 591–596 (2001)
[2]	ARANI, B. M. S., CARPENTER, S. R., LAHTI, L., VAN NES, E. H., and SCHEFFER, M. Exit time as a measure of ecological resilience. Science, 372(6547), eaay4895 (2021)
[3]	MA, J. Z., LIU, Q., XU, Y., and KURTHS, J. Early warning of noise-induced catastrophic high-amplitude oscillations in an airfoil model. Chaos, 32(3), 033119 (2022)
[4]	DITLEVSEN, P. D. and JOHNSEN, S. J. Tipping points: early warning and wishful thinking. Geophysical Research Letters, 37(19), L19703 (2010)
[5]	GHOLAMI, M. and EFTEKHARI, M. Nonlinear forced vibration in a subcritical regime of a porous functionally graded pipe conveying fluid with a retaining clip. Applied Mathematics and Mechanics (English Edition), 46(1), 101–122 (2025) https://doi.org/10.1007/s10483-025-3206-9
[6]	ZHANG, Y. M. Critical transition Reynolds number for plane channel flow. Applied Mathematics and Mechanics (English Edition), 38(10), 1415–1424 (2017) https://doi.org/10.1007/s10483-017-2245-6
[7]	MA, J. Z., XU, Y., XU, W., LI, Y. G., and KURTHS, J. Slowing down critical transitions via Gaussian white noise and periodic force. Science China Technological Sciences, 62(12), 2144–2152 (2019)
[8]	SCHEFFER, M., BASCOMPTE, J., BROCK, W. A., BROVKIN, V., CARPENTER, S. R., DAKOS, V., HELD, H., VAN NES, E. H., RIETKERK, M., and SUGIHARA, G. Early-warning signals for critical transitions. nature, 461(7260), 53–59 (2009)
[9]	DAKOS, V., CARPENTER, S. R., BROCK, W. A., ELLISON, A. M., GUTTAL, V., IVES, A. R., KÉFI, S., LIVINA, V., SEEKELL, D. A., VAN NES, E. H., and SCHEFFER, M. Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data. PLoS One, 7(7), e41010 (2012)
[10]	CARPENTER, S. R. and BROCK, W. A. Rising variance: a leading indicator of ecological transition. Ecology Letters, 9(3), 311–318 (2006)
[11]	ZHANG, X. Y., XU, Y., LIU, Q., KURTHS, J., and GREBOGI, C. Rate-dependent tipping and early warning in a thermoacoustic system under extreme operating environment. Chaos, 31(11), 113115 (2021)
[12]	GUTTAL, V. and JAYAPRAKASH, C. Changing skewness: an early warning signal of regime shifts in ecosystems. Ecology Letters, 11(5), 450–460 (2008)
[13]	BROCK, W. A. and CARPENTER, S. R. Interacting regime shifts in ecosystems: implication for early warnings. Ecological Monographs, 80(3), 353–367 (2010)
[14]	DAKOS, V., VAN NES, E. H., DONANGELO, R., FORT, H., and SCHEFFER, M. Spatial correlation as leading indicator of catastrophic shifts. Theoretical Ecology, 3, 163–174 (2010)
[15]	MA, J. Z., XU, Y., KURTHS, J., WANG, H. Y., and XU, W. Detecting early-warning signals in periodically forced systems with noise. Chaos, 28(11), 113601 (2018)
[16]	BOULTON, C. A., ALLISON, L. C., and LENTON, T. M. Early warning signals of Atlantic meridional overturning circulation collapse in a fully coupled climate model. Nature Communications, 5(1), 5752 (2014)
[17]	GOPALAKRISHNAN, E. A., SHARMA, Y., JOHN, T., DUTTA, P. S., and SUJITH R. I. Early warning signals for critical transitions in a thermoacoustic system. Scientific Reports, 6(1), 35310 (2016)
[18]	ZHANG, X. Y., XU, Y., LIU, Q., and KURTHS, J. Rate-dependent tipping-delay phenomenon in a thermoacoustic system with colored noise. Science China Technological Sciences, 63(11), 2315–2327 (2020)
[19]	STOLBOVA, V., SUROVYATKINA, E., BOOKHAGEN, B., and KURTHS, J. Tipping elements of the Indian monsoon: prediction of onset and withdrawal. Geophysical Research Letters, 43(8), 3982–3990 (2016)
[20]	SOUTHALL, E., TILDESLEY, M. J., and DYSON, L. Prospects for detecting early warning signals in discrete event sequence data: application to epidemiological incidence data. PLoS Computational Biology, 16(9), e1007836 (2020)
[21]	LI, H. T., ZHENG, T. Y., QIN, W. Y., TIAN, R. L., DING, H., JI, J. C., and CHEN, L. Q. Theoretical and experimental study of a bi-stable piezoelectric energy harvester under hybrid galloping and band-limited random excitations. Applied Mathematics and Mechanics (English Edition), 45(3), 461–478 (2024) https://doi.org/10.1007/s10483-024-3098-5
