A translator learns not from a dictionary but from speech. 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 data and learns to predict their behavior.

Previously, to predict a system's behavior, you had to fit a model, that is, guess its equation. It 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 recover the equation but directly estimates how the probability distribution of system states changes.

In practice, it looks like this. From the time series, you build a transition matrix between values of basis functions at neighboring steps. Then you use this matrix to evolve the coefficients of the initial density. All this works for any discretization step and does not require knowing the vector field or diffusion tensor.

On the Lorenz model, known for its chaotic behavior, diffusion forecasting showed significant improvement over locally linear methods. At a step of 0.1, the iterative local forecast accumulated bias and overestimated variance. Diffusion forecasting remained consistent with the reference obtained by averaging 50,000 trajectories.

On real El Niño 3.4 index data, the 14-month forecast achieved a correlation of 0.64 and an RMSE of 0.60. This is better than a previously published result with a correlation of 0.4 and an RMSE of 1.4. Such a result shows that the method can be applied to forecasting climate indices and assessing uncertainty in chaotic systems.

A translator who learns from speech does not know the rules but understands the meaning. Likewise, the new method does not know the equations but can predict. It opens the way to probabilistic forecasts for systems with unknown dynamics.