17,136 research outputs found
Dynamics of Learning with Restricted Training Sets I: General Theory
We study the dynamics of supervised learning in layered neural networks, in
the regime where the size of the training set is proportional to the number
of inputs. Here the local fields are no longer described by Gaussian
probability distributions and the learning dynamics is of a spin-glass nature,
with the composition of the training set playing the role of quenched disorder.
We show how dynamical replica theory can be used to predict the evolution of
macroscopic observables, including the two relevant performance measures
(training error and generalization error), incorporating the old formalism
developed for complete training sets in the limit as a
special case. For simplicity we restrict ourselves in this paper to
single-layer networks and realizable tasks.Comment: 39 pages, LaTe
Dynamics of Supervised Learning with Restricted Training Sets
We study the dynamics of supervised learning in layered neural networks, in
the regime where the size of the training set is proportional to the number
of inputs. Here the local fields are no longer described by Gaussian
probability distributions. We show how dynamical replica theory can be used to
predict the evolution of macroscopic observables, including the relevant
performance measures, incorporating the old formalism in the limit
as a special case. For simplicity we restrict ourselves
to single-layer networks and realizable tasks.Comment: 36 pages, latex2e, 12 eps figures (to be publ in: Proc Newton Inst
Workshop on On-Line Learning '97
The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries
We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body
Non-negativity preserving numerical algorithms for stochastic differential equations
Construction of splitting-step methods and properties of related
non-negativity and boundary preserving numerical algorithms for solving
stochastic differential equations (SDEs) of Ito-type are discussed. We present
convergence proofs for a newly designed splitting-step algorithm and simulation
studies for numerous numerical examples ranging from stochastic dynamics
occurring in asset pricing theory in mathematical finance (SDEs of CIR and CEV
models) to measure-valued diffusion and superBrownian motion (SPDEs) as met in
biology and physics.Comment: 23 pages, 7 figures. Figures 6.2 and 6.3 in low resolution due to
upload size restrictions. Original resolution at
http://gisc.uc3m.es/~moro/profesional.htm
An application of the Maslov complex germ method to the 1D nonlocal Fisher-KPP equation
A semiclassical approximation approach based on the Maslov complex germ
method is considered in detail for the 1D nonlocal
Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak
diffusion. In terms of the semiclassical formalism developed, the original
nonlinear equation is reduced to an associated linear partial differential
equation and some algebraic equations for the coefficients of the linear
equation with a given accuracy of the asymptotic parameter. The solutions of
the nonlinear equation are constructed from the solutions of both the linear
equation and the algebraic equations. The solutions of the linear problem are
found with the use of symmetry operators. A countable family of the leading
terms of the semiclassical asymptotics is constructed in explicit form.
The semiclassical asymptotics are valid by construction in a finite time
interval. We construct asymptotics which are different from the semiclassical
ones and can describe evolution of the solutions of the
Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. In the example
considered, an initial unimodal distribution becomes multimodal, which can be
treated as an example of a space structure.Comment: 28 pages, version accepted for publication in Int. J. Geom. Methods
Mod. Phy
Computation of three-dimensional nozzle-exhaust flow fields with the GIM code
A methodology is introduced for constructing numerical analogs of the partial differential equations of continuum mechanics. A general formulation is provided which permits classical finite element and many of the finite difference methods to be derived directly. The approach, termed the General Interpolants Method (GIM), can combined the best features of finite element and finite difference methods. A quasi-variational procedure is used to formulate the element equations, to introduce boundary conditions into the method and to provide a natural assembly sequence. A derivation is given in terms of general interpolation functions from this procedure. Example computations for transonic and supersonic flows in two and three dimensions are given to illustrate the utility of GIM. A three-dimensional nozzle-exhaust flow field is solved including interaction with the freestream and a coupled treatment of the shear layer. Potential applications of the GIM code to a variety of computational fluid dynamics problems is then discussed in terms of existing capability or by extension of the methodology
The Minimum-Uncertainty Squeezed States for for Atoms and Photons in a Cavity
We describe a six-parameter family of the minimum-uncertainty squeezed states
for the harmonic oscillator in nonrelativistic quantum mechanics. They are
derived by the action of corresponding maximal kinematical invariance group on
the standard ground state solution. We show that the product of the variances
attains the required minimum value 1/4 only at the instances that one variance
is a minimum and the other is a maximum, when the squeezing of one of the
variances occurs. The generalized coherent states are explicitly constructed
and their Wigner function is studied. The overlap coefficients between the
squeezed, or generalized harmonic, and the Fock states are explicitly evaluated
in terms of hypergeometric functions. The corresponding photons statistics are
discussed and some applications to quantum optics, cavity quantum
electrodynamics, and superfocusing in channeling scattering are mentioned.
Explicit solutions of the Heisenberg equations for radiation field operators
with squeezing are found.Comment: 27 pages, no figures, 174 references J. Phys. B: At. Mol. Opt. Phys.,
Special Issue celebrating the 20th anniversary of quantum state engineering
(R. Blatt, A. Lvovsky, and G. Milburn, Guest Editors), May 201
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