54 research outputs found
On the lifting and reconstruction of nonlinear systems with multiple attractors
The Koopman operator provides a linear perspective on non-linear dynamics by
focusing on the evolution of observables in an invariant subspace. Observables
of interest are typically linearly reconstructed from the Koopman
eigenfunctions. Despite the broad use of Koopman operators over the past few
years, there exist some misconceptions about the applicability of Koopman
operators to dynamical systems with more than one fixed point. In this work, an
explanation is provided for the mechanism of lifting for the Koopman operator
of nonlinear systems with multiple attractors. Considering the example of the
Duffing oscillator, we show that by exploiting the inherent symmetry between
the basins of attraction, a linear reconstruction with three degrees of freedom
in the Koopman observable space is sufficient to globally linearize the system.Comment: 8 page
Characterizing and Improving Predictive Accuracy in Shock-Turbulent Boundary Layer Interactions Using Data-driven Models
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143030/1/6.2017-0314.pd
Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data
High-dimensional spatio-temporal dynamics can often be encoded in a
low-dimensional subspace. Engineering applications for modeling,
characterization, design, and control of such large-scale systems often rely on
dimensionality reduction to make solutions computationally tractable in
real-time. Common existing paradigms for dimensionality reduction include
linear methods, such as the singular value decomposition (SVD), and nonlinear
methods, such as variants of convolutional autoencoders (CAE). However, these
encoding techniques lack the ability to efficiently represent the complexity
associated with spatio-temporal data, which often requires variable geometry,
non-uniform grid resolution, adaptive meshing, and/or parametric dependencies.
To resolve these practical engineering challenges, we propose a general
framework called Neural Implicit Flow (NIF) that enables a mesh-agnostic,
low-rank representation of large-scale, parametric, spatial-temporal data. NIF
consists of two modified multilayer perceptrons (MLPs): (i) ShapeNet, which
isolates and represents the spatial complexity, and (ii) ParameterNet, which
accounts for any other input complexity, including parametric dependencies,
time, and sensor measurements. We demonstrate the utility of NIF for parametric
surrogate modeling, enabling the interpretable representation and compression
of complex spatio-temporal dynamics, efficient many-spatial-query tasks, and
improved generalization performance for sparse reconstruction.Comment: 56 page
Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics
Recently developed physics-informed neural network (PINN) has achieved
success in many science and engineering disciplines by encoding physics laws
into the loss functions of the neural network, such that the network not only
conforms to the measurements, initial and boundary conditions but also
satisfies the governing equations. This work first investigates the performance
of PINN in solving stiff chemical kinetic problems with governing equations of
stiff ordinary differential equations (ODEs). The results elucidate the
challenges of utilizing PINN in stiff ODE systems. Consequently, we employ
Quasi-Steady-State-Assumptions (QSSA) to reduce the stiffness of the ODE
systems, and the PINN then can be successfully applied to the converted
non/mild-stiff systems. Therefore, the results suggest that stiffness could be
the major reason for the failure of the regular PINN in the studied stiff
chemical kinetic systems. The developed Stiff-PINN approach that utilizes QSSA
to enable PINN to solve stiff chemical kinetics shall open the possibility of
applying PINN to various reaction-diffusion systems involving stiff dynamics
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