[22]	WEI, Z. C., LI, Y. X., KAPITANIAK, T., and ZHANG, W. Chaotic characteristics for a class of hydro-pneumatic near-zero frequency vibration isolators under dry friction and noise excitation. Applied Mathematics and Mechanics (English Edition), 46(4), 647–662 (2025) https://doi.org/10.1007/s10483-025-3243-6
[23]	LIU, R., CHEN, P., AIHARA, K., and CHEN, L. N. Identifying early-warning signals of critical transitions with strong noise by dynamical network markers. Scientific Reports, 5(1), 17501 (2015)
[24]	MA, J. Z., XU, Y., LI, Y. G., TIAN, R. L., and KURTHS, J. Predicting noise-induced critical transitions in bistable systems. Chaos, 29(8), 081102 (2019)
[25]	MA, J. Z., XU, Y., LI, Y. G., TIAN, R. L., CHEN, G. R., and KURTHS, J. Precursor criteria for noise-induced critical transitions in multi-stable systems. Nonlinear Dynamics, 101, 21–35 (2020)
[26]	MA, J. Z., XU, Y., LI, Y. G., TIAN, R. L., MA, S. J., and KURTHS, J. Quantifying the parameter dependent basin of the unsafe regime of asymmetric Lévy-noise-induced critical transitions. Applied Mathematics and Mechanics (English Edition), 42(1), 65–84 (2021) https://doi.org/10.1007/s10483-021-2672-8
[27]	BURY, T. M., SUJITH, R. I., PAVITHRAN, I., SCHEFFER, M., LENTON, T. M., ANAND, M., and BAUCH, C. T. Deep learning for early warning signals of tipping points. Proceedings of the National Academy of Sciences of the United States of America, 118(39), e2106140118 (2021)
[28]	GARCÍA, C. A., OTERO, A., FÉLIX, P., PRESEDO, J., and MÁRQUEZ, D. G. Nonparametric estimation of stochastic differential equations with sparse Gaussian processes. Physical Review E, 96(2), 022104 (2017)
[29]	BRUNTON, S. L., PROCTOR, J. L., and KUTZ, J. N. 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, 113(15), 3932–3937 (2016)
[30]	CHAMPION, K., LUSCH, B., KUTZ, J. N., and BRUNTON, S. L. Data-driven discovery of coordinates and governing equations. Proceedings of the National Academy of Sciences of the United States of America, 116(45), 22445–22451 (2019)
[31]	WILLIAMS, M. O., KEVREKIDIS, I. G., and ROWLEY, C. W. A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. Journal of Nonlinear Science, 25, 1307–1346 (2015)
[32]	RAFIQ, D. and BAZAZ, M. A. Nonlinear model order reduction via nonlinear moment matching with dynamic mode decomposition. International Journal of Non-Linear Mechanics, 128, 103625 (2021)
[33]	YIN, Q., CAI, J. T., GONG, X., and DING, Q. Local parameter identification with neural ordinary differential equations. Applied Mathematics and Mechanics (English Edition), 43(12), 1887–1900 (2022) https://doi.org/10.1007/s10483-022-2926-9
[34]	ZHU, J. A., XUE, Y. H., and LIU, Z. S. A transfer learning enhanced physics-informed neural network for parameter identification in soft materials. Applied Mathematics and Mechanics (English Edition), 45(10), 1685–1704 (2024) https://doi.org/10.1007/s10483-024-3178-9
[35]	FENG, J., WANG, X. L., LIU, Q., XU, Y., and KURTHS, J. Fusing deep learning features for parameter identification of a stochastic airfoil system. Nonlinear Dynamics, 113, 4211–4233 (2025)
[36]	AO, P., GALAS, D., HOOD, L., and ZHU, X. M. Cancer as robust intrinsic state of endogenous molecular-cellular network shaped by evolution. Medical Hypotheses, 70(3), 678–684 (2008)
[37]	LU, Y. B. and DUAN, J. Q. Discovering transition phenomena from data of stochastic dynamical systems with Lévy noise. Chaos, 30(9), 093110 (2020)
[38]	CHEN, X. L., WU, F. Y., DUAN, J. Q., KURTHS, J., and LI, X. F. Most probable dynamics of a genetic regulatory network under stable Lévy noise. Applied Mathematics and Computation, 348, 425–436 (2019)
[39]	WU, D. F., FU, M. M., and DUAN, J. Q. Discovering mean residence time and escape probability from data of stochastic dynamical systems. Chaos, 29(9), 093122 (2019)