Data-driven state-space and Koopman operator models of coherent state dynamics on invariant manifolds

Abstract

The accurate simulation of complex dynamics in fluid flows demands a substantial number of degrees of freedom, i.e. a high-dimensional state space. Nevertheless, the swift attenuation of small-scale perturbations due to viscous diffusion permits in principle the representation of these flows using a significantly reduced dimensionality. Over time, the dynamics of such flows evolve towards a finite-dimensional invariant manifold. Using only data from direct numerical simulations, in the present work we identify the manifold and determine evolution equations for the dynamics on it. We use an advanced autoencoder framework to automatically estimate the intrinsic dimension of the manifold and provide an orthogonal coordinate system. Then, we learn the dynamics by determining an equation on the manifold by using both a function space approach (approximating the Koopman operator) and a state space approach (approximating the vector field on the manifold). We apply this method to exact coherent states for Kolmogorov flow and minimal flow unit pipe flow. Fully resolved simulations for these cases require O(103) and O(105) degrees of freedom respectively, and we build models with two or three degrees of freedom that faithfully capture the dynamics of these flows. For these examples, both the state space and function space time evaluations provide highly accurate predictions of the long-time dynamics in manifold coordinates.

arXiv comment10 pages

References in corpus (20)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Deep learning for universal linear embeddings of nonlinear dynamics
• Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
• The Significance of Simple Invariant Solutions in Turbulent Flows
• Study of dynamics in post-transient flows using Koopman mode decomposition
• Data-driven discovery of intrinsic dynamics
• Relative periodic orbits form the backbone of turbulent pipe flow
• Deep learning to discover and predict dynamics on an inertial manifold
• Revealing the state space of turbulent pipe flow by symmetry reduction
• The Openpipeflow Navier--Stokes Solver
• Residual Dynamic Mode Decomposition: Robust and verified Koopmanism
• Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems
• Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations
• Revealing the state space of turbulence using machine learning
• Searching turbulence for periodic orbits with dynamic mode decomposition
• Nonlinear traveling waves as a framework for understanding turbulent drag reduction
• Data-driven low-dimensional dynamic model of Kolmogorov flow
• Spectral Analysis of the Koopman Operator for Partial Differential Equations
• Periodic orbit analysis of a system with continuous symmetry - a tutorial

Cited by in corpus (3)

• Dynamics of a Data-Driven Low-Dimensional Model of Turbulent Minimal Pipe Flow
• Elastoinertial turbulence: Data-driven reduced-order model based on manifold dynamics
• Compact representation of transonic airfoil buffet flows with observable-augmented machine learning