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Machine Learning for Prediction, Data Assimilation, and Uncertainty Quantification of Dynamical Systems

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Information Theory, Probability and Statistics".

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 17051

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Special Issue Editors

Prof. Dr. John Harlim

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Guest Editor

Department of Mathematics, Department of Meteorology, Institute for CyberScience, The Pennsylvania State University, University Park, PA 16802, USA
Interests: data assimilation; data-driven modelling; stochastic computational methods; machine learning

Prof. Dr. Dimitrios Giannakis

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Co-Guest Editor

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012-1185, USA
Interests: geometric data analysis; spectral analysis of dynamical systems; statistical forecasting; climate dynamics

Prof. Dr. Tyrus Berry

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Co-Guest Editor

Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS 3F2, Fairfax, VA 22030, USA
Interests: manifold learning; non-parametric and semiparametric modeling; dynamical systems; filtering; data assimilation; control

Special Issue Information

Dear Colleagues,

Modeling dynamical systems is ubiquitous in a large variety of applications in the sciences. As the capability to collect data advances, an important and growing challenge is to extract relevant information from large datasets in ways that can improve modeling. Recent empirical results suggest that various machine learning algorithms are effective tools in approximating the solution operator of the underlying dynamics without relying on a parametric modeling assumption, but instead leveraging the available datasets to learn the dynamics. While the empirical successes are an important first step, they naturally introduce many practical and theoretical questions.

In this Special Issue, we particularly welcome contributions that address the following problems (other contributions relevant to the topic are also welcome):

Capabilities and Limitations of Purely Data-Driven Models: A theoretical or thorough empirical study to understand the extent to which model-free (or nonparametric) techniques can be used to improve the prediction of high-dimensional complex dynamical systems. One possible line of inquiry would be how to use criteria from information theory to determine any such limitations.
Leveraging Information from Partial Models: Another pertinent question, particularly in scenarios with intrinsically high-dimensional dynamics, is how to efficiently augment a partial or imperfect first-principles (parametric) model with a data-driven model to correct model error or model unresolved degrees of freedom. Prominent applications include subgrid-scale modeling and closure schemes for complex systems.
Data Assimilation: High-quality prediction requires powerful data assimilation techniques in order to determine accurate initial conditions. How can one leverage machine learning to improve data assimilation? For example, an issue that is particularly prevalent in the data assimilation community is estimating non-stationary second-order statistics, and this may be an opportunity for model-free methods. Conversely, how can one leverage modern data assimilation methods to improve model-free forecasting techniques?
Uncertainty Quantification: It is crucially important to provide confidence in the prediction through a reliable uncertainty quantification (UQ). How can one leverage machine learning in this context? Methods of improving UQ in parametric modeling with machine learning or evaluating UQ techniques applied to model-free methods are welcomed.

Prof. Dimitrios Giannakis
Prof. John Harlim
Prof. Tyrus Berry
Guest Editors

Manuscript Submission Information

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Keywords

model-free prediction
nonparametric models
model error
machine learning
kernel methods
data assimilation
Bayesian inferences
uncertainty quantification

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Published Papers (5 papers)

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Research

21 pages, 1218 KiB

Open AccessArticle

Bayesian Update with Importance Sampling: Required Sample Size

by Daniel Sanz-Alonso and Zijian Wang

Entropy 2021, 23(1), 22; https://doi.org/10.3390/e23010022 - 26 Dec 2020

Cited by 5 | Viewed by 2263

Abstract

Importance sampling is used to approximate Bayes’ rule in many computational approaches to Bayesian inverse problems, data assimilation and machine learning. This paper reviews and further investigates the required sample size for importance sampling in terms of the

χ^{2}

-divergence between target [...] Read more.

