Learning to Predict Chaos: Curriculum-Driven Training for Robust Forecasting of Chaotic Dynamics

Abstract

Forecasting chaotic systems is a cornerstone challenge in many scientific fields, complicated by the exponential amplification of even infinitesimal prediction errors. Modern machine learning approaches often falter due to two opposing pitfalls: over-specializing on a single, well-known chaotic system (e.g., Lorenz-63), which limits generalizability, or indiscriminately mixing vast, unrelated time-series, which prevents the model from learning the nuances of any specific dynamical regime. We propose Curriculum Chaos Forecasting (CCF), a training paradigm that bridges this gap. CCF organizes training data based on fundamental principles of dynamical systems theory, creating a curriculum that progresses from simple, periodic behaviors to highly complex, chaotic dynamics. We quantify complexity using the largest Lyapunov exponent and attractor dimension, two well-established metrics of chaos. By first training a sequence model on predictable systems and gradually introducing more chaotic trajectories, CCF enables the model to build a robust and generalizable representation of dynamical behaviors. We curate a library of over 50 synthetic ODE/PDE systems to build this curriculum. Our experiments show that pre-training with CCF significantly enhances performance on unseen, real-world benchmarks. On datasets including Sunspot numbers, electricity demand, and human ECG signals, CCF extends the valid prediction horizon by up to 40% compared to random-order training and more than doubles it compared to training on real-world data alone. We demonstrate that this benefit is consistent across various neural architectures (GRU, Transformer) and provide extensive ablations to validate the importance of the curriculum's structure.