scholarly journals Data-driven discovery of coordinates and governing equations

2019 ◽  
Vol 116 (45) ◽  
pp. 22445-22451 ◽  
Author(s):  
Kathleen Champion ◽  
Bethany Lusch ◽  
J. Nathan Kutz ◽  
Steven L. Brunton
Keyword(s):  
Coordinate System ◽  
Nonlinear Dynamical System ◽  
Scientific Data ◽  
Model Complexity ◽  
Governing Equations ◽  
Modeling Framework ◽  
Nonlinear Dynamical ◽  
Parsimonious Models ◽  
Reduced Space ◽  
Low Dimensional

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 deep autoencoder network 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 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. This method places the discovery of coordinates and models on an equal footing.

scholarly journals Discovering governing equations from data by sparse identification of nonlinear dynamical systems

2016 ◽  
Vol 113 (15) ◽  
pp. 3932-3937 ◽  
Author(s):  
Steven L. Brunton ◽  
Joshua L. Proctor ◽  
J. Nathan Kutz
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Measurement Data ◽  
Climate Science ◽  
Model Complexity ◽  
Governing Equations ◽  
Canonical Systems ◽  
Nonlinear Dynamical ◽  
Balance Accuracy ◽  
Wide Range

Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.

scholarly journals Deterministic Chaos in the Microstructure of Radio Pulses of PSR 0809+74

1992 ◽  
Vol 128 ◽  
pp. 329-331
Author(s):  
V. I. Zhuravlev ◽  
M. V. Popov
Keyword(s):  
Radio Emission ◽  
Random Sequence ◽  
Nonlinear Dynamical System ◽  
Deterministic Chaos ◽  
Relativistic Plasma ◽  
Emission Region ◽  
Polar Cap ◽  
Nonlinear Dynamical ◽  
Developed Turbulence ◽  
Low Dimensional

Abstract480 single pulses from PSR 0809+74, recorded at 102.5 MHz with a time resolution of 10μs, have been analyzed by the time delay technique in order to look for the parameters of deterministic chaos in the microstructure of radio pulses. The correlation dimension n was shown to be less than 5 in more than 20% of the analyzed pulses. This means that on such occasions the microstructure of pulsar radio emission with the time scales 10 to 100μs may be determined by the behavior of a nonlinear dynamical system with comparatively small numbers of independent parameters. For example, the observed low dimensional chaos may result from a turbulence process associated with the outflow of plasma—as in versions of the polar cap model where microstructure can be interpreted as reflecting the spatial structure of relativistic plasma outflow in the radio emission region.However, the correlation-time distribution demonstrates the tendency for microstructure to consist of a random sequence of unresolved micropulses in the majority of cases, which means in the framework of polar cap models the presence of well developed turbulence in the relativistic plasma outflow.

scholarly journals Analytical CPG model driven by single-limb velocity input generates accurate temporal locomotor dynamics

2018 ◽  
Author(s):  
Sergiy Yakovenko ◽  
Anton Sobinov ◽  
Valeriya Gritsenko
Keyword(s):  
Full Range ◽  
Nonlinear Dynamical System ◽  
Underlying Mechanism ◽  
Neural Pathways ◽  
Nonlinear Dynamical ◽  
Model Driven ◽  
Body Velocity ◽  
Leaky Integration ◽  
Low Dimensional ◽  
Walking Speeds

The ability of vertebrates to generate rhythm within their spinal neural networks is essential for walking, running, and other rhythmic behaviors. The central pattern generator (CPG) network responsible for these behaviors is well-characterized with experimental and theoretical studies, and it can be formulated as a nonlinear dynamical system. The underlying mechanism responsible for locomotor behavior can be expressed as the process of leaky integration with resetting states generating appropriate phases for changing body velocity. The low-dimensional input to the CPG model generates the bilateral pattern of swing and stance modulation for each limb and is consistent with the desired limb speed as the input command. To test the minimal configuration of required parameters for this model, we reduced the system of equations representing CPG for a single limb and provided the analytical solution with two complementary methods. The analytical and empirical cycle durations were similar (R2=0.99) for the full range of walking speeds. The structure of solution is consistent with the use of limb speed as the input domain for the CPG network. Moreover, the reciprocal interaction between two leaky integration processes representing a CPG for two limbs was sufficient to capture fundamental experimental dynamics associated with the control of heading direction. This analysis provides further support for the embedded velocity or limb speed representation within spinal neural pathways involved in rhythm generation.

CHAOTIC ATTRACTOR IN THE KURAMOTO MODEL

2005 ◽  
Vol 15 (11) ◽  
pp. 3457-3466 ◽  
Author(s):  
YURI L. MAISTRENKO ◽  
OLEKSANDR V. POPOVYCH ◽  
PETER A. TASS
Keyword(s):  
Chaotic Attractor ◽  
Nonlinear Dynamical System ◽  
Kuramoto Model ◽  
Phase Oscillators ◽  
Bifurcation And Chaos ◽  
Nonlinear Dynamical ◽  
Transition To Chaos ◽  
Quasiperiodic Dynamics ◽  
The Kuramoto Model ◽  
Low Dimensional

