1,900 research outputs found
Quantum Graphs: A simple model for Chaotic Scattering
We connect quantum graphs with infinite leads, and turn them to scattering
systems. We show that they display all the features which characterize quantum
scattering systems with an underlying classical chaotic dynamics: typical
poles, delay time and conductance distributions, Ericson fluctuations, and when
considered statistically, the ensemble of scattering matrices reproduce quite
well the predictions of appropriately defined Random Matrix ensembles. The
underlying classical dynamics can be defined, and it provides important
parameters which are needed for the quantum theory. In particular, we derive
exact expressions for the scattering matrix, and an exact trace formula for the
density of resonances, in terms of classical orbits, analogous to the
semiclassical theory of chaotic scattering. We use this in order to investigate
the origin of the connection between Random Matrix Theory and the underlying
classical chaotic dynamics. Being an exact theory, and due to its relative
simplicity, it offers new insights into this problem which is at the fore-front
of the research in chaotic scattering and related fields.Comment: 28 pages, 13 figures, submitted to J. Phys. A Special Issue -- Random
Matrix Theor
Enhancing Predictive Capabilities in Data-Driven Dynamical Modeling with Automatic Differentiation: Koopman and Neural ODE Approaches
Data-driven approximations of the Koopman operator are promising for
predicting the time evolution of systems characterized by complex dynamics.
Among these methods, the approach known as extended dynamic mode decomposition
with dictionary learning (EDMD-DL) has garnered significant attention. Here we
present a modification of EDMD-DL that concurrently determines both the
dictionary of observables and the corresponding approximation of the Koopman
operator. This innovation leverages automatic differentiation to facilitate
gradient descent computations through the pseudoinverse. We also address the
performance of several alternative methodologies. We assess a 'pure' Koopman
approach, which involves the direct time-integration of a linear,
high-dimensional system governing the dynamics within the space of observables.
Additionally, we explore a modified approach where the system alternates
between spaces of states and observables at each time step -- this approach no
longer satisfies the linearity of the true Koopman operator representation. For
further comparisons, we also apply a state space approach (neural ODEs). We
consider systems encompassing two and three-dimensional ordinary differential
equation systems featuring steady, oscillatory, and chaotic attractors, as well
as partial differential equations exhibiting increasingly complex and intricate
behaviors. Our framework significantly outperforms EDMD-DL. Furthermore, the
state space approach offers superior performance compared to the 'pure' Koopman
approach where the entire time evolution occurs in the space of observables.
When the temporal evolution of the Koopman approach alternates between states
and observables at each time step, however, its predictions become comparable
to those of the state space approach
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