Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNeural Network Approximation
109
0
0.0
(
0
)
Added by
Guergana Petrova
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Neural Networks (NNs) are the method of choice for building learning algorithms. Their popularity stems from their empirical success on several challenging learning problems. However, most scholars agree that a convincing theoretical explanation for this success is still lacking. This article surveys the known approximation properties of the outputs of NNs with the aim of uncovering the properties that are not present in the more traditional methods of approximation used in numerical analysis. Comparisons are made with traditional approximation methods from the viewpoint of rate distortion. Another major component in the analysis of numerical approximation is the computational time needed to construct the approximation and this in turn is intimately connected with the stability of the approximation algorithm. So the stability of numerical approximation using NNs is a large part of the analysis put forward. The survey, for the most part, is concerned with NNs using the popular ReLU activation function. In this case, the outputs of the NNs are piecewise linear functions on rather complicated partitions of the domain of $f$ into cells that are convex polytopes. When the architecture of the NN is fixed and the parameters are allowed to vary, the set of output functions of the NN is a parameterized nonlinear manifold. It is shown that this manifold has certain space filling properties leading to an increased ability to approximate (better rate distortion) but at the expense of numerical stability. The space filling creates a challenge to the numerical method in finding best or good parameter choices when trying to approximate.
rate research
Read More
In this paper, we introduce adaptive neuron enhancement (ANE) method for the best least-squares approximation using two-layer ReLU neural networks (NNs). For a given function f(x), the ANE method generates a two-layer ReLU NN and a numerical integration mesh such that the approximation accuracy is within the prescribed tolerance. The ANE method provides a natural process for obtaining a good initialization which is crucial for training nonlinear optimization problems. Numerical results of the ANE method are presented for functions of two variables exhibiting either intersecting interface singularities or sharp interior layers.
In this paper, we study adaptive neuron enhancement (ANE) method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The ANE method is a self-adaptive method generating a two-layer spline NN and a numerical integration mesh such that the approximation accuracy is within the prescribed tolerance. Moreover, the ANE method provides a natural process for obtaining a good initialization which is crucial for training nonlinear optimization problem. The underlying PDE is discretized by the Ritz method using a two-layer spline neural network based on either the primal or dual formulations that minimize the respective energy or complimentary functionals. Essential boundary conditions are imposed weakly through the functionals with proper norms. It is proved that the Ritz approximation is the best approximation in the energy norm; moreover, effect of numerical integration for the Ritz approximation is analyzed as well. Two estimators for adaptive neuron enhancement method are introduced, one is the so-called recovery estimator and the other is the least-squares estimator. Finally, numerical results for diffusion problems with either corner or intersecting interface singularities are presented.
In this paper, we develop a new neural network family based on power series expansion, which is proved to achieve a better approximation accuracy in comparison with existing neural networks. This new set of neural networks embeds the power series expansion (PSE) into the neural network structure. Then it can improve the representation ability while preserving comparable computational cost by increasing the degree of PSE instead of increasing the depth or width. Both theoretical approximation and numerical results show the advantages of this new neural network.
Deep learning is a powerful tool for solving nonlinear differential equations, but usually, only the solution corresponding to the flattest local minimizer can be found due to the implicit regularization of stochastic gradient descent. This paper proposes a network-based structure probing deflation method to make deep learning capable of identifying multiple solutions that are ubiquitous and important in nonlinear physical models. First, we introduce deflation operators built with known solutions to make known solutions no longer local minimizers of the optimization energy landscape. Second, to facilitate the convergence to the desired local minimizer, a structure probing technique is proposed to obtain an initial guess close to the desired local minimizer. Together with neural network structures carefully designed in this paper, the new regularized optimization can converge to new solutions efficiently. Due to the mesh-free nature of deep learning, the proposed method is capable of solving high-dimensional problems on complicated domains with multiple solutions, while existing methods focus on merely one or two-dimensional regular domains and are more expensive in operation counts. Numerical experiments also demonstrate that the proposed method could find more solutions than exiting methods.
We establish in this work approximation results of deep neural networks for smooth functions measured in Sobolev norms, motivated by recent development of numerical solvers for partial differential equations using deep neural networks. The error bounds are explicitly characterized in terms of both the width and depth of the networks simultaneously. Namely, for $fin C^s([0,1]^d)$, we show that deep ReLU networks of width $mathcal{O}(Nlog{N})$ and of depth $mathcal{O}(Llog{L})$ can achieve a non-asymptotic approximation rate of $mathcal{O}(N^{-2(s-1)/d}L^{-2(s-1)/d})$ with respect to the $mathcal{W}^{1,p}([0,1]^d)$ norm for $pin[1,infty)$. If either the ReLU function or its square is applied as activation functions to construct deep neural networks of width $mathcal{O}(Nlog{N})$ and of depth $mathcal{O}(Llog{L})$ to approximate $fin C^s([0,1]^d)$, the non-asymptotic approximation rate is $mathcal{O}(N^{-2(s-n)/d}L^{-2(s-n)/d})$ with respect to the $mathcal{W}^{n,p}([0,1]^d)$ norm for $pin[1,infty)$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
