Coarse-Graining of Overdamped Langevin Dynamics via the Mori--Zwanzig Formalism

Hudson, Thomas E.; Li, Xingjie Helen

doi:10.1137/18m1222533

Cited by 26 publications

(34 citation statements)

References 54 publications

(42 reference statements)

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“…These developments demand a systematic understanding of model reduction from the perspectives of dynamical systems (see, e.g. [7,19,20]), numerical approximation [21,22], and statistical learning [23].…”

Section: Introductionmentioning

confidence: 99%

Data-driven Model Reduction for Stochastic Burgers Equations

2020

Preprint

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We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variables’ trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal and optimal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step where the K-mode Galerkin system’s mean CFL number agrees with the full model’s.

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Section: Introductionmentioning

confidence: 99%

Data-driven Model Reduction for Stochastic Burgers Equations

2020

Preprint

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“…This approach has not been applied before for porous media flows although it can be regarded as a special case of coarse-graining or model reduction (Givon et al 2004) of the high-dimensional Langevin governing equations for particle position and velocity. While this has been extensively studied for Hamiltonian-type systems (Hijón et al 2006;Di Pasquale et al 2019), the coarse-graining of general advection-diffusion models has been only recently studied for the overdamped Langevin equation (Duong et al 2018;Legoll and Lelièvre 2010;Hudson and Li 2018).…”

Section: Introductionmentioning

confidence: 99%

Probability density function (PDF) models for particle transport in porous media

Icardi

Dentz

2020

Int J Geomath

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Mathematical models based on probability density functions (PDF) have been extensively used in hydrology and subsurface flow problems, to describe the uncertainty in porous media properties (e.g., permeability modelled as random field). Recently, closer to the spirit of PDF models for turbulent flows, some approaches have used this statistical viewpoint also in pore-scale transport processes (fully resolved porous media models). When a concentration field is transported, by advection and diffusion, in a heterogeneous medium, in fact, spatial PDFs can be defined to characterise local fluctuations and improve or better understand the closures performed by classical upscaling methods. In the study of hydrodynamical dispersion, for example, PDE-based PDF approach can replace expensive and noisy Lagrangian simulations (e.g., trajectories of drift-diffusion stochastic processes). In this work we derive a joint position-velocity Fokker–Planck equation to model the motion of particles undergoing advection and diffusion in in deterministic or stochastic heterogeneous velocity fields. After appropriate closure assumptions, this description can help deriving rigorously stochastic models for the statistics of Lagrangian velocities. This is very important to be able to characterise the dispersion properties and can, for example, inform velocity evolution processes in continuous time random walk dispersion models. The closure problem that arises when averaging the Fokker–Planck equation shows also interesting similarities with the mixing problem and can be used to propose alternative closures for anomalous dispersion.

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“…The past decades witness revolutionary developments of data-driven strategies, ranging from parametric models (see, e.g., [ 8 , 9 , 10 , 11 , 12 , 13 , 14 ] and the references therein) to nonparametric and machine learning methods (see, e.g., [ 15 , 16 , 17 , 18 ]). These developments demand a systematic understanding of model reduction from the perspectives of dynamical systems (see, e.g., [ 7 , 19 , 20 ]), numerical approximation [ 21 , 22 ], and statistical learning [ 17 , 23 ].…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Model Reduction for Stochastic Burgers Equations

2020

Entropy

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We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as functionals of the resolved variable’s trajectory. The reduced models are nonlinear autoregression (NAR) time series models, with coefficients estimated from data by least squares. The NAR models can accurately reproduce the energy spectrum, the invariant densities, and the autocorrelations. Taking advantage of the simplicity of the NAR models, we investigate maximal space-time reduction. Reduction in space dimension is unlimited, and NAR models with two Fourier modes can perform well. The NAR model’s stability limits time reduction, with a maximal time step smaller than that of the K-mode Galerkin system. We report a potential criterion for optimal space-time reduction: the NAR models achieve minimal relative error in the energy spectrum at the time step, where the K-mode Galerkin system’s mean Courant–Friedrichs–Lewy (CFL) number agrees with that of the full model.

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Coarse-Graining of Overdamped Langevin Dynamics via the Mori--Zwanzig Formalism

Cited by 26 publications

References 54 publications

Data-driven Model Reduction for Stochastic Burgers Equations

Data-driven Model Reduction for Stochastic Burgers Equations

Probability density function (PDF) models for particle transport in porous media

Data-Driven Model Reduction for Stochastic Burgers Equations

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