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Learning Stochastic Closures Using Ensemble Kalman Inversion

Schneider, Tapio and Stuart, Andrew M. and Wu, Jin-Long (2020) Learning Stochastic Closures Using Ensemble Kalman Inversion. . (Unpublished)

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Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions within the first principles description. Therefore researchers often seek simpler descriptions that describe complex phenomena without numerically resolving all the interacting components. Stochastic differential equations (SDEs) arise naturally as models in this context. The growth in data acquisition provides an opportunity for the systematic derivation of SDE models in many disciplines. However, inconsistencies between SDEs and real data at small time scales often cause problems, when standard statistical methodology is applied to parameter estimation. The incompatibility between SDEs and real data can be addressed by deriving sufficient statistics from the time-series data and learning parameters of SDEs based on these. Following this approach, we formulate the fitting of SDEs to sufficient statistics from real data as an inverse problem and demonstrate that this inverse problem can be solved by using ensemble Kalman inversion (EKI). Furthermore, we create a framework for non-parametric learning of drift and diffusion terms by introducing hierarchical, refineable parameterizations of unknown functions, using Gaussian process regression. We demonstrate the proposed methodology for the fitting of SDE models, first in a simulation study with a noisy Lorenz 63 model, and then in other applications, including dimension reduction starting from various deterministic chaotic systems arising in the atmospheric sciences, large-scale pattern modeling in climate dynamics, and simplified models for key observables arising in molecular dynamics. The results confirm that the proposed methodology provides a robust and systematic approach to fitting SDE models to real data.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Schneider, Tapio0000-0001-5687-2287
Additional Information:The authors thank Dr. Yvo Pokern at University College London for providing the butane dihedral angle data. All authors are supported by the generosity of Eric and Wendy Schmidt by recommendation of the Schmidt Futures program, by Earthrise Alliance, Mountain Philanthropies, the Paul G. Allen Family Foundation, and the National Science Foundation (NSF, award AGS1835860). A.M.S. is also supported by NSF (award DMS-1818977) and by the Office of Naval Research (award N00014-17-1-2079).
Funding AgencyGrant Number
Schmidt Futures ProgramUNSPECIFIED
Earthrise AllianceUNSPECIFIED
Mountain PhilanthropiesUNSPECIFIED
Paul G. Allen Family FoundationUNSPECIFIED
Office of Naval Research (ONR)N00014-17-1-2079
Subject Keywords:Stochastic differential equation; inverse problem; Ensemble Kalman inversion; Gaussian process regression; hierarchical parameterization
Record Number:CaltechAUTHORS:20201109-140955956
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106558
Deposited By: George Porter
Deposited On:09 Nov 2020 23:06
Last Modified:09 Nov 2020 23:06

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