Anticipating tipping in spatiotemporal systems with machine learning

Smita Deb; Zheng-Meng Zhai; Mulugeta Haile; Ying-Cheng Lai

nlin.CDApr 7, 2026

Anticipating tipping in spatiotemporal systems with machine learning

Authors: Smita Deb, Zheng-Meng Zhai, Mulugeta Haile, Ying-Cheng Lai

Organizations: School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287, USA · DEVCOM Army Research Office, 6340 Rodman Road, Aberdeen Proving Ground, MD 21005-5069, USA · Department of Physics, Arizona State University, Tempe, Arizona 85287, USA

Abstract

In nonlinear dynamical systems, tipping refers to a critical transition from one steady state to another, typically catastrophic, steady state, often resulting from a saddle-node bifurcation. Recently, the machine-learning framework of parameter-adaptable reservoir computing has been applied to predict tipping in systems described by low-dimensional stochastic differential equations. However, anticipating tipping in complex spatiotemporal dynamical systems remains a significant open problem. The ability to forecast not only the occurrence but also the precise timing of such tipping events is crucial for providing the actionable lead time necessary for timely mitigation. By utilizing the mathematical approach of non-negative matrix factorization to generate dimensionally reduced spatiotemporal data as input, we exploit parameter-adaptable reservoir computing to accurately anticipate tipping. We demonstrate that the tipping time can be identified within a narrow prediction window across a variety of spatiotemporal dynamical systems, as well as in CMIP5 (Coupled Model Intercomparison Project 5) climate projections. Furthermore, we show that this reservoir-computing framework, utilizing reduced input data, is robust against common forecasting challenges and significantly alleviates the computational overhead associated with processing full spatiotemporal data.

Figures & tables

Figure 1: Schematic representation of the parameter-adaptable reservoir-computing framework for estimating tipping points. (a) Spatiotemporal fields (heatmaps) are processed and dimensionally reduced via NMF to form the reservoir input. This is combined with an external parameter channel fed with the bifurcation parameter corresponding to the training samples. (b) Workflow of the parameter-adaptable reservoir computer. The reservoir encodes the input dynamics and produces an output trajectory. A TGAM (Threshold Generalized Additive Model) is then applied to the reservoir output to estimate the timing of the tipping point. Arrows indicate the flow of data from the raw inputs to the final estimated tipping point.

Figure 2: Reservoir-computing prediction of the tipping interval in the spatiotemporal vegetation-turbidity model. (a) The spatial mean density of the state variable

v

versus the bifurcation parameter

c

. The blue dashed curve indicates the tipping time obtained using the TGAM method. (b) Mean-field plots of the training, validation, and testing datasets after applying NMF to the snapshots. The dashed orange and black vertical lines mark the end of training and start of testing, respectively. In the testing phase, the green curve represents the ground truth. (c) Box plots representing the tipping point in each of the

n_{x}(=40)

spatial cells. (d) A histogram plot displaying the predicted tipping point of the spatiotemporal system from 1000 statistical realizations.

Figure 3: Reservoir-computing predictions with varying training data lengths. Panels (a–c), (d–f), (g–i), and (j–l) use 200, 400, 600, and 1000 training snapshots, respectively. Within each triplet, from left to right: mean curves for the training, validation, and testing datasets of the NMF-reduced spatial system; box plots of tipping points for each of the 40 spatial cells; and a histogram of predicted system tipping times from 1000 realizations. Dashed orange and black vertical lines indicate the end of training and the start of testing periods. In the first panel of each row, the green curve represents the ground truth trajectory.

Figure 4: Impact of varying the training data on tipping prediction. (a) Choice of training windows by shifting either their end point or their starting point to test the robustness of tipping prediction. Results framed in blue correspond to training windows that terminate at different distances away from the tipping point; those framed in orange correspond to windows whose starting time is progressively moved toward tipping while their length is held fixed. Within each boxed subsection, the top panel displays box plots of the predicted tipping times for all 40 spatial cells, and the bottom panel shows a histogram of system-wide tipping times derived from 1000 reservoir realizations for the setup in (a).

