A translator who learns to correct their own slips works more accurately. In the world of forecasting noisy processes, a new method takes on this role: a reservoir computer that adds a random correction to its prediction at each step. Previously, such systems quickly went off track because noise accumulated errors, like a translator who doesn't notice their own slips, and by the end of the speech the meaning was lost. Now the error is modeled separately and turned into a tool.

A reservoir computer is a network with fixed internal connections that learns only at the output. It predicted chaos without noise well, but as soon as randomness was added, the forecasts diverged from reality within a few steps. The authors of the RC-NF method took a different approach: first, they trained the reservoir on a set of trajectories, and then, using a normalizing flow, they estimated the error it makes at each step. A normalizing flow is a generative model that transforms a simple distribution into a complex one, and here it captures the error distribution.

Tests on seven systems showed that the method reproduces not only the average behavior but also rare events. For a double-well process, where the system sometimes jumps from one stable point to another, RC-NF reproduced the transition rate with an error of less than 0.001. For the stochastic Lorenz system, the Wasserstein distance, a measure of dissimilarity between distributions, was 0.0266 versus 0.0479 for a pure reservoir computer. The median of the maximal Lyapunov exponent, which characterizes chaoticity, was 2.5622, compared to the true value of 2.5557.

A translator who hears their own slips and corrects them on the fly preserves the meaning of a long speech. Similarly, RC-NF maintains the statistics of the process over long intervals, opening the way to forecasts of climate fluctuations, biological rhythms, and financial series without knowing their equations.