neural networks for functional approximation and system identification we construct generalized translation networks to uniformly approximate a class of nonlinear continuous functionals defined on l p 1 1 s for integer s 1 1 p 1 or c 1 1 s we obtain lower bounds on the possible order of approximation for such functionals in terms of any approximation process depending continuously upon a given number of parameters our networks almost achieve this order of approximation in terms of the number of parameters neurons involved in the network the training is simple and noniterative in particular we avoid any optimization such as that involved in the usual back propagation

on the partitioning capabilities of feedforward neural networks with sigmoid nodes in this letter the capabilities of feedforward neural networks fnns on the realization and approximation of functions of the form g rl a which partition the rl space into polyhedral sets each one being assigned to one out of the c classes of a are investigated more specifically a constructive proof is given for the fact that fnns consisting of nodes having sigmoid output functions are capable of approximating any function g with arbitrary accuracy also the capabilities of fnns consisting of nodes having the hard limiter as output function are reviewed in both cases the two class as well as the multiclass cases are considered

local models and integrability of certain almost kahler 4 manifolds we classify up to a local isometry all non kahler almost kahler 4 manifolds for which the fundamental 2 form is an eigenform of the weyl tensor and whose ricci tensor is invariant with respect to the almost complex structure equivalently such almost kahler 4 manifolds satisfy the third curvature condition of a gray we use our local classification to show that in the compact case the third curvature condition of gray is equivalent to the integrability of the corresponding almost complex structure

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems the purpose of this paper is to investigate neural network capability systematically the main results are 1 every tauber wiener function is qualified as an activation function in the hidden layer of a three layered neural network 2 for a continuous function in s r sup 1 to be a tauber wiener function the necessary and sufficient condition is that it is not a polynomial 3 the capability of approximating nonlinear functionals defined on some compact set of a banach space and nonlinear operators has been shown and 4 the possibility by neural computation to approximate the output as a whole not at a fixed point of a dynamical system thus identifying the system

the wilson loop observables of chern simons theory on ℝ3 in axial gauge we define the wilson loop observables wlos for pure chern simons models with base manifold m 3 rigorously as infinite dimensional oscillatory integrals by exploiting an axial gauge fixing and applying certain regularization techniques like loop smearing and framing the values of the wlos can be computed explicitly if the structure group g of the model considered is abelian one obtains well known linking number expressions for the wlos if g is non abelian one obtains expressions which are similar but not identical to the state sum representations for the homfly and kauffman polynomials by jones and turaev

on the approximation by neural networks with bounded number of neurons in hidden layers abstract in the current note we show that a two hidden layer neural network with d inputs d neurons in the first hidden layer 2 d 2 neurons in the second hidden layer and with a specifically constructed sigmoidal and infinitely differentiable activation function can approximate any continuous multivariate function with arbitrary accuracy

approximation by neural networks with weights varying on a finite set of directions abstract approximation properties of the mlp multilayer feedforward perceptron model of neural networks have been investigated in a great deal of works over the last 30 years it has been shown that for a large class of activation functions a neural network can approximate arbitrarily well any given continuous function the most significant result on this problem belongs to leshno lin pinkus and schocken they proved that the necessary and sufficient condition for any single hidden layer network to have the u a p universal approximation property is that its activation function not be a polynomial some authors white stinchcombe ito and others showed that a single hidden layer perceptron with some bounded weights can also have the u a p thus the weights required for u a p are not necessary to be of an arbitrarily large magnitude but what if they are too restricted how can one learn approximation properties of networks with arbitrarily restricted set of weights the current paper makes a first step in solving this general problem we consider neural networks with sets of weights consisting of a finite number of directions our purpose is to characterize compact sets x in the d dimensional space such that the network can approximate any continuous function over x in a special case when weights vary only on two directions we give a lower bound for the approximation error and find a sufficient condition for the network to be a best approximation

universal approximation of multiple nonlinear operators by neural networks recently there has been interest in the observed capabilities of some classes of neural networks with fixed weights to model multiple nonlinear dynamical systems while this property has been observed in simulations open questions exist as to how this property can arise in this article we propose a theory that provides a possible mechanism by which this multiple modeling phenomenon can occur

a mathematical construction of the non abelian chern simons functional integral we construct rigorously an infinite dimensional distribution which corresponds to the chern simons cs functional integral associated with a principal fiber bundle over r 3 with structure group a compact connected lie group we determine the moments of the cs distribution and show that these coincide with those used in informal studies of the cs integral a locality property of the cs distribution is proven the complexified theory of frohlich and king is also discussed within our framework

