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Deep learning for predicting the occurrence of tipping points

Chengzuo Zhuge, Jiawei Li, Wei Chen

TL;DR

This work tackles the challenge of predicting tipping points from time series, including irregularly sampled data, by introducing a deep learning predictor that leverages the normal-form structure of bifurcations. The model extracts dynamical features from irregular samples via an embedding-based CNN-LSTM architecture and uses training labels defined by recovery-rate changes rather than deterministic bifurcation points, enabling robustness to rate-delayed tipping and noise. The authors demonstrate superior performance over traditional autoregressive and feature-based methods on regular data and show accurate tipping-point prediction for irregularly-sampled and empirical time series across fold, Hopf, transcritical, and pitchfork bifurcations, with implications for risk mitigation in social, engineering, and biological systems. The supplementary analyses connect theory and practice through critical slowing-down considerations, fast-slow dynamics, irregular sampling embeddings, normal-form relationships, competing baselines, and control experiments, underlining the method’s reliability and potential impact in real-world monitoring and intervention tasks.

Abstract

Tipping points occur in many real-world systems, at which the system shifts suddenly from one state to another. The ability to predict the occurrence of tipping points from time series data remains an outstanding challenge and a major interest in a broad range of research fields. Particularly, the widely used methods based on bifurcation theory are neither reliable in prediction accuracy nor applicable for irregularly-sampled time series which are commonly observed from real-world systems. Here we address this challenge by developing a deep learning algorithm for predicting the occurrence of tipping points in untrained systems, by exploiting information about normal forms. Our algorithm not only outperforms traditional methods for regularly-sampled model time series but also achieves accurate predictions for irregularly-sampled model time series and empirical time series. Our ability to predict tipping points for complex systems paves the way for mitigation risks, prevention of catastrophic failures, and restoration of degraded systems, with broad applications in social science, engineering, and biology.

Deep learning for predicting the occurrence of tipping points

TL;DR

This work tackles the challenge of predicting tipping points from time series, including irregularly sampled data, by introducing a deep learning predictor that leverages the normal-form structure of bifurcations. The model extracts dynamical features from irregular samples via an embedding-based CNN-LSTM architecture and uses training labels defined by recovery-rate changes rather than deterministic bifurcation points, enabling robustness to rate-delayed tipping and noise. The authors demonstrate superior performance over traditional autoregressive and feature-based methods on regular data and show accurate tipping-point prediction for irregularly-sampled and empirical time series across fold, Hopf, transcritical, and pitchfork bifurcations, with implications for risk mitigation in social, engineering, and biological systems. The supplementary analyses connect theory and practice through critical slowing-down considerations, fast-slow dynamics, irregular sampling embeddings, normal-form relationships, competing baselines, and control experiments, underlining the method’s reliability and potential impact in real-world monitoring and intervention tasks.

Abstract

Tipping points occur in many real-world systems, at which the system shifts suddenly from one state to another. The ability to predict the occurrence of tipping points from time series data remains an outstanding challenge and a major interest in a broad range of research fields. Particularly, the widely used methods based on bifurcation theory are neither reliable in prediction accuracy nor applicable for irregularly-sampled time series which are commonly observed from real-world systems. Here we address this challenge by developing a deep learning algorithm for predicting the occurrence of tipping points in untrained systems, by exploiting information about normal forms. Our algorithm not only outperforms traditional methods for regularly-sampled model time series but also achieves accurate predictions for irregularly-sampled model time series and empirical time series. Our ability to predict tipping points for complex systems paves the way for mitigation risks, prevention of catastrophic failures, and restoration of degraded systems, with broad applications in social science, engineering, and biology.
Paper Structure (20 sections, 3 theorems, 56 equations, 16 figures, 3 tables)

This paper contains 20 sections, 3 theorems, 56 equations, 16 figures, 3 tables.

