Inducing fast-slow separation enables robust learning of complex dynamics

Abstract

Many real-world systems share a fast-slow structure: most degrees of freedom relax quickly, while a few slow modes govern the long-term evolution. Their dynamics often collapse onto low-dimensional complex structures such as chaotic attractors, and a goal of nonlinear science is to predict and faithfully reproduce them from observed time series. Machine-learning models and data-driven approaches can embed a chaotic attractor in high-dimensional spaces, but embedding alone does not guarantee faithful reproduction. Training can create spurious slow modes, excess modes unnecessary for the target dynamics, which destabilize the reconstruction. To prevent this instability, we introduce input-layer designs for reservoir computing, a framework suited to physical implementation. Through restriction of network controllability, the designs limit the number of slow modes available to learning and anchor the remaining modes to stay fast in advance, even for a black-box model. The reconstruction is thereby confined to a transversally attracting subspace, and spurious slow modes are suppressed. Across diverse chaotic systems, the designs robustly reproduce the attractors and their dynamical invariants, extend the horizon of accurate prediction, and withstand perturbations of the internal weights, without extensive tuning. Inducing fast-slow separation and reconstructing attractors in attracting subspaces offers a design principle for reliable data-driven modeling.

40 pages, 16 figures