χ^{2}

-divergence between target and proposal. We illustrate through examples the roles that dimension, noise-level and other model parameters play in approximating the Bayesian update with importance sampling. Our examples also facilitate a new direct comparison of standard and optimal proposals for particle filtering. Full article

(This article belongs to the Special Issue Machine Learning for Prediction, Data Assimilation, and Uncertainty Quantification of Dynamical Systems)

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22 pages, 2705 KiB

Open AccessArticle

Data-Driven Model Reduction for Stochastic Burgers Equations

by Fei Lu

Entropy 2020, 22(12), 1360; https://doi.org/10.3390/e22121360 - 30 Nov 2020

Cited by 13 | Viewed by 2372

Abstract

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. Full article

(This article belongs to the Special Issue Machine Learning for Prediction, Data Assimilation, and Uncertainty Quantification of Dynamical Systems)

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21 pages, 686 KiB

Open AccessArticle

Data-Driven Corrections of Partial Lotka–Volterra Models

by Rebecca E. Morrison

Entropy 2020, 22(11), 1313; https://doi.org/10.3390/e22111313 - 18 Nov 2020

Cited by 4 | Viewed by 2601

Abstract

In many applications of interacting systems, we are only interested in the dynamic behavior of a subset of all possible active species. For example, this is true in combustion models (many transient chemical species are not of interest in a given reaction) and in epidemiological models (only certain subpopulations are consequential). Thus, it is common to use greatly reduced or partial models in which only the interactions among the species of interest are known. In this work, we explore the use of an embedded, sparse, and data-driven discrepancy operator to augment these partial interaction models. Preliminary results show that the model error caused by severe reductions—e.g., elimination of hundreds of terms—can be captured with sparse operators, built with only a small fraction of that number. The operator is embedded within the differential equations of the model, which allows the action of the operator to be interpretable. Moreover, it is constrained by available physical information and calibrated over many scenarios. These qualities of the discrepancy model—interpretability, physical consistency, and robustness to different scenarios—are intended to support reliable predictions under extrapolative conditions. Full article

(This article belongs to the Special Issue Machine Learning for Prediction, Data Assimilation, and Uncertainty Quantification of Dynamical Systems)

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24 pages, 1162 KiB

Open AccessArticle

Can Short and Partial Observations Reduce Model Error and Facilitate Machine Learning Prediction?

by Nan Chen

Entropy 2020, 22(10), 1075; https://doi.org/10.3390/e22101075 - 24 Sep 2020

Cited by 2 | Viewed by 2814

Abstract

Predicting complex nonlinear turbulent dynamical systems is an important and practical topic. However, due to the lack of a complete understanding of nature, the ubiquitous model error may greatly affect the prediction performance. Machine learning algorithms can overcome the model error, but they are often impeded by inadequate and partial observations in predicting nature. In this article, an efficient and dynamically consistent conditional sampling algorithm is developed, which incorporates the conditional path-wise temporal dependence into a two-step forward-backward data assimilation procedure to sample multiple distinct nonlinear time series conditioned on short and partial observations using an imperfect model. The resulting sampled trajectories succeed in reducing the model error and greatly enrich the training data set for machine learning forecasts. For a rich class of nonlinear and non-Gaussian systems, the conditional sampling is carried out by solving a simple stochastic differential equation, which is computationally efficient and accurate. The sampling algorithm is applied to create massive training data of multiscale compressible shallow water flows from highly nonlinear and indirect observations. The resulting machine learning prediction significantly outweighs the imperfect model forecast. The sampling algorithm also facilitates the machine learning forecast of a highly non-Gaussian climate phenomenon using extremely short observations. Full article

(This article belongs to the Special Issue Machine Learning for Prediction, Data Assimilation, and Uncertainty Quantification of Dynamical Systems)

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22 pages, 1381 KiB

Open AccessArticle

Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator

by Stefan Klus, Feliks Nüske and Boumediene Hamzi

Entropy 2020, 22(7), 722; https://doi.org/10.3390/e22070722 - 30 Jun 2020

Cited by 37 | Viewed by 6188

Abstract

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems. Full article

(This article belongs to the Special Issue Machine Learning for Prediction, Data Assimilation, and Uncertainty Quantification of Dynamical Systems)

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