Nonlinear stochastic modelling with Langevin regression.

Area of Science:

• Physics
• Applied Mathematics
• Data Science

Background:

• Many physical systems exhibit nonlinear multiscale interactions, often modeled by treating unresolved degrees of freedom as random fluctuations.
• Deriving accurate stochastic models consistent with observations is challenging, particularly for systems like turbulence with non-Gaussian, non-Markovian noise.
• Existing methods struggle to capture the complex dynamics arising from unresolved scales.

Purpose of the Study:

• To develop a robust framework for identifying interpretable stochastic nonlinear dynamics directly from experimental data.
• To address the challenges of non-Gaussian and non-Markovian behavior in stochastic modeling.
• To create a method that enforces statistical consistency between models and observations.

Main Methods:

• Utilized forward and adjoint Fokker-Planck equations to ensure statistical consistency.
• Employed a sparsifying procedure to determine the functional form of unknown Langevin equations.
• Applied Langevin regression to experimental data from a turbulent bluff body wake.

Main Results:

• Successfully learned stochastic models in two artificial examples, including a nonlinear Langevin equation with colored noise.
• Approximated second-order dynamics in a double-well potential system using a first-order bifurcation normal form.
• Demonstrated that the center of pressure in a turbulent wake follows dynamics driven by nonlinear state-dependent noise.

Conclusions:

• The developed framework effectively identifies interpretable stochastic nonlinear dynamics from experimental data.
• The method provides a powerful tool for modeling complex systems where traditional stochastic approaches fail.
• This approach advances the understanding and prediction of physical systems characterized by nonlinear multiscale interactions and complex noise characteristics.