Chaos Meets Attention: Transformers for Large-Scale Dynamical Prediction

Generating long-term trajectories of dissipative chaotic systems autoregressively is a highly challenging task. The inherent positive Lyapunov exponents amplify prediction errors over time. Many chaotic systems possess a crucial property -ergodicity on their attractors, which makes long-term prediction possible. State-of-the-art methods address ergodicity by preserving statistical properties using optimal transport techniques. However, these methods face scalability challenges due to the curse of dimensionality when matching distributions. To overcome this bottleneck, we propose a scalable transformerbased framework capable of stably generating long-term high-dimensional and high-resolution chaotic dynamics while preserving ergodicity. Our method is grounded in a physical perspective, revisiting the Von Neumann mean ergodic theorem to ensure the preservation of long-term statistics in the L 2 space. We introduce novel modifications to the attention mechanism, making the transformer architecture well-suited for learning large-scale chaotic systems. Compared to operator-based and transformer-based methods, our model achieves better performances across five metrics, from short-term prediction accuracy to long-term statistics. In addition to our methodological contributions, we introduce new chaotic system benchmarks: a machine learning dataset of 140k snapshots of turbulent channel flow and a processed high-dimensional Kolmogorov Flow dataset, along with various evaluation metrics for both short-and long-term performances. Both are well-suited for machine learning research on chaotic systems.