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

Linot, Alec J.; Graham, Michael D.

doi:10.48550/arxiv.2109.00060

Cited by 1 publication

(2 citation statements)

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“…In general the linear term is unknown, so the filter width can be viewed as an additional tuning parameter that is similar to selecting the stencil size for a finite difference approximation. This result agrees with what was found in [20]. Predictions from the stabilized neural ODEs appear in Figs.…”

Section: B Kuramoto-sivashinsky Equationsupporting

confidence: 92%

“…Neural ODEs have been successfully used for short-time prediction of Burgers equation [24], for the evolution of dissipation in decaying isotropic turbulence [28], and for flow around a cylinder [30]. Additionally, in [20] we showed that the long-time dynamics of the KSE could be captured using neural ODEs. However, this only works when the dimension of the problem is reduced.…”

Section: Introductionmentioning

confidence: 82%

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Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems

Linot¹,

Burby²,

Qi³

et al. 2022

Preprint

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In data-driven modeling of spatiotemporal phenomena careful consideration often needs to be made in capturing the dynamics of the high wavenumbers. This problem becomes especially challenging when the system of interest exhibits shocks or chaotic dynamics. We present a data-driven modeling method that accurately captures shocks and chaotic dynamics by proposing a novel architecture, stabilized neural ordinary differential equation (ODE). In our proposed architecture, we learn the right-hand-side (RHS) of an ODE by adding the outputs of two NN together where one learns a linear term and the other a nonlinear term. Specifically, we implement this by training a sparse linear convolutional NN to learn the linear term and a dense fully-connected nonlinear NN to learn the nonlinear term. This is in contrast with the standard neural ODE which involves training only a single NN for learning the RHS. We apply this setup to the viscous Burgers equation, which exhibits shocked behavior, and show better short-time tracking and prediction of the energy spectrum at high wavenumbers than a standard neural ODE. We also find that the stabilized neural ODE models are much more robust to noisy initial conditions than the standard neural ODE approach. We also apply this method to chaotic trajectories of the Kuramoto-Sivashinsky equation. In this case, stabilized neural ODEs keep long-time trajectories on the attractor, and are highly robust to noisy initial conditions, while standard neural ODEs fail at achieving either of these results. We conclude by demonstrating how stabilizing neural ODEs provide a natural extension for use in reduced-order modeling by projecting the dynamics onto the eigenvectors of the learned linear term.

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Section: B Kuramoto-sivashinsky Equationsupporting

confidence: 92%