Table of Contents

  1. Supplementary Figures
  2. Supplementary Notes
  3. Supplementary Note 1. Critical slowing down
  4. Supplementary Note 2. Fast-slow systems and rate-delayed tipping
  5. Fast-slow systems in critical transition
  6. Rate-delayed tipping
  7. Training labels of tipping points
  8. Supplementary Note 3. Embedding theorem for irregular sampling
  9. Supplementary Note 4. Normal form
  10. The formula for the recovery rate in the normal form
  11. Dimension reduction near a bifurcation
  12. Supplementary Note 5. Competing algorithms
  13. Degenerate fingerprinting
  14. BB method
  15. Dynamical eigenvalue
  16. ...and 5 more sections

Key Result

Lemma 1

Let $X$ be a $C^r$$(r\geq 1)$ vector field on a compact connected manifold $M$, and $\tau:M\rightarrow \mathbb{R}^+$ a $C^r$ function. The mapping $g:M \rightarrow M$ defined by $g(x)=\phi(x,\tau(x))$, (where $\phi:M \times \mathbb{R}^+ \rightarrow M$ is the flow arising from $X$), is a $C^r$ diffeo

Figures (16)

  • Figure : Figure S1.The mean relative error of tipping points prediction between the DL algorithm and LSTM on regularly-sampled model time series in the ablation study. The horizontal axis represents the initial values of the bifurcation parameter, and the vertical axis represents the mean relative error of prediction. The area covered by the polyline represents the 90% confidence interval for the relative error of tipping points prediction. (a-c) Three ecological model time series with white noise, which undergo fold, Hopf, and transcritical bifurcation, respectively. (d-f) Three climate model time series with red noise, which undergo fold, Hopf, and transcritical bifurcation, respectively.
  • Figure : Figure S2.The mean relative error of tipping points prediction between the DL algorithm and LSTM on irregularly-sampled model time series in the ablation study. The horizontal axis represents the initial values of the bifurcation parameter, and the vertical axis represents the mean relative error of prediction. The area covered by the polyline represents the 90% confidence interval for the relative error of tipping points prediction. (a-c) Three ecological model time series with white noise, which undergo fold, Hopf, and transcritical bifurcation, respectively. (d-f) Three climate model time series with red noise, which undergo fold, Hopf, and transcritical bifurcation, respectively.
  • Figure : Figure S3.The mean relative error of tipping points prediction between the DL algorithm and competing algorithms on regularly-sampled model time series. The horizontal axis represents the distance between the final value of bifurcation parameter time series and the value of the tipping point, and the vertical axis represents the mean relative error of prediction. The area covered by the polyline represents the 90% confidence interval for the relative error of tipping points prediction. (a-c) We compared the DL algorithm (red lines) with degenerate fingerprinting (blue lines), DEV (green lines) and LSTM (purple lines) on three ecological model time series with white noise. These model time series undergo fold, Hopf, and transcritical bifurcation, respectively. (d-f) The DL algorithm (red lines) is compared with BB method (blue lines), DEV (green lines) and LSTM (purple lines) on three climate model time series with red noise. These model time series undergo fold, Hopf, and transcritical bifurcation, respectively.
  • Figure : Figure S4.The mean relative error of tipping points prediction between the DL algorithm and competing algorithms on irregularly-sampled model time series. The horizontal axis represents the distance between the final value of bifurcation parameter time series and the value of the tipping point, and the vertical axis represents the mean relative error of prediction. The area covered by the polyline represents the 90% confidence interval for the relative error of tipping points prediction. (a-c) We compared the DL algorithm (red lines) with degenerate fingerprinting (blue lines), DEV (green lines) and LSTM (purple lines) on three ecological model time series with white noise. These model time series undergo fold, Hopf, and transcritical bifurcation, respectively. (d-f) The DL algorithm (red lines) is compared with BB method (blue lines), DEV (green lines) and LSTM (purple lines) on three climate model time series with red noise. These model time series undergo fold, Hopf, and transcritical bifurcation, respectively.
  • Figure : Figure S5.The mean relative error of predicted tipping points by the DL algorithm on regularly-sampled model time series of different changing rates of bifurcation parameter. The horizontal axis represents the initial values of the bifurcation parameter, and the vertical axis represents the mean relative error of prediction. The area covered by the polyline represents the 90% confidence interval for the relative error of tipping points prediction. (a-c) Three ecological model time series with white noise, which undergo fold, Hopf, and transcritical bifurcation, respectively. (d-f) Three climate model time series with red noise, which undergo fold, Hopf, and transcritical bifurcation, respectively.
  • ...and 11 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1
  • Definition 1: bifurcation
  • Definition 2: $n_-,n_0,n_+$
  • Definition 3: hyperbolic equilibrium
  • Theorem 2