universal approximations of continuous fuzzy valued functions by multi layer fuzzy neural networks abstract the fact that four layer feedforward regular fuzzy neural networks with sigmoid function in the first hidden layer are capable of approximately representing continuous fuzzy valued functions on any compact set of r is shown at first bernstein polynomials associated with fuzzy valued functions are employed to approximate continuous fuzzy valued function defined on a compact set secondly by the conclusions related to standard feedforward networks universal approximations of continuous fuzzy valued functions by regular fuzzy neural networks are obtained

the chern simons theory and knot polynomials the chern simons gauge theory is studied using a functional integral quantization this leads to a differential equation for expectations of wilson lines the solution of this differential equation is shown to be simply related to the two variable jones polynomial of the corresponding link in the case where the gauge group issu n a similar equation has been used before to get the jones polynomial from a braid representation of the link the main novelty of our approach is that we get the jones polynomial from a plat representation of the link

from simplicial chern simons theory to the shadow invariant ii this is the second of a series of papers in which we introduce and study a rigorous simplicial realization of the non abelian chern simons path integral for manifolds m of the form m s1 and arbitrary simply connected compact structure groups g more precisely we introduce for general links l in m a rigorous simplicial version wlorig l of the corresponding wilson loop observable wlo l in the so called torus gauge by blau and thompson nucl phys b 408 2 345 390 1993 for a simple class of links l we then evaluate wlorig l explicitly in a non perturbative way finding agreement with turaev s shadow invariant l

knot theory and the heuristics of functional integration abstract this paper is an exposition of the relationship between the heuristics of witten s functional integral and the theory of knots and links in three dimensional space

fuzzy neural networks a survey abstract in this paper a fuzzy neural network will be a layered feedforward neural net that has fuzzy signals and or fuzzy weights we survey recent results on learning algorithms and applications for fuzzy neural networks

complete memory structures for approximating nonlinear discrete time mappings this paper introduces a general structure that is capable of approximating input output maps of nonlinear discrete time systems the structure is comprised of two stages a dynamical stage followed by a memoryless nonlinear stage a theorem is presented which gives a simple necessary and sufficient condition for a large set of structures of this form to be capable of modeling a wide class of nonlinear discrete time systems in particular we introduce the concept of a complete memory a structure with a complete memory dynamical stage and a sufficiently powerful memoryless stage is shown to be capable of approximating arbitrarily wide class of continuous causal time invariant approximately finite memory mappings between discrete time signal spaces furthermore we show that any bounded input bounded output time invariant causal memory structure has such an approximation capability if and only if it is a complete memory several examples of linear and nonlinear complete memories are presented the proposed complete memory structure provides a template for designing a wide variety of artificial neural networks for nonlinear spatiotemporal processing

approximation of nonlinear systems with radial basis function neural networks a technique for approximating a continuous function of n variables with a radial basis function rbf neural network is presented the method uses an n dimensional raised cosine type of rbf that is smooth yet has compact support the rbf network coefficients are low order polynomial functions of the input a simple computational procedure is presented which significantly reduces the network training and evaluation time storage space is also reduced by allowing for a nonuniform grid of points about which the rbfs are centered the network output is shown to be continuous and have a continuous first derivative when the network is used to approximate a nonlinear dynamic system the resulting system is bounded input bounded output stable for the special case of a linear system the rbf network representation is exact on the domain over which it is defined and it is optimal in terms of the number of distinct storage parameters required several examples are presented which illustrate the effectiveness of this technique

abelian chern simons theory and linking numbers via oscillatory integrals a rigorous mathematical model of abelian chern simons theory based on the theory of infinite dimensional oscillatory integrals developed by albeverio and ho egh krohn is introduced a gauge fixed chern simons path integral is constructed as a fresnel integral in a certain hilbert space wilson loop variables are defined as fresnel integrable functions and it is shown in this context that the expectation value of products of wilson loops with respect to the chern simons path integral is a topological invariant which can be computed in terms of pairwise linking numbers of the loops as conjectured by witten furthermore a lattice chern simons action is proposed which converges to the continuum limit

ridge polynomial networks this paper presents a polynomial connectionist network called ridge polynomial network rpn that can uniformly approximate any continuous function on a compact set in multidimensional input space r sup d with arbitrary degree of accuracy this network provides a more efficient and regular architecture compared to ordinary higher order feedforward networks while maintaining their fast learning property the ridge polynomial network is a generalization of the pi sigma network and uses a special form of ridge polynomials it is shown that any multivariate polynomial can be represented in this form and realized by an rpn approximation capability of the rpn s is shown by this representation theorem and the weierstrass polynomial approximation theorem the rpn provides a natural mechanism for incremental network growth simulation results on a surface fitting problem the classification of high dimensional data and the realization of a multivariate polynomial function are given to highlight the capability of the network in particular a constructive learning algorithm developed for the network is shown to yield smooth generalization and steady learning