The Kuramoto model of globally coupled phase oscillators is an essentially nonlinear dynamical system with a rich dynamics including synchronization and chaos. We study the Kuramoto model from the standpoint of bifurcation and chaos theory of low-dimensional dynamical systems. We find a chaotic attractor in the four-dimensional Kuramoto model and study its origin. The torus destruction scenario is one of the major mechanisms by which chaos arises. L. P. Shilnikov has made decisive contributions to its discovery. We show also that in the Kuramoto model the transition to chaos is in accordance with the torus destruction scenario. We present the general bifurcation diagram containing phase chaos, Cherry flow as well as periodic and quasiperiodic dynamics.

scholarly journals Analytical CPG model driven by single-limb velocity input generates accurate temporal locomotor dynamics

2018 ◽  
Author(s):  
Sergiy Yakovenko ◽  
Anton Sobinov ◽  
Valeriya Gritsenko
Keyword(s):  
Full Range ◽  
Nonlinear Dynamical System ◽  
Underlying Mechanism ◽  
Neural Pathways ◽  
Nonlinear Dynamical ◽  
Model Driven ◽  
Body Velocity ◽  
Leaky Integration ◽  
Low Dimensional ◽  
Walking Speeds

The ability of vertebrates to generate rhythm within their spinal neural networks is essential for walking, running, and other rhythmic behaviors. The central pattern generator (CPG) network responsible for these behaviors is well-characterized with experimental and theoretical studies, and it can be formulated as a nonlinear dynamical system. The underlying mechanism responsible for locomotor behavior can be expressed as the process of leaky integration with resetting states generating appropriate phases for changing body velocity. The low-dimensional input to the CPG model generates the bilateral pattern of swing and stance modulation for each limb and is consistent with the desired limb speed as the input command. To test the minimal configuration of required parameters for this model, we reduced the system of equations representing CPG for a single limb and provided the analytical solution with two complementary methods. The analytical and empirical cycle durations were similar (R2=0.99) for the full range of walking speeds. The structure of solution is consistent with the use of limb speed as the input domain for the CPG network. Moreover, the reciprocal interaction between two leaky integration processes was sufficient to capture fundamental experimental dynamics. This analysis provides further support for the embedded velocity or limb speed representation within spinal neural pathways involved in rhythm generation.

scholarly journals Analytical CPG model driven by limb velocity input generates accurate temporal locomotor dynamics

PeerJ ◽  
2018 ◽  
Vol 6 ◽  
pp. e5849
Author(s):  
Sergiy Yakovenko ◽  
Anton Sobinov ◽  
Valeriya Gritsenko
Keyword(s):  
Full Range ◽  
Nonlinear Dynamical System ◽  
Underlying Mechanism ◽  
Neural Pathways ◽  
Nonlinear Dynamical ◽  
Model Driven ◽  
Body Velocity ◽  
Leaky Integration ◽  
Low Dimensional ◽  
Walking Speeds

The ability of vertebrates to generate rhythm within their spinal neural networks is essential for walking, running, and other rhythmic behaviors. The central pattern generator (CPG) network responsible for these behaviors is well-characterized with experimental and theoretical studies, and it can be formulated as a nonlinear dynamical system. The underlying mechanism responsible for locomotor behavior can be expressed as the process of leaky integration with resetting states generating appropriate phases for changing body velocity. The low-dimensional input to the CPG model generates the bilateral pattern of swing and stance modulation for each limb and is consistent with the desired limb speed as the input command. To test the minimal configuration of required parameters for this model, we reduced the system of equations representing CPG for a single limb and provided the analytical solution with two complementary methods. The analytical and empirical cycle durations were similar (R2 = 0.99) for the full range of walking speeds. The structure of solution is consistent with the use of limb speed as the input domain for the CPG network. Moreover, the reciprocal interaction between two leaky integration processes representing a CPG for two limbs was sufficient to capture fundamental experimental dynamics associated with the control of heading direction. This analysis provides further support for the embedded velocity or limb speed representation within spinal neural pathways involved in rhythm generation.

scholarly journals Analytical CPG model driven by single-limb velocity input generates accurate temporal locomotor dynamics

2018 ◽  
Author(s):  
Sergiy Yakovenko ◽  
Anton Sobinov ◽  
Valeriya Gritsenko
Keyword(s):  
Full Range ◽  
Nonlinear Dynamical System ◽  
Underlying Mechanism ◽  
Neural Pathways ◽  
Nonlinear Dynamical ◽  
Model Driven ◽  
Body Velocity ◽  
Leaky Integration ◽  
Low Dimensional ◽  
Walking Speeds

The ability of vertebrates to generate rhythm within their spinal neural networks is essential for walking, running, and other rhythmic behaviors. The central pattern generator (CPG) network responsible for these behaviors is well-characterized with experimental and theoretical studies, and it can be formulated as a nonlinear dynamical system. The underlying mechanism responsible for locomotor behavior can be expressed as the process of leaky integration with resetting states generating appropriate phases for changing body velocity. The low-dimensional input to the CPG model generates the bilateral pattern of swing and stance modulation for each limb and is consistent with the desired limb speed as the input command. To test the minimal configuration of required parameters for this model, we reduced the system of equations representing CPG for a single limb and provided the analytical solution with two complementary methods. The analytical and empirical cycle durations were similar (R2=0.99) for the full range of walking speeds. The structure of solution is consistent with the use of limb speed as the input domain for the CPG network. Moreover, the reciprocal interaction between two leaky integration processes representing a CPG for two limbs was sufficient to capture fundamental experimental dynamics associated with the control of heading direction. This analysis provides further support for the embedded velocity or limb speed representation within spinal neural pathways involved in rhythm generation.

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