Figure 5: Comparison of success rates in tipping point anticipation using reservoir computing, a convolution neural network, a time-delay feedforward network, and classical spatial indicators (spatial standard deviation, spatial skewness, spatial correlation). The spatial data are of size

40\times 40

, obtained from Eq. ( 1 ), with its coarse-grained reduced subsystem subject to multiplicative noise of varying amplitude

\sigma

. Panels (a-f) correspond to

\sigma=0.05,0.1,0.15,0.2,0.25,0.3

, respectively.

Figure 6: Predicting tipping in the spatially extended vegetation grazing system, Eq. ( 4 ), and cellular automata model, Eq. ( 5 ). (a) The spatial mean density of the state variable

v

plotted against a gradually changing bifurcation parameter for the vegetation grazing model. The blue dashed curve marks the ground-truth tipping time identified by the TGAM method. (b) Mean plots for the training, validation, and testing datasets after applying NMF to the snapshots for the vegetation grazing model. The orange and black dashed vertical lines indicate the end of training and the start of testing periods, respectively. In the testing phase, the green curve represents the ground truth trajectory. (c,d) Box plots representing the tipping point for each of the

40

spatial cells and histogram showing the predicted tipping points based on

1000

realizations, respectively, for the vegetation grazing system. (e-h) The results corresponding to (a-d) but for the cellular automata model.

Figure 7: Tipping point analysis for CMIP5 climate projections. Data are presented for three variables: (1) annual percentage sea ice cover in the Southern Ocean from MRI‑CGCM3, (2) annual percentage sea ice cover in the Arctic Ocean from CSIRO-MK3-6-0, and (3) air temperature in April from MPI-ESM-LR for the Pacific sector of the Arctic Ocean. (a, e, i) Spatial mean density of the variables (temperature and sea ice cover) over time (in years). The blue dashed curve in each plot marks the ground-truth tipping time as determined by the TGAM method. (b, f, j) Mean plots for the training, validation, and testing datasets of the spatial system after applying NMF to the snapshots. The orange and black dashed vertical lines indicate the end of training and the start of testing periods, respectively. In the testing phase, the green curve represents the ground truth. (c, g, k) Box plots representing the tipping point for each of the

40

spatial cells. (d, h, l) Histograms showing the predicted tipping points for the three climate systems based on

1000

realizations.

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1: Test of false positives. Shown is the NMF-reduced density of the state variable

\mathbf{v}

versus a gradual change in the parameter

c

for each spatial cell. Plots are displayed for the training, validation, and testing datasets of the spatial system following the application of NMF to the data snapshots. The training dataset covers the parameter range

c\in(1,1.55)

, validation for

c\in(1.55,1.6)

, and testing is performed on the remaining parameter regime. The reservoir computer predicts no transition in each individual cell, attesting to its ability to eliminate false positives.

Figure S2: The spatial mean density of the state variable

\mathbf{v}

corresponding to the plots in Fig. S1 . The reservoir computer robustly produces true negatives.

Figure S3: Prediction of tipping in the vegetation-turbidity model, Eq. ( 1 ), when the parameter channel of the reservoir computer is nullified. Shown are the mean fields for the training, validation, and testing datasets of the system after applying NMF to the snapshots. The dashed orange and black vertical lines mark the end of the training and the start of the testing periods, respectively. In the testing phase, the green curve represents the ground truth. The setup is identical to Fig. 2 , except that it does not involve a parameter channel receiving the bifurcation parameter values corresponding to the input during training.

Figure S4: Anticipating tipping in the vegetation-turbidity model, Eq. ( 1 ), using a time-delay feedforward network (TDFN). Panels show two representative outcomes: (a) unsuccessful anticipation, where no clear increasing trend is detected, and (b) successful anticipation, characterized by a pronounced increasing trend prior to the tipping point. The mean trajectories for the training, validation, and testing datasets are shown for the spatial system after applying NMF to the snapshots. The dashed orange and black vertical lines indicate the end of the training and the start of the testing periods, respectively. During the testing phase, the green curve denotes the ground truth.

Figure S5: Anticipating tipping in the vegetation-turbidity model, Eq. ( 1 ), with a convolutional neural network (CNN). Two representative outcomes: (a) unsuccessful anticipation, where no clear increasing trend is detected prior to tipping, and (b) successful anticipation, characterized by a pronounced increasing trend prior to the tipping point. The mean trajectories for the training, validation, and testing datasets are shown for the spatial system after applying NMF to the snapshots. The dashed orange and black vertical lines indicate the end of the training and the start of the testing periods, respectively. During the testing phase, the green curve denotes the ground truth.

