Lyapunov exponents of the Kuramoto--Sivashinsky PDE

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

doi:10.21914/anziamj.v61i0.13939

Cited by 3 publications

(2 citation statements)

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“…with periodic boundary conditions. For system length L = 22 the Kuramoto-Sivashinsky equation is chaotic with a maximal Lyapunov exponent of λ max = 0.043 (Edson et al, 2019). We remark that although the Kuramoto-Sivashinsky equation is a partial-differential equation, its dynamics evolves on a finite dimensional manifold (Temam, 1997).…”

Section: Partial Differential Equations: Kuramoto-sivashinsky Equationmentioning

confidence: 83%

Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation

Gottwald,

Reich

2020

Preprint

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Data-driven prediction and physics-agnostic machine-learning methods have attracted increased interest in recent years achieving forecast horizons going well beyond those to be expected for chaotic dynamical systems. In a separate strand of research data-assimilation has been successfully used to optimally combine forecast models and their inherent uncertainty with incoming noisy observations. The key idea in our work here is to achieve increased forecast capabilities by judiciously combining machine-learning algorithms and data assimilation. We combine the physics-agnostic data-driven approach of random feature maps as a forecast model within an ensemble Kalman filter data assimilation procedure. The machine-learning model is learned sequentially by incorporating incoming noisy observations. We show that the obtained forecast model has remarkably good forecast skill while being computationally cheap once trained. Going beyond the task of forecasting, we show that our method can be used to generate reliable ensembles for probabilistic forecasting as well as to learn effective model closure in multi-scale systems.

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Section: Partial Differential Equations: Kuramoto-sivashinsky Equationmentioning

confidence: 83%

Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation

Gottwald,

Reich

2020

Preprint

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“…(Left) Comparison of the power spectral density (PSD) of the ground-truth solution and the STENCIL-NET prediction on a coarser grid. (Right) Growth of the distance between nearby trajectories in the STENCIL-NET prediction, characterizing the maximum Lyaponuv exponent (slope of the dashed red line), compared to the ground truth value 38 . …”

Section: Numerical Experimentsmentioning

confidence: 99%

STENCIL-NET for equation-free forecasting from data

Maddu,

Sturm,

Cheeseman

et al. 2023

Sci Rep

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We present an artificial neural network architecture, termed STENCIL-NET, for equation-free forecasting of spatiotemporal dynamics from data. STENCIL-NET works by learning a discrete propagator that is able to reproduce the spatiotemporal dynamics of the training data. This data-driven propagator can then be used to forecast or extrapolate dynamics without needing to know a governing equation. STENCIL-NET does not learn a governing equation, nor an approximation to the data themselves. It instead learns a discrete propagator that reproduces the data. It therefore generalizes well to different dynamics and different grid resolutions. By analogy with classic numerical methods, we show that the discrete forecasting operators learned by STENCIL-NET are numerically stable and accurate for data represented on regular Cartesian grids. A once-trained STENCIL-NET model can be used for equation-free forecasting on larger spatial domains and for longer times than it was trained for, as an autonomous predictor of chaotic dynamics, as a coarse-graining method, and as a data-adaptive de-noising method, as we illustrate in numerical experiments. In all tests, STENCIL-NET generalizes better and is computationally more efficient, both in training and inference, than neural network architectures based on local (CNN) or global (FNO) nonlinear convolutions.

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