Lyapunov Exponents of the Kuramoto–sivashinsky Pde

Edson, Russell A.; Bunder, J. E.; Mattner, Trent W.; Roberts, Anthony John

doi:10.1017/s1446181119000105

Cited by 23 publications

(11 citation statements)

References 35 publications

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“…The relevant timescale for each of these system is the Lyapunov timescale (inverse of the largest Lyapunov exponent), which we previously found to be be very close to the integral time for KSE data [25]. For these domain sizes, the Lyapunov times are τ L = 22.2, 12.3, and 11.6, respectively [7,9]. We use 10 5 time units of data for training at each domain size, and vary how frequently we sample this dataset in time.…”

Section: Frameworksupporting

confidence: 60%

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

Linot,

Graham

2021

Preprint

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Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced order modeling method that capitalizes on this fact by finding the coordinates of this manifold and finding an ordinary differential equation (ODE) describing the dynamics in this coordinate system. The manifold coordinates are discovered using an undercomplete autoencoder -a neural network (NN) that reduces then expands dimension. Then the ODE, in these coordinates, is approximated by a NN using the neural ODE framework. Both of these methods only require snapshots of data to learn a model, and the data can be widely and/or unevenly spaced. We apply this framework to the Kuramoto-Sivashinsky for different domain sizes that exhibit chaotic dynamics. With this system, we find that dimension reduction improves performance relative to predictions in the ambient space, where artifacts arise. Then, with the low-dimensional model, we vary the training data spacing and find excellent short-and long-time statistical recreation of the true dynamics for widely spaced data (spacing of ∼0.7 Lyapunov times). We end by comparing performance with various degrees of dimension reduction, and find a "sweet spot" in terms of performance vs. dimension.

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Section: Frameworksupporting

confidence: 60%

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

Linot,

Graham

2021

Preprint

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“…x u is hyper-diffussion (thus stabilizing). This combination leads to small-scale dissipations and large-scale instabilities, transferring energy from the large to small scales [55,56]. The domain is periodic, u (0, t) = u (L, t), where L, the domain length, determines the dimension of the attractor [57]; larger L is expected to increase the complexity of the model error discovery.…”

Section: Test Case: Ks Equationmentioning

confidence: 99%

Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case

Mojgani,

Chattopadhyay,

Hassanzadeh

2021

Preprint

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Models used for many important engineering and natural systems are imperfect. The discrepancy between the mathematical representations of a true physical system and its imperfect model is called the model error. These model errors can lead to substantial difference between the numerical solutions of the model and the observations of the system, particularly in those involving nonlinear, multi-scale phenomena. Thus, there is substantial interest in reducing model errors, particularly through understanding their physics and sources and leveraging the rapid growth of observational data. Here we introduce a framework named MEDIDA: Model Error Discovery with Interpretability and Data Assimilation. MEDIDA only requires a working numerical solver of the model and a small number of noise-free or noisy sporadic observations of the system. In MEDIDA, first the model error is estimated from differences between the observed states and model-predicted states (the latter are obtained from a number of one-time-step numerical integrations from the previous observed states). If observations are noisy, a data assimilation (DA) technique such as ensemble Kalman filter (EnKF) is first used to provide a noise-free analysis state of the system, which is then used in estimating the model error. Finally, an equation-discovery technique, such as the relevance vector machine (RVM), a sparsity-promoting Bayesian method, is used to identify an interpretable, parsimonious, closedform representation of the model error. Using the chaotic Kuramoto-Sivashinsky (KS) system as the test case, we demonstrate the excellent performance of MEDIDA in discovering different types of structural/parametric model errors, representing different types of missing physics, using noise-free and noisy observations.

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“…We impose periodic boundary conditions on the spatial domain [0, 2πL). The chaos of the system depends on both λ and L. Higher values of L or λ increase the exhibited chaos [43,26,16,13]. We choose L = 16, λ = 1 for all simulations as this represents a sufficiently chaotic state [28].…”

Section: Numerical Considerationsmentioning

confidence: 99%

“…Here we examine the efficacy of this algorithm for the Kuramoto-Sivashinsky equation (KSE) numerically. KSE is selected because it presents chaotic spatial-temporal dynamics at a relatively low computational cost (see [43,13,8,16] for example studies of the complicated dynamics that arise in KSE). In addition, KSE naturally provides the context for introducing several artificial parameters whose estimation is of significant import.…”

Section: Introductionmentioning

confidence: 99%

Concurrent multi-parameter learning demonstrated on the Kuramoto-Sivashinsky equation

Pachev¹,

Whitehead²,

McQuarrie³

2021

Preprint

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We develop an algorithm for the concurrent (on-the-fly) estimation of parameters for a system of evolutionary dissipative partial differential equations in which the state is partially observed. The intuitive nature of the algorithm makes its extension to several different systems immediate, and it allows for recovery of multiple parameters simultaneously. We test this algorithm on the Kuramoto-Sivashinsky equation in one dimension and demonstrate its efficacy in this context.

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Lyapunov Exponents of the Kuramoto–sivashinsky Pde

Cited by 23 publications

References 35 publications

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case

Concurrent multi-parameter learning demonstrated on the Kuramoto-Sivashinsky equation

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