Parametric Reduced Models for the Nonlinear Schrödinger Equation

Abstract

Reduced models for the (defocusing) nonlinear Schrödinger equation are developed. In particular, we develop reduced models that only involve the low-frequency modes given noisy observations of these modes. The ansatz of the reduced parametric models are obtained by employing a rational approximation and a colored noise approximation, respectively, on the memory terms and the random noise of a generalized Langevin equation that is derived from the standard Mori-Zwanzig formalism. The parameters in the resulting reduced models are inferred from noisy observations with a recently developed ensemble Kalman filter-based parameterization method. The forecasting skill across different temperature regimes are verified by comparing the moments up to order four, a two-time correlation function statistics, and marginal densities of the coarse-grained variables.

References in corpus (3)

• Computation of the Memory Functions in the Generalized Langevin Models for Collective Dynamics of Macromolecules
• Linear theory for filtering nonlinear multiscale systems with model error
• Adaptive Error Covariances Estimation Methods for Ensemble Kalman Filters

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• Kernel-based Prediction of Non-Markovian Time Series
• Forecasting Turbulent Modes with Nonparametric Diffusion Models: Learning from noisy data
• Data-driven model reduction for stochastic Burgers equations
• Modeling of Missing Dynamical Systems: Deriving Parametric Models using a Nonparametric Framework
• Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Itô Diffusions