21,094 research outputs found
Controlling nonlinear dynamical systems into arbitrary states using machine learning
Controlling nonlinear dynamical systems is a central task in many different areas of science and
engineering. Chaotic systems can be stabilized (or chaotified) with small perturbations, yet existing
approaches either require knowledge about the underlying system equations or large data sets as they
rely on phase space methods. In this work we propose a novel and fully data driven scheme relying on
machine learning (ML), which generalizes control techniques of chaotic systems without requiring a
mathematical model for its dynamics. Exploiting recently developed ML-based prediction capabilities,
we demonstrate that nonlinear systems can be forced to stay in arbitrary dynamical target states
coming from any initial state. We outline and validate our approach using the examples of the Lorenz
and the Rössler system and show how these systems can very accurately be brought not only to
periodic, but even to intermittent and different chaotic behavior. Having this highly flexible control
scheme with little demands on the amount of required data on hand, we briefly discuss possible
applications ranging from engineering to medicine
State-Space Inference and Learning with Gaussian Processes
State-space inference and learning with Gaussian processes (GPs) is an unsolved problem. We propose a new, general methodology for inference and learning in nonlinear state-space models that are described probabilistically by non-parametric GP models. We apply the expectation maximization algorithm to iterate between inference in the latent state-space and learning the parameters of the underlying GP dynamics model. Copyright 2010 by the authors
Data-driven discovery of coordinates and governing equations
The discovery of governing equations from scientific data has the potential
to transform data-rich fields that lack well-characterized quantitative
descriptions. Advances in sparse regression are currently enabling the
tractable identification of both the structure and parameters of a nonlinear
dynamical system from data. The resulting models have the fewest terms
necessary to describe the dynamics, balancing model complexity with descriptive
ability, and thus promoting interpretability and generalizability. This
provides an algorithmic approach to Occam's razor for model discovery. However,
this approach fundamentally relies on an effective coordinate system in which
the dynamics have a simple representation. In this work, we design a custom
autoencoder to discover a coordinate transformation into a reduced space where
the dynamics may be sparsely represented. Thus, we simultaneously learn the
governing equations and the associated coordinate system. We demonstrate this
approach on several example high-dimensional dynamical systems with
low-dimensional behavior. The resulting modeling framework combines the
strengths of deep neural networks for flexible representation and sparse
identification of nonlinear dynamics (SINDy) for parsimonious models. It is the
first method of its kind to place the discovery of coordinates and models on an
equal footing.Comment: 25 pages, 6 figures; added acknowledgment
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