137 research outputs found
Carleman Linearization of Nonlinear Systems and Its Finite-Section Approximations
The Carleman linearization is one of the mainstream approaches to lift a
finite-dimensional nonlinear dynamical system into an infinite-dimensional
linear system with the promise of providing accurate approximations of the
original nonlinear system over larger regions around the equilibrium for longer
time horizons with respect to the conventional first-order linearization
approach. Finite-section approximations of the lifted system has been widely
used to study dynamical and control properties of the original nonlinear
system. In this context, some of the outstanding problems are to determine
under what conditions, as the finite-section order (i.e., truncation length)
increases, the trajectory of the resulting approximate linear system from the
finite-section scheme converges to that of the original nonlinear system and
whether the time interval over which the convergence happens can be quantified
explicitly. In this paper, we provide explicit error bounds for the
finite-section approximation and prove that the convergence is indeed
exponential with respect to the finite-section order. For a class of nonlinear
systems, it is shown that one can achieve exponential convergence over the
entire time horizon up to infinity. Our results are practically plausible as
our proposed error bound estimates can be used to compute proper truncation
lengths for a given application, e.g., determining proper sampling period for
model predictive control and reachability analysis for safety verifications. We
validate our theoretical findings through several illustrative simulations.Comment: 25 Pages, 10 figure
Koopman Spectral Linearization vs. Carleman Linearization: A Computational Comparison Study
Nonlinearity presents a significant challenge in problems involving dynamical
systems, prompting the exploration of various linearization techniques,
including the well-known Carleman Linearization. In this paper, we introduce
the Koopman Spectral Linearization method tailored for nonlinear autonomous
dynamical systems. This innovative linearization approach harnesses the
Chebyshev differentiation matrix and the Koopman Operator to yield a lifted
linear system. It holds the promise of serving as an alternative approach that
can be employed in scenarios where Carleman linearization is traditionally
applied. Numerical experiments demonstrate the effectiveness of this
linearization approach for several commonly used nonlinear dynamical systems.Comment: 17 pages, 7 figure
Potential quantum advantage for simulation of fluid dynamics
Numerical simulation of turbulent fluid dynamics needs to either parameterize
turbulence-which introduces large uncertainties-or explicitly resolve the
smallest scales-which is prohibitively expensive. Here we provide evidence
through analytic bounds and numerical studies that a potential quantum
exponential speedup can be achieved to simulate the Navier-Stokes equations
governing turbulence using quantum computing. Specifically, we provide a
formulation of the lattice Boltzmann equation for which we give evidence that
low-order Carleman linearization is much more accurate than previously believed
for these systems and that for computationally interesting examples. This is
achieved via a combination of reformulating the nonlinearity and accurately
linearizing the dynamical equations, effectively trading nonlinearity for
additional degrees of freedom that add negligible expense in the quantum
solver. Based on this we apply a quantum algorithm for simulating the
Carleman-linerized lattice Boltzmann equation and provide evidence that its
cost scales logarithmically with system size, compared to polynomial scaling in
the best known classical algorithms. This work suggests that an exponential
quantum advantage may exist for simulating fluid dynamics, paving the way for
simulating nonlinear multiscale transport phenomena in a wide range of
disciplines using quantum computing
A Class of Logistic Functions for Approximating State-Inclusive Koopman Operators
An outstanding challenge in nonlinear systems theory is identification or
learning of a given nonlinear system's Koopman operator directly from data or
models. Advances in extended dynamic mode decomposition approaches and machine
learning methods have enabled data-driven discovery of Koopman operators, for
both continuous and discrete-time systems. Since Koopman operators are often
infinite-dimensional, they are approximated in practice using
finite-dimensional systems. The fidelity and convergence of a given
finite-dimensional Koopman approximation is a subject of ongoing research. In
this paper we introduce a class of Koopman observable functions that confer an
approximate closure property on their corresponding finite-dimensional
approximations of the Koopman operator. We derive error bounds for the fidelity
of this class of observable functions, as well as identify two key learning
parameters which can be used to tune performance. We illustrate our approach on
two classical nonlinear system models: the Van Der Pol oscillator and the
bistable toggle switch.Comment: 8 page
Quantum algorithms for linear and non-linear fractional reaction-diffusion equations
High-dimensional fractional reaction-diffusion equations have numerous
applications in the fields of biology, chemistry, and physics, and exhibit a
range of rich phenomena. While classical algorithms have an exponential
complexity in the spatial dimension, a quantum computer can produce a quantum
state that encodes the solution with only polynomial complexity, provided that
suitable input access is available. In this work, we investigate efficient
quantum algorithms for linear and nonlinear fractional reaction-diffusion
equations with periodic boundary conditions. For linear equations, we analyze
and compare the complexity of various methods, including the second-order
Trotter formula, time-marching method, and truncated Dyson series method. We
also present a novel algorithm that combines the linear combination of
Hamiltonian simulation technique with the interaction picture formalism,
resulting in optimal scaling in the spatial dimension. For nonlinear equations,
we employ the Carleman linearization method and propose a block-encoding
version that is appropriate for the dense matrices that arise from the spatial
discretization of fractional reaction-diffusion equations
Towards provably efficient quantum algorithms for large-scale machine-learning models
Large machine learning models are revolutionary technologies of artificial
intelligence whose bottlenecks include huge computational expenses, power, and
time used both in the pre-training and fine-tuning process. In this work, we
show that fault-tolerant quantum computing could possibly provide provably
efficient resolutions for generic (stochastic) gradient descent algorithms,
scaling as , where is the size
of the models and is the number of iterations in the training, as long as
the models are both sufficiently dissipative and sparse, with small learning
rates. Based on earlier efficient quantum algorithms for dissipative
differential equations, we find and prove that similar algorithms work for
(stochastic) gradient descent, the primary algorithm for machine learning. In
practice, we benchmark instances of large machine learning models from 7
million to 103 million parameters. We find that, in the context of sparse
training, a quantum enhancement is possible at the early stage of learning
after model pruning, motivating a sparse parameter download and re-upload
scheme. Our work shows solidly that fault-tolerant quantum algorithms could
potentially contribute to most state-of-the-art, large-scale machine-learning
problems.Comment: 7+30 pages, 3+5 figure
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