22,428 research outputs found
On the Lagrangian and Hamiltonian aspects of infinite -dimensional dynamical systems and their finite-dimensional reductions
A description of Lagrangian and Hamiltonian formalisms naturally arisen from
the invariance structure of given nonlinear dynamical systems on the
infinite--dimensional functional manifold is presented. The basic ideas used to
formulate the canonical symplectic structure are borrowed from the Cartan's
theory of differential systems on associated jet--manifolds. The symmetry
structure reduced on the invariant submanifolds of critical points of some
nonlocal Euler--Lagrange functional is described thoroughly for both
differential and differential discrete dynamical systems. The Hamiltonian
representation for a hierarchy of Lax type equations on a dual space to the Lie
algebra of integral-differential operators with matrix coefficients, extended
by evolutions for eigenfunctions and adjoint eigenfunctions of the
corresponding spectral problems, is obtained via some special Backlund
transformation. The connection of this hierarchy with integrable by Lax
spatially two-dimensional systems is studied.Comment: 30 page
A new class of wavelet networks for nonlinear system identification
A new class of wavelet networks (WNs) is proposed for nonlinear system identification. In the new networks, the model structure for a high-dimensional system is chosen to be a superimposition of a number of functions with fewer variables. By expanding each function using truncated wavelet decompositions, the multivariate nonlinear networks can be converted into linear-in-the-parameter regressions, which can be solved using least-squares type methods. An efficient model term selection approach based upon a forward orthogonal least squares (OLS) algorithm and the error reduction ratio (ERR) is applied to solve the linear-in-the-parameters problem in the present study. The main advantage of the new WN is that it exploits the attractive features of multiscale wavelet decompositions and the capability of traditional neural networks. By adopting the analysis of variance (ANOVA) expansion, WNs can now handle nonlinear identification problems in high dimensions
Fitting Effective Diffusion Models to Data Associated with a "Glassy Potential": Estimation, Classical Inference Procedures and Some Heuristics
A variety of researchers have successfully obtained the parameters of low
dimensional diffusion models using the data that comes out of atomistic
simulations. This naturally raises a variety of questions about efficient
estimation, goodness-of-fit tests, and confidence interval estimation. The
first part of this article uses maximum likelihood estimation to obtain the
parameters of a diffusion model from a scalar time series. I address numerical
issues associated with attempting to realize asymptotic statistics results with
moderate sample sizes in the presence of exact and approximated transition
densities. Approximate transition densities are used because the analytic
solution of a transition density associated with a parametric diffusion model
is often unknown.I am primarily interested in how well the deterministic
transition density expansions of Ait-Sahalia capture the curvature of the
transition density in (idealized) situations that occur when one carries out
simulations in the presence of a "glassy" interaction potential. Accurate
approximation of the curvature of the transition density is desirable because
it can be used to quantify the goodness-of-fit of the model and to calculate
asymptotic confidence intervals of the estimated parameters. The second part of
this paper contributes a heuristic estimation technique for approximating a
nonlinear diffusion model. A "global" nonlinear model is obtained by taking a
batch of time series and applying simple local models to portions of the data.
I demonstrate the technique on a diffusion model with a known transition
density and on data generated by the Stochastic Simulation Algorithm.Comment: 30 pages 10 figures Submitted to SIAM MMS (typos removed and slightly
shortened
Trace formulas for stochastic evolution operators: Smooth conjugation method
The trace formula for the evolution operator associated with nonlinear
stochastic flows with weak additive noise is cast in the path integral
formalism. We integrate over the neighborhood of a given saddlepoint exactly by
means of a smooth conjugacy, a locally analytic nonlinear change of field
variables. The perturbative corrections are transfered to the corresponding
Jacobian, which we expand in terms of the conjugating function, rather than the
action used in defining the path integral. The new perturbative expansion which
follows by a recursive evaluation of derivatives appears more compact than the
standard Feynman diagram perturbation theory. The result is a stochastic analog
of the Gutzwiller trace formula with the ``hbar'' corrections computed an order
higher than what has so far been attainable in stochastic and
quantum-mechanical applications.Comment: 16 pages, 1 figure, New techniques and results for a problem we
considered in chao-dyn/980703
Functional expansion representations of artificial neural networks
In the past few years, significant interest has developed in using artificial neural networks to model and control nonlinear dynamical systems. While there exists many proposed schemes for accomplishing this and a wealth of supporting empirical results, most approaches to date tend to be ad hoc in nature and rely mainly on heuristic justifications. The purpose of this project was to further develop some analytical tools for representing nonlinear discrete-time input-output systems, which when applied to neural networks would give insight on architecture selection, pruning strategies, and learning algorithms. A long term goal is to determine in what sense, if any, a neural network can be used as a universal approximator for nonliner input-output maps with memory (i.e., realized by a dynamical system). This property is well known for the case of static or memoryless input-output maps. The general architecture under consideration in this project was a single-input, single-output recurrent feedforward network
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