Figure S6: Trends in spatial indicators for the vegetation-turbidity model, Eq. ( 1 ). (a–c) Spatial standard deviation, spatial skewness, and spatial lag-1 correlation, respectively, computed from the original system. (d–f) The corresponding indicators for the reduced system obtained via coarse-graining using

5\times 5

submatrices. The orange and black dashed vertical lines indicate the time interval over which the spatial indicators are used to compute Kendall’s-

\tau

to compare the strength of the observed trends.

Spatial indicator	$\tau_{1}$	$\tau_{2}$
S.S.D	0.79	0.85
S.S	0.43	0.34
S.C	0.54	0.68
$S.S.D_{red}$	0.72	0.80
$S.S_{red}$	0.27	0.29
$S.C_{red}$	0.12	0.17

Appendix

Table S1: Kendall’s-

\tau

values corresponding to the orange (

\tau_{1}

) and black (

\tau_{2}

) dashed lines, respectively, in Fig. S6 . The values are presented for the three spatial indicators computed from snapshots along the gradients of the control parameter by solving Eq. ( 1 ).

Noise amplitude ( $\sigma$ )	$CI_{W}$
0.05	[83.8,88]
0.1	[82.5,87]
0.15	[81,86]
0.2	[77.4,82]
0.25	[73.2,78.5]
0.3	[69.5,75]

Appendix

Table S2: Wilson confidence interval (

CI_{W}

) computed for predictions from

1000

realizations of the parameter-adaptable reservoir computer, using spatial data simulated at different noise amplitudes. This interval provides an estimate of the fraction of realizations in which the predicted tipping point falls within the true tipping interval, calculated at a

95\%

confidence level.

Figure S7: Estimating the tipping point of the vegetation-turbidity system using a TGAM. (a) Estimated tipping point for the vegetation-turbidity model, Eq. ( 1 ). (b) The bifurcation parameter as a function of time. The orange marker indicates the true tipping threshold of the bifurcation parameter (

c=1.784

t=735

) as estimated using the TGAM.

Figure S8: Estimating the tipping point of the vegetation-grazing system using a TGAM. (a) Estimated tipping point for the vegetation-grazing model, Eq. ( 4 ). (b) The bifurcation parameter as a function of time. The orange marker indicates the true tipping threshold of the bifurcation parameter (

c=24.6812

t=750

) as estimated using the TGAM.

Figure S9: Estimating the tipping point of the cellular automata model using a TGAM. (a) Estimated tipping point for the cellular automata model, Eq. ( 5 ). (b) The bifurcation parameter as a function of time. The orange marker indicates the true tipping threshold of the bifurcation parameter (

c=0.718

t=709

) as estimated using the TGAM.

Figure S10: Estimating the tipping point of the first CMIP5 dataset. Shown is the tipping point estimated using a TGAM for the CMIP5 projected climate variable: percentage sea ice cover in the high-latitude oceans from MRI‑CGCM3.

Figure S11: Estimating the tipping point of the second CMIP5 dataset. Shown is the tipping point estimated using a TGAM for the CMIP5 projected climate variable: percentage sea ice cover in the Arctic Ocean from CSIRO-MK3-6-0.

Figure S12: Estimating the tipping point of the third CMIP5 dataset. Shown is the tipping point estimated using a TGAM for the CMIP5 projected climate variable: temperature in the month of April for MPI-ESM-LR for the Pacific sector of the Arctic Ocean.

Figure S13: Training, validation, and testing by the reservoir computer on the spatial vegetation-turbidity system following the application of NMF to the data snapshots for each spatial cell. The mean values are presented in Fig. 2 (b).

Figure S14: Training, validation, and testing by the reservoir computer on the spatial vegetation-grazing system following the application of NMF to the data snapshots for each spatial cell. The mean values are shown in Fig. 6 (b).

Figure S15: Training, validation, and testing by the reservoir computer on the spatial cellular automata system following the application of NMF to the data snapshots for each spatial cell. The mean values are presented in Fig. 6 (f).

Figure S16: The NMF-reduced sea ice cover in the high latitudes from MRI‑CGCM3 versus time for each spatial cell. Plots are displayed for the training, validation, and testing datasets of the spatial system following the application of NMF to the data snapshots. The mean values for the training, validation, and testing datasets of the spatial system after applying NMF to the snapshots are shown in Fig. 7 (b).

