Organizations: College of Science, National University of Defense Technology, Changsha, Hunan 410073, China · Research Institute of Intelligent Complex Systems and MOE Frontiers Center for Brain Science, Fudan University, Shanghai 200433, China · School of Mathematical Sciences, SCMS, SCAM, and CCSB, Fudan University, Shanghai 200433, China · Shanghai Artificial Intelligence Laboratory, Shanghai 200232, China.
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.
Figures & tables
Figure 1: Schematic diagram for the proposed RCDyM method. ( A ) The basic structure of RC consists of the input layer, the hidden layer, and the output layer. The continuous-time, autonomous RC is obtained by replacing the input with the predicted state using the learned RC. ( B ) Three RCDyMs are depicted with the application scenarios for the tipping point prediction. ( C ) An illustrative example of tipping point prediction using the DEJ measure, one of the RCDyMs: Utilizing trend analysis for ultra-early prediction of tipping points.
Figure 2: Tipping points prediction for representative continuous/discrete-time dynamical systems, including four bifurcation systems and two chaotic systems. ( A ) Ultra-early prediction of tipping points in systems with different bifurcations, viz., fold, period-doubling, pitchfork, and Hopf bifurcations. The time interval Δt was set to 0.1 for pitchfork bifurcation and 0.05 for Hopf bifurcation. For each experiment, the early-warning parameter ϵ is taken as 15% of the total DEJ measure range. ( B ) The estimated DEJ measures using the RCDyM method and the ground truth (GT) calculated directly using the original system equations are distributed around the line y=x . ( C ) Predicting the critical transition from a limit cycle to an equilibrium in the Hopf bifurcation system, as well as the critical transitions from period-4 to period-2 and period-8 dynamics in the system governed by the Logistic map. ( D ) Predicting the critical transitions from an equilibrium to chaos and from chaos to an equilibrium in the Lorenz63 system.
Figure 3: Tipping points prediction for the KS system. ( A ) and ( B ) illustrate the predicted transitions in the Kuramoto-Sivashinsky (KS) system: from periodic behavior to chaotic behavior, and from chaotic behavior to periodic behavior, respectively. Here, RC-para (no parameter) refers to the tipping point prediction directly through the RC extrapolation prediction, and the green dashed line illustrates the change in system parameters.
Figure 4: Robustness demonstration of the RCDyM method. ( A ) Predicting tipping points in the experiment of period-doubling bifurcation for different noise intensities. ( B ) Upper bound of the lead time for achieving ultra-early prediction of the tipping points for four bifurcation systems. ( C ) The estimations of the MLE and the DEJ in the Lorenz63 system for different window length d , compared to the ground-truth values of the DEJ at −0.596 ( p=10 ) and the MLE at 0.91 ( p=28 ), respectively. ( D ) The estimation of the MLE in the KS system for different n , with the ground-truth value of the MLE at 0.05 ( p=1 and L=22 ). ( E ) Modeling dynamics near an equilibrium using the observational time series data from the Lorenz63 system. ( F ) Modeling the chaotic behavior of the KS system using the RC under an elevated noise level with ω=0.05 .
Figure 5: Performance of the RCDyM method in predicting tipping points across eight real-world datasets. ( A )-( H ) The experimental results for voice production, sedimentary archives, APT dynamics, and thermoacoustic transitions. ( I )-( P ) The experimental results for power grid failure, greenhouse Earth conditions, cyanobacteria population dynamics, and the Cariaco Basin climate, respectively. Here, we denote the time interval corresponding to the top 30% of indicators closest to the tipping point as [tw,tp] and label this segment as 1, while all other regions are labeled 0. Thus, we obtain the ROC curve and AUC value, which are used to quantitatively evaluate the performance of our method in comparison to baseline approaches.
Figure 6: Predicting the tipping point of the AMOC using the RCDyM method. ( A ) The simplified schematic of the AMOC on an Earth map, with a color bar representing salinity levels in the ocean from 300 meters depth to the surface, covering the first four months of 2024. ( B ) The upper panel displays the mean salinity and SST indices within the quadrilateral region highlighted in subfigure A , spanning the period from January 1900 to April 2024. The lower panel presents the EWS calculated using the continuous RCDyM method, where the red line represents the regression results using all the RCDyMs shown in the subfigure, while the blue line represents the regression results based on indicators from 1996 onward.
