Integrating Score-Based Generative Modeling and Neural ODEs for Accurate Representation of Multiscale Chaotic Dynamics

Abstract

Multiscale dynamical systems characterized by interacting fast and slow processes are ubiquitous across scientific domains, from climate dynamics to fluid mechanics. Accurate modeling of such systems requires capturing both the long-term statistical properties governed by slow variables and the short-term transient dynamics driven by fast chaotic processes. We present a hybrid data-driven framework that integrates score-based generative modeling with Neural Ordinary Differential Equations (NODEs) to construct reduced-order models (ROMs) capable of reproducing both regimes. The slow dynamics are represented by a Langevin equation whose drift is informed by a score function learned via the K-means Gaussian Mixture Model (KGMM) method, ensuring faithful reproduction of the system's invariant measure. The fast chaotic forcing is modeled by a NODE trained on delay-embedded residuals extracted from observed trajectories, replacing conventional Gaussian noise approximations. We validate this approach on a hierarchy of prototypical metastable systems driven by Lorenz 63 dynamics, including bistable potentials with additive and multiplicative forcing, and tristable non-autonomous systems with cycloperiodic components. Our results demonstrate that the hybrid framework maintains statistical consistency over long time horizons while accurately forecasting rare critical transitions between metastable states with lead times approaching the Lyapunov time of the chaotic driver. This work establishes a principled methodology for combining statistical closure techniques with explicit surrogate models of fast dynamics, offering a pathway toward predictive modeling of complex multiscale phenomena where both long-term statistics and short-term transients are essential.