Fast-Slow Neural Networks for Learning Singularly Perturbed Dynamical Systems

Singularly perturbed dynamical systems play a crucial role in climate dynamics and plasma physics. A powerful and well-known tool to address these systems is the Fenichel normal form, which significantly simplifies fast dynamics near slow manifolds through a transformation. However, this normal form is difficult to realize in conventional numerical algorithms. In this work, we explore an alternative way of realizing it through structure-preserving machine learning. Specifically, a fast-slow neural network (FSNN) is proposed for learning data-driven models of singularly perturbed dynamical systems with dissipative fast timescale dynamics. Our method enforces the existence of a trainable, attracting invariant slow manifold as a hard constraint. Closed-form representation of the slow manifold enables efficient integration on the slow time scale and significantly improves prediction accuracy beyond the training data. We demonstrate the FSNN on examples including the Grad moment system, two-scale Lorenz96 equations, and Abraham-Lorentz dynamics modeling radiation reaction of electrons.