Nonlinear stochastic modelling with Langevin regression

Abstract

Many physical systems characterized by nonlinear multiscale interactions can be modelled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behaviour 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 Gaussian white noise, introducing non-Markovian behaviour to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using 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 learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by coloured 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 Langevin regression to experimental measurements of a turbulent bluff body wake and show that the statistical behaviour of the centre of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.

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“…In particular [6] used a robust method termed SR3 (sparse relaxed regularized regression) to extract the sparse, nonzero coefficients of the dynamical model. Since its introduction, SINDy has been applied to a wide range of systems, including for reduced-order models of fluid dynamics [7]- [14] and plasma dynamics [15], [16], turbulence closures [17]- [19], nonlinear optics [20], numerical integration schemes [21], discrepancy modeling [22], [23], boundary value problems [24], multiscale dynamics [25], identifying dynamics on Poincare maps [26], [27], tensor formulations [28], and systems with stochastic dynamics [29], [30]. It can also be used to jointly discovery coordinates and dynamics simultaneously [31], [32].…”

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