Nonlinear stochastic modeling with Langevin regression

Abstract

Many physical systems characterized by nonlinear multiscale interactions can be effectively modeled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behavior are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like deltacorrelated Gaussian white noise, introducing non-Markovian behavior to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using both forward and adjoint Fokker-Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can effectively learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by colored noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply the proposed method to experimental measurements of a turbulent bluff body wake and show that the statistical behavior of the center of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.

Fluctuating landscapes and heavy tails in animal behavior

“…Technically speaking, we build an overdamped description of a partially-observed system with metastable dynamics, combining ideas from reduced order modeling [25,[58][59][60] and stochastic model inference [32,61,62]. Our contribution here is mostly conceptual: instead of assuming structure a priori, we reveal it by carefully analyzing the time series.…”

Maximally predictive ensemble dynamics from data

“…(D-right) Estimated effective diffusion coefficients obtained from simulations Dsim and the data D data . We estimate D by fitting the slope of the mean square displacement curves in the interval τ ∈ [60,100] s and find operator-predicted values (blue) closely match those obtained from the data. Symbolic sequences that preserve the steady-state distribution but are otherwise random (green) fail to capture the diffusive properties of the centroid trajectories.…”

“…Nonetheless, a τparametrized family of coherent sets can capture moving regions of the state space that remain coherent within a time scale τ [56,[94][95][96][97][98]. Even for non-Markovian systems our approach can help identify and isolate the longlived dynamics [99,100].…”

Sparse nonlinear models of chaotic electroconvection

“…Of particular note are its uses in identifying Lorenz-like dynamics from a thermosyphon simulation by Loiseau [70] and to identify a model for a nonlinear magnetohydrodynamic plasma system by Kaptanoglu et al [88]. It has also been extended to handle more complex modeling scenarios such as partial differential equations [89,90], systems with inputs or control [91], to enforce physical constraints [68], to identify models from corrupt or limited data [92,93] and ensembles of initial conditions [94], and extending the formulation to include integral terms [95,96], tensor representations [97,98], deep autoencoders [99], and stochastic forcing [100,101]; an open-source software package, PySINDy, has been developed to integrate a number of these innovations [102]. We will enforce the symmetries observed above as constraints, as in Loiseau and Brunton [68].…”