Figure S17: The NMF-reduced annual sea ice cover in the Arctic Ocean from CSIRO-MK3-6-0 versus time for each spatial cell. Plots are displayed for the training, validation, and testing datasets of the spatial system following the application of NMF to the data snapshots. The mean values for the training, validation, and testing datasets of the spatial system after applying NMF to the snapshots are presented in Fig. 7 (f).

Figure S18: The NMF-reduced temperature in the month of April for MPI-ESM-LR for the Pacific sector of the Arctic Ocean versus time for each spatial cell. Plots are displayed for the training, validation, and testing datasets of the spatial system following the application of NMF to the data snapshots. The mean values for the training, validation, and testing datasets of the spatial system after applying NMF to the snapshots are presented in Fig. 7 (j).

Spatial Data	$\rho$	$\gamma$	$\alpha$	$\beta$	$d$	$b_{0}$
Vegetation turbidity model	0.682	0.0235	0.572	$10^{-3.9404}$	0.2535	1.4794
Vegetation grazing model	0.637	0.0235	0.1523	$10^{-3.3672}$	0.247	0.5252
Cellular automata model	0.4713	0.02829	0.4147	$10^{-3.1260}$	0.2164	0.7245
Temperature (MPI-ESM-LR)	0.68	0.0007	0.57	$10^{-3.5}$	0.2535	0.1
$\%$ sea ice cover (CSIRO-MK3-6-0)	0.6	4.009	0.65	$10^{-3.4.45}$	0.25	1
$\%$ sea ice cover (MRI-CGCM3)	0.65	0.0029	0.5965	$10^{-3.65}$	0.35	3

Appendix

Table S3: Hyperparameter values for the spatial data across the different models used in the reservoir-computing predictions. The size of the reservoir network is

2000

for all cases.

Details	CMIP5 data 1	CMIP5 data 2	CMIP5 data 3
Model	MPI-ESM-LR	CSIRO-MK3-6-0	MRI-CGCM3
Variable	ta	sic	sic
Scenario	RCP85	RCP85	RCP26
Time	2020-2100	1850-2100	1850-2100
Time frequency	mon	mon	mon
Ensemble	r1i1p1	r5i1p1	r1i1p1

Appendix

Table S4: Details of the CMIP5 climate projection data sampling used to evaluate the generality of the parameter-adaptable reservoir computer on real-world datasets. All datasets are sampled at a monthly resolution. For the MPI-ESM-LR data, only the temperature data for the month of April is used. For all remaining cases, monthly data for all

12

months are sampled and averaged to obtain the annual mean values of the respective variables. Data from

1850–2005

correspond to the historical experiment, while data from

2006–2100

correspond to the RCP scenario specified for each case in the table.

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Complex dynamical systems-such as climate, ecosystems, and economics-can undergo catastrophic and potentially irreversible regime changes, often triggered by environmental parameter drift and stochastic disturbances. These critical thresholds, known as tipping points, pose a prediction problem of both theoretical and practical significance, yet remain largely unresolved. To address this, we articulate a model-free framework that integrates the measures characterizing the stability and sensitivity of dynamical systems with the reservoir computing (RC), a lightweight machine learning technique, using only observational time series data. The framework consists of two stages. The first stage involves using RC to robustly learn local complex dynamics from observational data segmented into windows. The second stage focuses on accurately detecting early warning signals of tipping points by analyzing the learned autonomous RC dynamics through dynamical measures, including the dominant eigenvalue of the Jacobian matrix, the maximum Floquet multiplier, and the maximum Lyapunov exponent. Furthermore, when these dynamical measures exhibit trend-like patterns, their extrapolation enables ultra-early prediction of tipping points significantly prior to the occurrence of critical transitions. We conduct a rigorous theoretical analysis of the proposed method and perform extensive numerical evaluations on a series of representative synthetic systems and eight real-world datasets, as well as quantitatively predict the tipping time of the Atlantic Meridional Overturning Circulation system. Experimental results demonstrate that our framework exhibits advantages over the baselines in comprehensive evaluations, particularly in terms of dynamical interpretability, prediction stability and robustness, and ultra-early prediction capability.

Abstract

Spatiotemporal Forecasting

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