Taming Chaos with Physics: Predicting Extreme Events from Sparse Data

Imagine a translator who learns the grammar of a chaotic language from textbook examples and then becomes fluent in real speech, complete with accents and noise. That is how the new KDL method works: first, a neural network is trained on synthetic data generated by the Liénard equation, then it is transferred to real time series where extreme events are rare and unpredictable.

Extreme events are rogue waves in the ocean, El Niño episodes, sudden downpours. They are rare, so training data are scarce, and their dynamics are nonlinear and sensitive to noise. Classical physics-based models require precise initial conditions and quickly diverge from reality, while purely data-driven approaches need huge samples. KDL overcomes this limitation: it learns from synthetic data, where extreme events can be generated in abundance, and transfers that knowledge to real data.

The secret of KDL lies in physical regularization: the network predicts not only the value itself but also its derivatives, and the loss function includes a term that enforces the differential equation. It is as if the translator not only memorized phrases but also learned the grammar of the language, and can now recognize even unfamiliar constructions. On El Niño data, KDL achieved the lowest RMSE (0.428 versus 0.433 for LSTM with 20% training data) and the lowest physical residual (276.654 versus 283.340). When the training fraction is increased to 80%, the advantage persists but narrows.

For dengue fever, transferring knowledge from synthetic data worsened the result, so the authors turned off pretraining and kept only physical regularization: KDL achieved the best physical residual at 40% and 60% of the data, although it lagged behind LSTM in RMSE. On precipitation data from Bjørnøya Island (15320 observations), KDL won on RMSE at 20%, 60%, and 80% of training data, but lost on physical residual to reservoir computing. On average across all datasets and splits, KDL ranked first in RMSE (rank 1.12) and MAE (rank 1.29), outperforming LSTM, FFNN, CNN, and ESN.

A translator who learned grammar from textbook examples now understands live speech with its accent and noise. KDL can already improve early warning systems for climate anomalies, infectious disease outbreaks, and extreme precipitation in regions with short observation records. In engineering, it is applicable to predicting rare failures in chaotic systems, such as in power grids.