https://cnam.hal.science/hal-03605461Callaham, J.J.CallahamDepartment of Mechanical Engineering [University of Washington] - University of Washington [Seattle]Loiseau, J.-C.J.-C.LoiseauDynFluid - Laboratoire de Dynamique des Fluides - CNAM - Conservatoire National des Arts et Métiers [CNAM] - HESAM - HESAM Université - Communauté d'universités et d'établissements Hautes écoles Sorbonne Arts et métiers université - Arts et Métiers Sciences et Technologies - HESAM - HESAM Université - Communauté d'universités et d'établissements Hautes écoles Sorbonne Arts et métiers universitéRigas, G.G.RigasDepartment of Aeronautics - Imperial College LondonBrunton, S.S.BruntonDepartment of Mechanical Engineering [University of Washington] - University of Washington [Seattle]Nonlinear stochastic modelling with Langevin regressionHAL CCSD2021mathematical modellingfluid mechanicsstatistical physics system identificationdata-driven modellingstochastic modellingLangevin equationFokker-Planck equationsparse regression[PHYS.MECA.MEFL] Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph][MATH] Mathematics [math]Bertram, Marie-Liesse2022-03-11 09:39:462022-09-28 05:59:402022-03-16 10:51:13enJournal articleshttps://cnam.hal.science/hal-03605461/document10.1098/rspa.2021.0092application/pdf1Many 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.