ArticleProceedings of the National Academy of Sciences of the United States of America2017
Beating the curse of dimension with accurate statistics for the Fokker-Planck equation in complex turbulent systems.
Article in Proceedings of the National Academy of Sciences of the United States of America, 2017. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 9 papers.
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9 citing papers in PubMed.
- A Martingale-Free Introduction to Conditional Gaussian Nonlinear Systems.Entropy (Basel, Switzerland) · 2024Article
- Unambiguous Models and Machine Learning Strategies for Anomalous Extreme Events in Turbulent Dynamical System.Entropy (Basel, Switzerland) · 2024Article
- Addressing the curse of dimensionality in stochastic dynamics: a Wiener path integral variational formulation with free boundaries.Proceedings. Mathematical, physical, and engineering sciences · 2020Article
- Can Short and Partial Observations Reduce Model Error and Facilitate Machine Learning Prediction?Entropy (Basel, Switzerland) · 2020Article
- Using machine learning to predict extreme events in complex systems.Proceedings of the National Academy of Sciences of the United States of America · 2020Article
- Reaction Time Improvements by Neural Bistability.Behavioral sciences (Basel, Switzerland) · 2019Article
- Statistical dynamical model to predict extreme events and anomalous features in shallow water waves with abrupt depth change.Proceedings of the National Academy of Sciences of the United States of America · 2019Article
- Model Error, Information Barriers, State Estimation and Prediction in Complex Multiscale Systems.Entropy (Basel, Switzerland) · 2018Article
- Conditional Gaussian Systems for Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification.Entropy (Basel, Switzerland) · 2018Article
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Abstract
Solving the Fokker-Planck equation for high-dimensional complex dynamical systems is an important issue. Recently, the authors developed efficient statistically accurate algorithms for solving the Fokker-Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures, which contain many strong non-Gaussian features such as intermittency and fat-tailed probability density functions (PDFs). The algorithms involve a hybrid strategy with a small number of samples [Formula: see text], where a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious Gaussian kernel density estimation in the remaining low-dimensional subspace. In this article, two effective strategies are developed and incorporated into these algorithms. The first strategy involves a judicious block decomposition of the conditional covariance matrix such that the evolutions of different blocks have no interactions, which allows an extremely efficient parallel computation due to the small size of each individual block. The second strategy exploits statistical symmetry for a further reduction of [Formula: see text] The resulting algorithms can efficiently solve the Fokker-Planck equation with strongly non-Gaussian PDFs in much higher dimensions even with orders in the millions and thus beat the curse of dimension. The algorithms are applied to a [Formula: see text]-dimensional stochastic coupled FitzHugh-Nagumo model for excitable media. An accurate recovery of both the transient and equilibrium non-Gaussian PDFs requires only [Formula: see text] samples! In addition, the block decomposition facilitates the algorithms to efficiently capture the distinct non-Gaussian features at different locations in a [Formula: see text]-dimensional two-layer inhomogeneous Lorenz 96 model, using only [Formula: see text] samples.
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