Predicting chaotic time series with a partial model

Hamilton, Franz; Berry, Tyrus; Sauer, Tim

doi:10.1103/physreve.92.010902

Cited by 35 publications

(25 citation statements)

References 35 publications

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“…When a priori knowledge on the physical system is available, one of the standard approaches to reconstruct the nonlinear dynamics from an incomplete data set is to design a statistical model that incorporates the physical knowledge [15,16,17]. In [18], it is shown that a high-resolution temporal dynamics can be reconstructed from a low-resolution time series data, which has a 10 times lower time resolution, by using only a subset of the governing equations. Incorporating the physics knowledge on the training of ANN, [6] proposed "physics informed neural network", which can recover the complex nonlinear dynamics from only a small fraction of the data.It is challenging to reconstruct nonlinear dynamics without having any prior knowledge on the underlying process.…”

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confidence: 99%

Data-driven reconstruction of nonlinear dynamics from sparse observation

Yeo

2019

Journal of Computational Physics

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We present a data-driven model to reconstruct nonlinear dynamics from a very sparse times series data, which relies on the strength of the echo state network (ESN) in learning nonlinear representation of data. With an assumption of the universal function approximation capability of ESN, it is shown that the reconstruction problem can be formulated as a fixed-point problem, in which the trajectory of the dynamical system is a fixed point of the ESN. An under-relaxed fixed-point iteration is proposed to reconstruct the nonlinear dynamics from a sparse observation. The proposed fixed-point ESN is tested against both univariate and multivariate chaotic dynamical systems by randomly removing up to 95% of the data. It is shown that the fixed-point ESN is able to reconstruct the complex dynamics from only 5 ∼ 10% of the data. For a relatively simple non-chaotic dynamical system, the numerical experiments on a forced van der Pol oscillator show that it is possible to reconstruct the nonlinear dynamics from only 1∼2% of the data. particular, in [8,9], it is shown that ANN is capable of predicting the dynamics of a chaotic attractor. When training an ANN for the modeling of a dynamical system, it is typical to use a sufficiently long time series data, so that the ANN fully explores the phase space. Given the huge amount of training data, it is unclear whether the long-term forecast capability of ANN is due to a high dimensional interpolation in a phase space, similar to the delay-coordinate embedding [10,11], or because ANN really learns the nonlinear dynamics. As a first step to answer this question, we first consider the problem, where only a partial observation of a time series data is available. We investigate if ANN can reconstruct the nonlinear dynamics from the incomplete information.Reconstructing dynamics from an incomplete observation has practical applications in many physical, biological, and engineering problems, where the state of the system is accessible only through a sensor network. For example, in geophysics, it is not uncommon to find time series data, which contains a large amount of missing information or is irregularly sampled, due to sensor malfunction, atmospheric conditions, or physical limitations of the sensors [12,13,14]. When a priori knowledge on the physical system is available, one of the standard approaches to reconstruct the nonlinear dynamics from an incomplete data set is to design a statistical model that incorporates the physical knowledge [15,16,17]. In [18], it is shown that a high-resolution temporal dynamics can be reconstructed from a low-resolution time series data, which has a 10 times lower time resolution, by using only a subset of the governing equations. Incorporating the physics knowledge on the training of ANN, [6] proposed "physics informed neural network", which can recover the complex nonlinear dynamics from only a small fraction of the data.It is challenging to reconstruct nonlinear dynamics without having any prior knowledge on the underlying process. There are three ...

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confidence: 99%

Data-driven reconstruction of nonlinear dynamics from sparse observation

Yeo

2019

Journal of Computational Physics

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“…Nowadays, the Lorenz system is a popular test model for new methods in such fields as machine learning and forecasting (e.g. [13,30,18]).…”

Section: A Brief Overview Of Lorenz'63mentioning

confidence: 99%

Computing the quasipotential for highly dissipative and chaotic SDEs an application to stochastic Lorenz’63

Cameron¹,

Yang²

2019

Commun. Appl. Math. Comput. Sci.

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The study of noise-driven transitions occurring rarely on the time-scale of systems modeled by SDEs is of crucial importance for understanding such phenomena as genetic switches in living organisms and magnetization switches of the Earth. For a gradient SDE, the predictions for transition times and paths between its metastable states are done using the potential function. For a nongradient SDE, one needs to decompose its forcing into a gradient of the so-called quasipotential and a rotational component, which cannot be done analytically in general.We propose a methodology for computing the quasipotential for highly dissipative and chaotic systems built on the example of Lorenz'63 with an added stochastic term. It is based on the ordered line integral method, a Dijkstra-like quasipotential solver, and combines 3D computations in whole regions, a dimensional reduction technique, and 2D computations on radial meshes on manifolds or their unions. Our collection of source codes is available on M. Cameron's web page and on GitHub.

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“…For the nonlinear filtering with known transition functions, particle filters, or sequential Monte Carlo methods, provide a very powerful tool for the modeling of non-Gaussian distributions [13,14]. While most of these nonlinear filtering models require at least a partial knowledge of the dynamical system [15], in many problems, we do not have knowledge about the underlying physical processes, or the system is too complex to develop a model from the first principles [16].…”

Section: Introductionmentioning

confidence: 99%

Deep learning algorithm for data-driven simulation of noisy dynamical system

Yeo

Melnyk

2019

Journal of Computational Physics

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We present a deep learning model, DE-LSTM, for the simulation of a stochastic process with an underlying nonlinear dynamics. The deep learning model aims to approximate the probability density function of a stochastic process via numerical discretization and the underlying nonlinear dynamics is modeled by the Long Short-Term Memory (LSTM) network. It is shown that, when the numerical discretization is used, the function estimation problem can be solved by a multi-label classification problem. A penalized maximum log likelihood method is proposed to impose a smoothness condition in the prediction of the probability distribution. We show that the time evolution of the probability distribution can be computed by a high-dimensional integration of the transition probability of the LSTM internal states. A Monte Carlo algorithm to approximate the high-dimensional integration is outlined. The behavior of DE-LSTM is thoroughly investigated by using the Ornstein-Uhlenbeck process and noisy observations of nonlinear dynamical systems; Mackey-Glass time series and forced Van der Pol oscillator. It is shown that DE-LSTM makes a good prediction of the probability distribution without assuming any distributional properties of the stochastic process. For a multiple-step forecast of the Mackey-Glass time series, the prediction uncertainty, denoted by the 95% confidence interval, first grows, then dynamically adjusts following the evolution of the system, while in the simulation of the forced Van der Pol oscillator, the prediction uncertainty does not grow in time even for a 3,000-step forecast.

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Predicting chaotic time series with a partial model

Cited by 35 publications

References 35 publications

Data-driven reconstruction of nonlinear dynamics from sparse observation

Data-driven reconstruction of nonlinear dynamics from sparse observation

Computing the quasipotential for highly dissipative and chaotic SDEs an application to stochastic Lorenz’63

Deep learning algorithm for data-driven simulation of noisy dynamical system

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