Machine learning for prediction with missing dynamics

Harlim, John; Jiang, Shixiao W.; Liang, Senwei; Yang, Haizhao

doi:10.1016/j.jcp.2020.109922

Cited by 61 publications

(66 citation statements)

References 45 publications

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“…Secondly, in the model development under various situations, e.g., when additional couplings need to be included, the issue of model closure becomes important. This has to be done in a dynamic manner, and one of the possible routes for completing this task lies through a reformulation of the problem as a supervised machine learning (SML) process [216]. In this context, we would like to mention the Nakajima-Zwanzig equation which belongs to the Mori-Zwanzig theory within the statistical mechanics of irreversible processes.…”

Section: Modelling With Nonlocality In Data-driven Environmentsmentioning

confidence: 99%

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Sytnyk

Melnik

2021

MCA

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Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.

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Section: Modelling With Nonlocality In Data-driven Environmentsmentioning

confidence: 99%

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Sytnyk

Melnik

2021

MCA

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“…Data-driven approaches, which are based on statistical learning methods, provide useful and practical tools for model reduction. The past decades witness revolutionary developments of data-driven strategies, ranging from parametric models (see, e.g., [ 8 , 9 , 10 , 11 , 12 , 13 , 14 ] and the references therein) to nonparametric and machine learning methods (see, e.g., [ 15 , 16 , 17 , 18 ]). These developments demand a systematic understanding of model reduction from the perspectives of dynamical systems (see, e.g., [ 7 , 19 , 20 ]), numerical approximation [ 21 , 22 ], and statistical learning [ 17 , 23 ].…”

Section: Introductionmentioning

confidence: 99%

“…When the nonlinearity is complicated, a linear-in-parameter ansatz may be out of reach. One can overcome this limitation by nonparametric techniques [ 23 , 29 ] and machine learning methods (see, e.g., [ 16 , 17 , 30 ]).…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Model Reduction for Stochastic Burgers Equations

2020

Entropy

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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.

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“…And we will use the multiscale Lorenz 96 model [28] to provide an example of an ODE to which the averaging principle may be applied to effect dimension reduction, as pioneered and exploited in [14]. The work of Jiang and Harlim [21] studies data-informed model-driven prediction in partially observed ODEs, using ideas from kernel based approximation; and in the paper [20] the idea is generalized to discrete time dynamical systems, and neural networks and LSTM modeling is used in place of kernel methods. In both the papers [21,20] multiscale systems are used to test their methods in certain regimes.…”

mentioning

confidence: 99%

“…The work of Jiang and Harlim [21] studies data-informed model-driven prediction in partially observed ODEs, using ideas from kernel based approximation; and in the paper [20] the idea is generalized to discrete time dynamical systems, and neural networks and LSTM modeling is used in place of kernel methods. In both the papers [21,20] multiscale systems are used to test their methods in certain regimes.…”

mentioning

confidence: 99%

Kernel Analog Forecasting: Multiscale Test Problems

Burov¹,

Giannakis²,

Manohar³

et al. 2021

Multiscale Model. Simul.

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Data-driven prediction is becoming increasingly widespread as the volume of data available grows and as algorithmic development matches this growth. The nature of the predictions made, and the manner in which they should be interpreted, depends crucially on the extent to which the variables chosen for prediction are Markovian, or approximately Markovian. Multiscale systems provide a framework in which this issue can be analyzed. In this work kernel analog forecasting methods are studied from the perspective of data generated by multiscale dynamical systems. The problems chosen exhibit a variety of different Markovian closures, using both averaging and homogenization; furthermore, settings where scale-separation is not present and the predicted variables are non-Markovian, are also considered. The studies provide guidance for the interpretation of data-driven prediction methods when used in practice.

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Machine learning for prediction with missing dynamics

Cited by 61 publications

References 45 publications

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Data-Driven Model Reduction for Stochastic Burgers Equations

Kernel Analog Forecasting: Multiscale Test Problems

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