Data-assisted reduced-order modeling of extreme events in complex dynamical systems

Abstract

Dynamical systems with high intrinsic dimensionality are often characterized by extreme events having the form of rare transitions several standard deviations away from the mean. For such systems, order-reduction methods through projection of the governing equations have limited applicability due to the large intrinsic dimensionality of the underlying attractor but also the complexity of the transient events. An alternative approach is data-driven techniques that aim to quantify the dynamics of specific modes utilizing data-streams. Several of these approaches have improved performance by expanding the state representation using delayed coordinates. However, such strategies are limited in regions of the phase space where there is a small amount of data available, as is the case for extreme events. In this work, we develop a blended framework that integrates an imperfect model, obtained from projecting equations into a subspace that still contains crucial dynamical information, with data-streams through a recurrent neural network (RNN) architecture. In particular, we employ the long-short-term memory (LSTM), to model portions of the dynamics which cannot be accounted by the equations. The RNN is trained by analyzing the mismatch between the imperfect model and the data-streams, projected in the reduced-order space. In this way, the data-driven model improves the imperfect model in regions where data is available, while for locations where data is sparse the imperfect model still provides a baseline for the prediction of the system dynamics. We assess the developed framework on two challenging prototype systems exhibiting extreme events and show that the blended approach has improved performance compared with methods that use either data streams or the imperfect model alone. The improvement is more significant in regions associated with extreme events, where data is sparse.

arXiv commentSubmitted to PLOS ONE on March 8, 2018

References in corpus (18)

• Adam: A Method for Stochastic Optimization
• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
• Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
• A Physics Informed Machine Learning Approach for Reconstructing Reynolds Stress Modeling Discrepancies Based on DNS Data
• Machine Learning of Linear Differential Equations using Gaussian Processes
• Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks
• Hybrid Forecasting of Chaotic Processes: Using Machine Learning in Conjunction with a Knowledge-Based Model
• Nonparametric forecasting of low-dimensional dynamical systems
• Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures
• A variational approach to probing extreme events in turbulent dynamical systems
• Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems
• Reduced-space Gaussian Process Regression for Data-Driven Probabilistic Forecast of Chaotic Dynamical Systems
• A statistical learning strategy for closed-loop control of fluid flows
• Forecasting Turbulent Modes with Nonparametric Diffusion Models: Learning from noisy data
• Physics-Based Learning Models for Ship Hydrodynamics
• Reduced order prediction of rare events in unidirectional nonlinear water waves

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