Imagine a map where each patch of ocean is colored with its own shade: in some places the color is saturated, meaning the future of the system is predetermined; in others it is pale, meaning the outcome depends on a random gust of wind. Climate scientists build such a probability map to understand when to expect a strong El Niño, the warm current off the coast of South America that alters weather across the entire planet.

The problem is that the climate system is chaotic: on short timescales forecasting is still possible, and on very long timescales one can only speak of average values. But the intermediate range, from a few months to a couple of decades, remains a gray zone. This is where the committor function comes into play: it shows the probability that a system starting from a given point will reach the threshold for a strong El Niño within a specified time.

To build such a map, scientists use the Jin and Timmermann model, which consists of three equations describing the interaction between the ocean and the atmosphere in the tropical Pacific. This model has two intertwined attractors, that is, regions toward which the system tends: one corresponds to periodic strong El Niño events, the other to chaotic behavior without such events. Adding random noise allows the system to switch between them.

At an intermediate noise level, four regions emerged in the state space. Two of them have probabilities of 0 or 1, meaning the outcome is predetermined. The third region has a smoothly varying probability: forecasting is possible but not precise. The fourth region has an extremely sensitive dependence on initial conditions: a tiny change in the starting point changes the probability from 0 to 1. This is practical unpredictability.

The neural network approach opens the way to estimating the committor in more complex models and from real data. Instead of running thousands of trajectories for each point, one can train a network on pairs of initial state and outcome, and then quickly compute the probability for any state. This is especially valuable when direct simulation is exponentially expensive.

The probability map mentioned at the beginning is no longer a metaphor. The committor function provides a way to draw it. And although we are still far from an accurate El Niño forecast, at least now it is clear where on the map lie the regions where forecasting is possible in principle.