Predicting discrete-time bifurcations with deep learning.

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 for critical transitions by learning generic features of bifurcations 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 early warning signal for the five local discrete-time bifurcations of codimension-one. 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 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. 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.

Universal early warning signals of phase transitions in climate systems.

The potential for complex systems to exhibit tipping points in which an equilibrium state undergoes a sudden and often irreversible shift is well established, but prediction of these events using standard forecast modelling techniques is quite difficult. This has led to the development of an alternative suite of methods that seek to identify signatures of critical phenomena in data, which are expected to occur in advance of many classes of dynamical bifurcation. Crucially, the manifestations of these critical phenomena are generic across a variety of systems, meaning that data-intensive deep learning methods can be trained on (abundant) synthetic data and plausibly prove effective when transferred to (more limited) empirical datasets. This paper provides a proof of concept for this approach as applied to lattice phase transitions: a deep neural network trained exclusively on two-dimensional Ising model phase transitions is tested on a number of real and simulated climate systems with considerable success. Its accuracy frequently surpasses that of conventional statistical indicators, with performance shown to be consistently improved by the inclusion of spatial indicators. Tools such as this may offer valuable insight into climate tipping events, as remote sensing measurements provide increasingly abundant data on complex geospatially resolved Earth systems.

Rhythms from Two Competing Periodic Sources Embedded in an Excitable Medium.

In an excitable medium, a stimulus generates a wave that propagates in space until it reaches the boundary or collides with another wave and annihilates. We study the dynamics generated by two periodic sources with different frequencies in excitable cardiac tissue culture using optogenetic techniques. The observed rhythms, which can be modeled using cellular automata and studied analytically, show unexpected regularities related to classic results in number theory. We apply the results to identify cardiac arrhythmias in people that are due to a putative mechanism of two competing pacemakers.

Deep learning for early warning signals of tipping points.

Many natural systems exhibit tipping points where slowly changing environmental conditions spark a sudden shift to a new and sometimes very different state. As the tipping point is approached, the dynamics of complex and varied systems simplify down to a limited number of possible "normal forms" that determine qualitative aspects of the new state that lies beyond the tipping point, such as whether it will oscillate or be stable. In several of those forms, indicators like increasing lag-1 autocorrelation and variance provide generic early warning signals (EWS) of the tipping point by detecting how dynamics slow down near the transition. But they do not predict the nature of the new state. Here we develop a deep learning algorithm that provides EWS in systems it was not explicitly trained on, by exploiting information about normal forms and scaling behavior of dynamics near tipping points that are common to many dynamical systems. The algorithm provides EWS in 268 empirical and model time series from ecology, thermoacoustics, climatology, and epidemiology with much greater sensitivity and specificity than generic EWS. It can also predict the normal form that characterizes the oncoming tipping point, thus providing qualitative information on certain aspects of the new state. Such approaches can help humans better prepare for, or avoid, undesirable state transitions. The algorithm also illustrates how a universe of possible models can be mined to recognize naturally occurring tipping points.

When conflicts get heated, so does the planet: coupled social-climate dynamics under inequality.

Climate dynamics are inextricably linked to processes in social systems that are highly unequal. This suggests a need for coupled social-climate models that capture pervasive real-world asymmetries in the population distribution of the consequences of anthropogenic climate change and climate (in)action. Here, we use evolutionary game theory to develop a social-climate model with group structure to investigate how anthropogenic climate change and population heterogeneity coevolve. We find that greater homophily and resource inequality cause an increase in the global peak temperature anomaly by as much as 0.7°C. Also, climate change can structure human populations by driving opinion polarization. Finally, climate mitigation achieved by reducing the cost of mitigation measures paid by individuals tends to be contingent upon socio-economic conditions, whereas policies that achieve communication between different strata of society show climate mitigation benefits across a broad socio-economic regime. We conclude that advancing climate change mitigation efforts can benefit from a social-climate systems perspective.

Charting pathways to climate change mitigation in a coupled socio-climate model.

Geophysical models of climate change are becoming increasingly sophisticated, yet less effort is devoted to modelling the human systems causing climate change and how the two systems are coupled. Here, we develop a simple socio-climate model by coupling an Earth system model to a social dynamics model. We treat social processes endogenously-emerging from rules governing how individuals learn socially and how social norms develop-as well as being influenced by climate change and mitigation costs. Our goal is to gain qualitative insights into scenarios of potential socio-climate dynamics and to illustrate how such models can generate new research questions. We find that the social learning rate is strongly influential, to the point that variation of its value within empirically plausible ranges changes the peak global temperature anomaly by more than 1°C. Conversely, social norms reinforce majority behaviour and therefore may not provide help when we most need it because they suppress the early spread of mitigative behaviour. Finally, exploring the model's parameter space for mitigation cost and social learning suggests optimal intervention pathways for climate change mitigation. We find that prioritising an increase in social learning as a first step, followed by a reduction in mitigation costs provides the most efficient route to a reduced peak temperature anomaly. We conclude that socio-climate models should be included in the ensemble of models used to project climate change.

Critical dynamics in population vaccinating behavior.

Vaccine refusal can lead to renewed outbreaks of previously eliminated diseases and even delay global eradication. Vaccinating decisions exemplify a complex, coupled system where vaccinating behavior and disease dynamics influence one another. Such systems often exhibit critical phenomena-special dynamics close to a tipping point leading to a new dynamical regime. For instance, critical slowing down (declining rate of recovery from small perturbations) may emerge as a tipping point is approached. Here, we collected and geocoded tweets about measles-mumps-rubella vaccine and classified their sentiment using machine-learning algorithms. We also extracted data on measles-related Google searches. We find critical slowing down in the data at the level of California and the United States in the years before and after the 2014-2015 Disneyland, California measles outbreak. Critical slowing down starts growing appreciably several years before the Disneyland outbreak as vaccine uptake declines and the population approaches the tipping point. However, due to the adaptive nature of coupled behavior-disease systems, the population responds to the outbreak by moving away from the tipping point, causing "critical speeding up" whereby resilience to perturbations increases. A mathematical model of measles transmission and vaccine sentiment predicts the same qualitative patterns in the neighborhood of a tipping point to greatly reduced vaccine uptake and large epidemics. These results support the hypothesis that population vaccinating behavior near the disease elimination threshold is a critical phenomenon. Developing new analytical tools to detect these patterns in digital social data might help us identify populations at heightened risk of widespread vaccine refusal.