Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systems

TL;DR

The paper addresses learning the effective slow dynamics of unknown slow-fast multiscale stochastic systems from observations of the slow state alone. It proposes a stochastic flow map learning (sFML) framework that uses a conditional normalizing flow to build a one-step generative model G for the slow variable, delivering a weak distributional match to the true slow dynamics. The contributions include extending sFML to multiscale stochastic settings without memory terms, training on short bursts of slow data, and validating across five numerical examples that demonstrate accurate slow-variable distributions and trajectory statistics. This data-driven approach enables reliable forecasting and analysis of slow dynamics when fast variables are unobserved or inaccessible, providing a practical tool for complex multiscale systems.

Abstract

We present a numerical method for learning the dynamics of slow components of unknown multiscale stochastic dynamical systems. While the governing equations of the systems are unknown, bursts of observation data of the slow variables are available. By utilizing the observation data, our proposed method is capable of constructing a generative stochastic model that can accurately capture the effective dynamics of the slow variables in distribution. We present a comprehensive set of numerical examples to demonstrate the performance of the proposed method.

Figures (16)

• Figure 1: Illustration of how the fast variables $\mathbf{y}$ quickly become slaved to the slow variables $\mathbf{x}$ in multiscale system.
• Figure 1: Sample trajectories of Example \ref{['s:ex1']} slow variable with initial condition $x_0=1.5$ and $y_0$ sampling from stationary distribution. Left: ground truth; Right: Simulation using the trained sFML model.
• Figure 2: The DNN model structure for the proposed normalizing flow sFML method \ref{['x1']}.
• Figure 2: Left: One simulated sample of full system of Example \ref{['s:ex1']} with initial condition $x_0=1.5$ and $y_0$ sampling from stationary distribution. Right: Comparison for mean and standard deviation (STD) of ground truth and learned sFL model for the slow variable.
• Figure 3: Sample trajectories of Example \ref{['s:ex2']} slow variable with initial condition $x_0=1.5$ and $y_0$ sampling from stationary distribution. Left: ground truth; Right: Simulation using the trained sFML model.