Deep Learning for predicting rate-induced tipping

TL;DR

This is the first attempt to develop a deep learning framework predicting the transition probabilities of dynamical systems ahead of rate-induced and noise-induced tipping, and demonstrates the predictability of rate-induced and noise-induced tipping.

Abstract

Nonlinear dynamical systems exposed to changing forcing can exhibit catastrophic transitions between alternative and often markedly different states. The phenomenon of critical slowing down (CSD) can be used to anticipate such transitions if caused by a bifurcation and if the change in forcing is slow compared to the internal time scale of the system. However, in many real-world situations, these assumptions are not met and transitions can be triggered because the forcing exceeds a critical rate. For example, given the pace of anthropogenic climate change in comparison to the internal time scales of key Earth system components, such as the polar ice sheets or the Atlantic Meridional Overturning Circulation, such rate-induced tipping poses a severe risk. Moreover, depending on the realisation of random perturbations, some trajectories may transition across an unstable boundary, while others do not, even under the same forcing. CSD-based indicators generally cannot distinguish these cases of noise-induced tipping versus no tipping. This severely limits our ability to assess the risks of tipping, and to predict individual trajectories. To address this, we make a first attempt to develop a deep learning framework to predict transition probabilities of dynamical systems ahead of rate-induced transitions. Our method issues early warnings, as demonstrated on three prototypical systems for rate-induced tipping, subjected to time-varying equilibrium drift and noise perturbations. Exploiting explainable artificial intelligence methods, our framework captures the fingerprints necessary for early detection of rate-induced tipping, even in cases of long lead times. Our findings demonstrate the predictability of rate-induced and noise-induced tipping, advancing our ability to determine safe operating spaces for a broader class of dynamical systems than possible so far.

Figures (5)

• Figure 1: Diagram depicting R-tipping through visualization of a system represented as a particle in a basin of attraction. (A) The comparison of basins of attraction before (black solid line) and after (black dash line) undergoing a shift due to the change in environmental forcing. When the time-varying rate of environmental forcing change (i.e., forcing rate $\epsilon$) is much slower than the critical threshold ($\epsilon_c$), the particle near to the basin (its quasi-equilibrium state) can recover timely and recover to the equilibrium state (i.e., the stable fixed point); in other words, the forcing rate is sufficiently low for the system to be able to track the basin of attraction associated with that equilibrium. (B) When the forcing rate is faster than the critical threshold, the particle cannot track the basin of attraction anymore, and R-tipping occurs. (C) When the forcing rate is rapid but stays slightly below the critical threshold, but the system additionally experiences slight noise perturbations, the particle will leave the basin of attraction for some noise realisations, but not for others, establishing a mixture of rate- and noise-induced tipping.
• Figure 2: Ensemble simulation results for prototype R-tipping systems. Ensemble simulations are conducted on the Saddle–node system (A), Bautin system (B), and Compost-bomb system (C), respectively. For each simulated system, the same time-varying forcing parameter but different noise perturbations are used for the simulations. Top panels: simulated time series that do not tip (blue) and that exhibit R-tipping (red). Middle panels: The time-varying forcing parameter. Bottom panels: The probability density of the observed occurrence time of R-tipping. Blue shading area denotes the 99% confidence intervals of the ensemble realizations which do not manifest R-tipping.
• Figure 3: Deep-learning based prediction of R-tipping for the paradigmatic systems such as the Saddle–node system (A), Bautin system (B), and Compost-bomb system (C), and comparison to classical early-warning indicators. Top panel: statistics of simulated time series for indicating the time-varying system states. Upper middle panel: estimated autocorrelation evolution. Lower middle panel: estimated variance. Bottom panel: DL-derived R-tipping probabilities as functions of time. The red solid lines represents the composite mean values of time series within the class exhibiting R-tipping, while light red and red shading areas depict the 99% and 75% confidence intervals, respectively. The non-tipping scenarios are indicated by blue correspondingly. The unit on the time axis adopts the time step used in numerical integration (see Materials and Methods).
• Figure 4: Saliency maps to interpret the fingerprint features that the trained DL models for predicting R-tipping have extracted. (A) Three example time series (from the Saddle–node system) approaching R-tipping (red solid, dashed and doted lines), and the 99% confidence intervals of R-tipping and non-tipping scenarios (red and blue shading areas). (B) The predicted R-tipping probabilities for the three example time series. (C) LRP scores for a lead time equal to 0 time steps, indicating the relative importance of each individual time series point for guiding the prediction. Blue solid line shows the LRP scores for the time series of example 1. Red and blue shading areas represent the 99% confidence intervals of the LRP scores for R-tipping and non-tipping scenarios, respectively. The left and right vertical axes denote the LRP values of R-tipping and non-tipping scenarios, respectively. (D), (E), and (F) mirror the configuration of (C), corresponding to lead times of 30, 50 and 150 time steps, respectively.
• Figure 5: Prediction accuracy of DL models for R-tipping with out-of-sample forcing rates. For the Saddle–node system, DL models were trained on time series with specific forcing rates $\epsilon=1.25$ (A) and $\epsilon=1.7$ (D), respectively, and subsequently used to predict R-tipping cases with previously unseen forcing rates, and the prediction accuracy as a function of forcing rate and forecast lead time is shown. For the Bautin system, DL models were trained on forcing rates $r=1.0$ (B) and $r=0.6$ (E), respectively. For the Compost-bomb system, DL models were trained on forcing rates $v=0.1$ (C) and $v=0.17$ (F), respectively.