A translator learns not from a dictionary but from speeches. They listen to native speakers and pick up rules that are written in no textbook. Similarly, a new method for forecasting complex systems, such as climate or turbulence, does not require knowing their equations. Instead, it looks at the data and learns to predict their behavior.
Previously, to predict a system's behavior, one had to fit a model, that is, guess its equation. This is like trying to describe a person's speech knowing only grammar but not words. The new approach works differently: it builds a forecast directly from the time series, like a translator who learns from live conversations. The method does not reconstruct the equation but directly estimates how the probability distribution of the system's states changes.
On the Lorenz model, known for its chaotic behavior, diffusion forecasting showed significant improvement over locally linear methods. On real El Niño 3.4 index data, the forecast for 14 months achieved a correlation of 0.64 and RMSE of 0.60. This is better than a previously published result, where the correlation was 0.4 and RMSE 1.4.
A translator who learns from speeches does not know the rules but understands the meaning. Similarly, the new method does not know the equations but can predict. It opens the way to probabilistic forecasts for systems with unknown dynamics.
The diffusion maps method underlying the forecast was originally developed for analyzing high-dimensional data, such as images, and only later adapted for dynamical systems.