Nonparametric forecasting of low-dimensional dynamical systems.

Area of Science:

• Dynamical Systems and Chaos Theory
• Stochastic Processes
• Data Science and Machine Learning

Background:

• Forecasting complex dynamical systems is challenging due to inherent stochasticity and nonlinearity.
• Existing methods often struggle with equation-free modeling and uncertainty quantification.
• Low-dimensional manifold representations are crucial for simplifying complex systems.

Purpose of the Study:

• To develop a nonparametric modeling approach for forecasting stochastic dynamical systems on low-dimensional manifolds.
• To enable equation-free modeling and uncertainty quantification for complex systems.
• To provide a robust framework for analyzing systems where underlying equations are unknown or too complex.

Main Methods:

• Utilizing diffusion maps to obtain a smooth basis representation of discrete shift maps.
• Applying Galerkin projection in the large data limit to approximate the semigroup solution.
• Implementing an equation-free modeling strategy for dynamical systems analysis.

Main Results:

• The proposed method converges to a Galerkin projection adapted to the invariant measure.
• Demonstrated ability to quantify uncertainties and evolve probability distributions for complex systems.
• Successfully applied to diverse examples including stochastic differential equations, the Lorenz model, and El Niño Southern Oscillation data.

Conclusions:

• The nonparametric approach offers a powerful tool for forecasting and understanding stochastic dynamical systems.
• This method enhances the capability for uncertainty quantification in equation-free modeling.
• The approach is versatile and effective across various complex scientific domains.