Chaotic prediction using weakly-synchronised trajectories and machine learning

Machine learning approaches for forecasting chaotic dynamical systems typically rely on neural networks trained on arbitrarily selected trajectory data from a target system. However, this conventional strategy often limits predictive performance to short horizons, due to the inherent sensitivity and complexity of chaotic dynamics. Here, we show that carefully curated ensembles of initial conditions spanning a large extent of an attractor, such as the Lorenz attractor, enable longer prediction horizons. This demonstrates that prediction quality depends not only on sampling more trajectories per se, but also on capturing the right range of dynamical behaviours, driven by shadowing and structural stability. As is well known, the Lorenz attractor is structurally stable, meaning small perturbations in initial conditions lead to trajectories that “shadow” each other, leading to higher synchronisation levels among them. Motivated by this insight, we introduce a novel data filtering method based on phase synchrony analysis, using the Kuramoto Order Parameter (KOP) to select training trajectories with low synchronisation levels, still evolving chaotically on the attractor. We demonstrate the approach on feed forward neural networks applied to the Lorenz system, comparing models trained on KOP-filtered datasets with those trained on unfiltered datasets. Evaluation across varying perturbation scales shows that KOP-based filtering substantially improves predictive performance. Models trained on low-KOP data achieve longer prediction horizons, exhibit superior agreement with ground truth dynamics, and produce output probability distributions closely approximating Gaussian behaviour, significantly outperforming traditional strategies that do not provide insight into the dynamical properties of the datasets. Our results highlight the importance of considering the properties of the dataset to increase dynamical diversity in the training datasets and suggest a principled pathway for enhancing machine learning-based forecasts of chaotic systems.