asymptotic expansion of the one loop approximation of the chern simons integral in an abstract wiener space setting abstract in an abstract wiener space setting we construct a rigorous mathematical model of the one loop approximation of the perturbative chern simons integral and derive its explicit asymptotic expansion for stochastic wilson lines

analyses of regular fuzzy neural networks for approximation capabilities abstract by introducing a trapezoidal fuzzy function and using the properties of the function we set up a continuously increasing fuzzy function that cannot be arbitrarily closely approximated on a compact set of f 0 r by the regular fuzzy neural network rfnn thus the conclusions in buckley and hayashi fuzzy sets and systems 61 1994 43 51 are improved and the problem if the rfnn is the universal approximator to the class of continuously increasing fuzzy functions is solved finally we obtain the universal approximation to the extended fuzzy functions by rfnn

approximations of continuous functionals by neural networks with application to dynamic systems the paper gives several strong results on neural network representation in an explicit form under very mild conditions a functional defined on a compact set in c a b or l sup p a b spaces of infinite dimensions can be approximated arbitrarily well by a neural network with one hidden layer the results are a significant development beyond earlier work where theorems of approximating continuous functions defined on a finite dimensional real space by neural networks with one hidden layer were given all the results are shown to be applicable to the approximation of the output of dynamic systems at any particular time

nonlinear structural response prediction based on support vector machines a support vector machines svm based two stage method is proposed to simulate and predict the nonlinear dynamic response of structures in the first stage an autoregressive moving average with exogenous input armax model is used to represent the acceleration response as the output of a single input single output siso system and the least square method is used to estimate the model parameters with which the linear acceleration response of the system can be simulated and predicted then the linear velocity and displacement are estimated using numerical integration of the predicted acceleration in the second stage by using the predicted linear responses acceleration velocity and displacement and the excitation to construct the input vector the svm is used to approximate nonlinear mapping from the input vector to system output and the trained svm can be used to simulate and predict the nonlinear dynamic response conveniently the nonlinear dynamic responses of a duffing oscillator and a frame structure are simulated and predicted using the proposed method as well as the neural network based method the results demonstrate that the svm based method provides superior performance in generalization and accuracy and can be a powerful tool for nonlinear system simulation and prediction

separating the vertices of n cubes by hyperplanes and its application to artificial neural networks a new sufficient condition that a region be classifiable by a two layer feedforward network using threshold activation functions is found briefly it is either a convex polytope or that minus the removal of convex polytope from its interior or recursively the author refers to these sets as convex recursive deletion regions the proof of implementability exploits the equivalence of this problem with that of characterizing two set partitions of the vertices of a hypercube which are separable by a hyperplane for which a new result is obtained

the feynman integrand as a hida distribution it is shown that feynman integrals are weighted averages over brownian paths with suitable hida distributions as complex weights for simple cases the construction of these distributions is be performed explicitly for a much larger class of interactions their existence is shown

existence and uniqueness results for neural network approximations some approximation theoretic questions concerning a certain class of neural networks are considered the networks considered are single input single output single hidden layer feedforward neural networks with continuous sigmoidal activation functions no input weights but with hidden layer thresholds and output layer weights specifically questions of existence and uniqueness of best approximations on a closed interval of the real line under mean square and uniform approximation error measures are studied a by product of this study is a reparametrization of the class of networks considered in terms of rational functions of a single variable this rational reparametrization is used to apply the theory of pade approximation to the class of networks considered in addition a question related to the number of local minima arising in gradient algorithms for learning is examined

symplectic curvature flow revisited abstract we continue studying a parabolic flow of almost kahler structures introduced by streets and tian which naturally extends kahler ricci flow onto symplectic manifolds in the system of primarily the symplectic form almost complex structure chern torsion and chern connection we establish new formulas for the evolutions of canonical quantities in particular those related to the chern connection using this we give an extended characterization of fixed points of the flow originally performed in 12

chern simons theory on r3 in axial gauge a rigorous approach we study pure chern simons models on m r3 using a functional integral quantization approach which is based on axial gauge fixing it is well known see e g comm math phys 126 1989 167 comm math phys 186 1997 563 that in axial gauge the chern simons action function is quadratic and that the faddeev popov determinant of this gauge fixing procedure is a constant function this means that the wilson loop observables wlos of the model considered can be obtained heuristically by integrating certain quantities against a functional measure of gaussian type we demonstrate that although these heuristic integral expressions look rather singular it is possible to give a rigorous meaning to them by combining constructions from white noise analysis with certain regularization techniques like loop smearing and framing for the special case g u 1 g being the group of the model we carry out the details of this approach which is also applicable to the case of non abelian g and find well known linking number expressions for the wlos