Method
Reference
Generality
Pre-tipping
Nonlinear
Bifurcation types
Periodic/Chaotic
Ultra-early prediction
Classical EWSs
Scheffer et al., 2009 [ 47 ]
✔
✔
N/A
✗
✗
✗
DL
Bury et al., 2021 [ 7 ]
✔
✗
N/A
✔
✗
✗
RC-para1
Patel & Ott, 2022 [ 41 ]
✗
✔
✔
✗
✔
✔
DEV
Grziwotz et al., 2023 [ 20 ]
✔
✔
✗
✔
✗
✗
RC-para2
Panahi et al., 2024 [ 40 ]
✗
✔
✔
✗
✔
✔
RCDyM
This work
✔
✔
✔
✔
✔
✔
Table 1: Comparison of the existing SOTA baselines and the proposed RCDyM method . Here, “Generality” refers to the applicability of the method to arbitrary systems without additional information about the time-varying parameters, “Pre-tipping” indicates whether the method requires data solely from the pre-tipping regime, “Nonlinear” indicates the model’s capacity to handle potential nonlinear dynamics, “Bifurcation types” signifies the method’s ability to identify various prevalent bifurcation types, “Periodic/Chaotic” describes the capability to predict tipping points in systems exhibiting periodic or chaotic behavior, and “Ultra-early prediction” refers to the capability of our RCDyM method to achieve quantitative prediction at a very early stage, well before the onset of critical transitions.
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.
Smita Deb, Zheng-Meng Zhai, Mulugeta Haile +1
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
Critical transitions - abrupt, often irreversible changes in system dynamics - arise across human and natural systems, often with catastrophic consequences. Real-world observations of such shifts remain scarce, preventing the development of reliable early warning systems. Conventional statistical and spectral indicators, such as increasing variance, tend to fail under realistic conditions of limited data and correlated noise, whereas existing deep learning classifiers do not extrapolate beyond their training data distribution. In this work, we introduce TipPFN, an in-context learning (ICL) framework that uses a prior-data fitted network to infer a system's proximity to a critical transition. Trained on our novel synthetic data generator, which is based on canonical bifurcation scenarios coupled to diverse, randomized stochastic dynamics, TipPFN flexibly capitalizes on contexts of various sizes, complexity and dimensionalities. We demonstrate robust, state-of-the-art early detection of critical transitions in previously unseen tipping regimes, sim-to-real examples, and real-world observations in both ICL and zero-shot settings.
Yunus Sevinchan, Juan Nathaniel, Kai Ueltzhöffer +8
kausable Heidelberg, Germany · Columbia University New York, USA · University of Edinburgh Scotland, UK
Abrupt transitions in complex systems are often preceded by early warning signals. However, most indicators rely on the notion of critical slowing down and do not generally extend to rate-induced tipping where transitions can occur without local loss of stability. This is problematic in stochastic, nonautonomous systems where internal variability and time-varying variables interact to shape tipping onset. We use Koopman operator theory to develop a unified early warning framework for both bifurcation and rate-induced tipping in stochastic systems. Our approach builds on residual Koopman mode decomposition that measures discrepancies between dynamics and their finite-dimensional approximation, and extends it to the control setting by augmenting the observable space with time-varying control variables. In idealized examples, the resulting indicators recover expected signatures near bifurcation points and improve detection in rate-induced regimes where classical indicators fail. We further show that learned embeddings through deep learning outperform prescribed dictionaries, especially in a high-dimensional setting. Applied to simulations of the Atlantic Meridional Overturning Circulation, our Koopman-based indicators distinguish tipping from non-tipping trajectories and reveal interpretable spectral signatures prior to critical transition.
Juan Nathaniel, Carla Roesch, Derek DeSantis +5
Columbia University, USA · University of Edinburgh, UK · Los Alamos National Laboratory, USA +3