1,400 research outputs found
Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era
This study presents a collection of purely data-driven workflows for
constructing reduced-order models (ROMs) for distributed dynamical systems. The
ROMs we focus on, are data-assisted models inspired by, and templated upon, the
theory of Approximate Inertial Manifolds (AIMs); the particular motivation is
the so-called post-processing Galerkin method of Garcia-Archilla, Novo and
Titi. Its applicability can be extended: the need for accurate truncated
Galerkin projections and for deriving closed-formed corrections can be
circumvented using machine learning tools. When the right latent variables are
not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a
manifold learning scheme) can be used to discover good sets of latent variables
and test their explainability. The proposed methodology can express the ROMs in
terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD
modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both
Black-Box and (theoretically-informed and data-corrected) Gray-Box models are
described; the necessity for the latter arises when truncated Galerkin
projections are so inaccurate as to not be amenable to post-processing. We use
the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative
partial differential equations to illustrate and successfully test the overall
framework.Comment: 27 pages, 22 figure
Data-Driven Modeling and Forecasting of Chaotic Dynamics on Inertial Manifolds Constructed as Spectral Submanifolds
We present a data-driven and interpretable approach for reducing the
dimensionality of chaotic systems using spectral submanifolds (SSMs). Emanating
from fixed points or periodic orbits, these SSMs are low-dimensional inertial
manifolds containing the chaotic attractor of the underlying high-dimensional
system. The reduced dynamics on the SSMs turn out to predict chaotic dynamics
accurately over a few Lyapunov times and also reproduce long-term statistical
features, such as the largest Lyapunov exponents and probability distributions,
of the chaotic attractor. We illustrate this methodology on numerical data sets
including a delay-embedded Lorenz attractor, a nine-dimensional Lorenz model,
and a Duffing oscillator chain. We also demonstrate the predictive power of our
approach by constructing an SSM-reduced model from unforced trajectories of a
buckling beam, and then predicting its periodically forced chaotic response
without using data from the forced beam.Comment: Submitted to Chao
Capturing the Edge of Chaos as a Spectral Submanifold in Pipe Flows
An extended turbulent state can coexist with the stable laminar state in pipe
flows. We focus here on short pipes with additional discrete symmetries
imposed. In this case, the boundary between the coexisting basins of
attraction, often called the edge of chaos, is the stable manifold of an edge
state, which is a lower-branch traveling wave solution. We show that a
low-dimensional submanifold of the edge of chaos can be constructed from
velocity data using the recently developed theory of spectral submanifolds
(SSMs). These manifolds are the unique smoothest nonlinear continuations of
nonresonant spectral subspaces of the linearized system at stationary states.
Using very low dimensional SSM-based reduced-order models, we predict
transitions to turbulence or laminarization for velocity fields near the edge
of chaos.Comment: Accepted for publication in the Journal of Fluid Mechanic
Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems
While many phenomena in physics and engineering are formally
high-dimensional, their long-time dynamics often live on a lower-dimensional
manifold. The present work introduces an autoencoder framework that combines
implicit regularization with internal linear layers and regularization
(weight decay) to automatically estimate the underlying dimensionality of a
data set, produce an orthogonal manifold coordinate system, and provide the
mapping functions between the ambient space and manifold space, allowing for
out-of-sample projections. We validate our framework's ability to estimate the
manifold dimension for a series of datasets from dynamical systems of varying
complexities and compare to other state-of-the-art estimators. We analyze the
training dynamics of the network to glean insight into the mechanism of
low-rank learning and find that collectively each of the implicit regularizing
layers compound the low-rank representation and even self-correct during
training. Analysis of gradient descent dynamics for this architecture in the
linear case reveals the role of the internal linear layers in leading to faster
decay of a "collective weight variable" incorporating all layers, and the role
of weight decay in breaking degeneracies and thus driving convergence along
directions in which no decay would occur in its absence. We show that this
framework can be naturally extended for applications of state-space modeling
and forecasting by generating a data-driven dynamic model of a spatiotemporally
chaotic partial differential equation using only the manifold coordinates.
Finally, we demonstrate that our framework is robust to hyperparameter choices
Nonlinear proper orthogonal decomposition for convection-dominated flows
Autoencoder techniques find increasingly common use in reduced order modeling as a means to create a latent space. This reduced order representation offers a modular data-driven modeling approach for nonlinear dynamical systems when integrated with a time series predictive model. In this Letter, we put forth a nonlinear proper orthogonal decomposition (POD) framework, which is an end-to-end Galerkin-free model combining autoencoders with long short-term memory networks for dynamics. By eliminating the projection error due to the truncation of Galerkin models, a key enabler of the proposed nonintrusive approach is the kinematic construction of a nonlinear mapping between the full-rank expansion of the POD coefficients and the latent space where the dynamics evolve. We test our framework for model reduction of a convection-dominated system, which is generally challenging for reduced order models. Our approach not only improves the accuracy, but also significantly reduces the computational cost of training and testing.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research under Award Number DE-SC0019290. O.S. gratefully acknowledges the Early Career Research Program (ECRP) support of the U.S. Department of Energy. O.S. also gratefully acknowledges the financial support of the National Science Foundation under Award No. DMS-2012255. T.I. acknowledges support through National Science Foundation Grant No. DMS-2012253.acceptedVersio
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