Predicting discrete-time bifurcations with deep learning

TL;DR

A deep learning classifier is trained to provide an early warning signal for the five local discrete-time bifurcations of codimension-one, and shows higher sensitivity and specificity than commonly used early warning signals under a wide range of noise intensities and rates of approach to the bIfurcation.

Abstract

Many natural and man-made systems are prone to critical transitions -- abrupt and potentially devastating changes in dynamics. Deep learning classifiers can provide an early warning signal (EWS) for critical transitions by learning generic features of bifurcations (dynamical instabilities) from large simulated training data sets. So far, classifiers have only been trained to predict continuous-time bifurcations, ignoring rich dynamics unique to discrete-time bifurcations. Here, we train a deep learning classifier to provide an EWS for the five local discrete-time bifurcations of codimension-1. We test the classifier on simulation data from discrete-time models used in physiology, economics and ecology, as well as experimental data of spontaneously beating chick-heart aggregates that undergo a period-doubling bifurcation. The classifier outperforms commonly used EWS under a wide range of noise intensities and rates of approach to the bifurcation. It also predicts the correct bifurcation in most cases, with particularly high accuracy for the period-doubling, Neimark-Sacker and fold bifurcations. Deep learning as a tool for bifurcation prediction is still in its nascence and has the potential to transform the way we monitor systems for critical transitions.

Figures (5)

• Figure 1: Period-doubling bifurcation in a spontaneously beating aggregate of embryonic chick heart cells following treatment with a potassium channel blocker (E-4031, 1.5$\upmu$mol). (a) Interbeat intervals (IBI) for consecutive beats. A period-doubling bifurcation occurs at approximately beat 230. Arrows correspond to traces plotted in lower panels. (b-d) Normalised signal from optical imaging of the aggregate's motion. Traces are from a section well before (b), just before (c), and after (d) the period-doubling bifurcation.
• Figure 2: Trends in indicators prior to five different bifurcations in the theoretical models. (a-e) Trajectory (gray) and smoothing (black) of a simulation going through a period-doubling, Neimark-Sacker, fold, transcritical and pitchfork bifurcation, respectively. (f-j) Variance of residual dynamics after smoothing, computed over a rolling window (arrow) of size 0.5 times the length of the pre-transition data. (k-o) Lag-1 autocorrelation. (p-t) Probabilities assigned by the deep learning (DL) classifier when given all preceding data. Orange line shows probability assigned to the true bifurcation. Gray lines show probabilities assigned to the other bifurcations. Blue line shows the sum of the five bifurcation probabilities.
• Figure 3: Trends in indicators prior to a period-doubling bifurcation in five chick heart aggregates treated with a potassium channel blocker. (a-e) Inter-beat interval (gray) and smoothing. (f-j) Variance of residual dynamics after smoothing, computed over a rolling window (arrow) of size 0.5 times the length of the pre-transition data. (k-o) Lag-1 autocorrelation. (p-t) Probabilities assigned by the deep learning (DL) classifier to the period-doubling bifurcation (orange) and the other bifurcations (gray). Blue line shows the sum of the five bifurcation probabilities.
• Figure 4: ROC curves for predictions of an upcoming transition in model and experimental data. ROC curves compare the performance of the deep learning classifier (DL, blue), variance (Var, red) and lag-1 autocorrelation (AC, green). For models (a-e), performance is assessed on 5,000 simulations with different noise amplitudes and rates of forcing. For experimental data (f), performance is assessed on 46 experimental runs. The area under the curve (AUC), abbreviated to A, is a measure of performance. Insets show the probabilities assigned by the classifier to each type of bifurcation (orange being the true bifurcation) among the trajectories approaching a transition. (a) Fox model going through a period-doubling bifurcation fox2002period. (b) Westerhoff model going through a Neimark-Sacker bifurcation westerhoff2008consumer. (c) Ricker model going through a fold bifurcation ricker1954stock. (d) Lotka-Volterra model going through a transcritical bifurcation smith1968mathematical. (e) Lorenz model going through a pitchfork bifurcation lorenz1989computational. (f) Chick heart aggregates going through a period-doubling bifurcation kim2009stochastic. Predictions are made 80% of the way through the pretransition data for the models, and 60-100% of the way for the experimental data. PD: period-doubling. NS: Neimark-Sacker. TC: transcritical. PF: pitchfork.
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