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There is a growing interest in the climate community to improve the prediction of high impact climate events, for instance ENSO (El-Niño-Southern Oscillation) or extreme events, using a combination of model and observation data. In this note we explain that, in a dynamical context, the relevant quantity for predicting a future event is a committor function. We explain the main mathematical properties of this probabilistic concept. We compute and discuss the committor function of the Jin and Timmerman model of El-Niño. Our first conclusion is that one should generically distinguish between states with either intrinsic predictability or intrinsic unpredictability. This predictability concept is markedly different from the deterministic unpredictability arising because of chaotic dynamics and exponential sensibility to initial conditions. The second aim of this work is to compare the inference of a committor function from data, either through a direct approach or through a machine learning approach using neural networks. We discuss the consequences of this study for future applications to more complex data sets.

Many atmosphere and climate phenomena lie in the gray zone between weather and climate: they are not amenable to deterministic forecast, but they still depend on the initial condition. A natural example is medium-range forecasting, which is inherently probabilistic because it lies beyond the deterministic predictability time of the atmosphere, but for which statistically significant prediction can be made, which depends on the current state of the system. Similarly, one may ask the probability of occurrence of an El Niño event several months ahead of time. We introduce a quantity that corresponds precisely to this type of prediction problem: the committor function is the probability that an event takes place within a given time window, as a function of the initial condition. We compute it in the case of a low-dimensional stochastic model for El Niño, the Jin and Timmermann model. In this context, we show that the ability to predict the probability of occurrence of the event of interest may differ strongly depending on the initial state. The main result is the new distinction between probabilistic predictability (when the committor function is smooth and probability can be computed, which does not depend sensitively on the initial condition) and probabilistic unpredictability (when the committor function depends sensitively on the initial condition). We also demonstrate that the Jin and Timmermann model might be the first example of a stochastic differential equation with weak noise for which transition between attractors does not follow the Arrhenius law, which is expected based on large deviation theory and generic hypothesis. Significance Statement A key problem for atmospheric and climate phenomena is to predict events beyond the time scale over which deterministic weather forecast is possible. In a simple model of El Niño, we demonstrate the existence of two regimes, depending on initial conditions. For initial conditions in the “probabilistic predictability” regime, the system is unpredictable deterministically because of chaos, but the probability of occurrence of the event can still be predicted because it depends only weakly on the initial condition. In the “probabilistic unpredictability” regime, even predicting probabilities is difficult, because the probability depends strongly on initial conditions. These new concepts of probabilistic predictability and unpredictability should be key in understanding the predictability potential for rare events in climate problems, as well as in other complex dynamics.