Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, and many other domains. Deep neural network architectures and computational issues have been ...well studied in machine learning. But there lacks a theoretical foundation for understanding the approximation or generalization ability of deep learning methods generated by the network architectures such as deep convolutional neural networks. Here we show that a deep convolutional neural network (CNN) is universal, meaning that it can be used to approximate any continuous function to an arbitrary accuracy when the depth of the neural network is large enough. This answers an open question in learning theory. Our quantitative estimate, given tightly in terms of the number of free parameters to be computed, verifies the efficiency of deep CNNs in dealing with large dimensional data. Our study also demonstrates the role of convolutions in deep CNNs.
The thermodynamic and magnetic properties of a spin-mixture of sup.40K atoms in a 1D harmonic trap, in the presence of an external magnetic field H, are studied using the static fluctuation ...approximation (SFA). The two-body interaction used is the contact potential. The thermodynamic properties include the chemical potential, energy per particle, specific heat capacity, and entropy per particle. The magnetic properties comprise the magnetization per particle and susceptibility per particle. A phase transition is observed for the repulsive interactions around 0.35T.sub.F and 0.5T.sub.F in the channels {-9/2, -7/2} and {-9/2, -5/2}, respectively. A peak appears in the entropy and susceptibility at effective scattering length absolute value of a.sub.1d approximately equal to 0.16 microm. For the attractive interactions, three different behaviors depending on H are found in the low-temperature limit. The first behavior is when H < H.sub.c, where H.sub.c is the critical magnetic field, and typical peaks emerge. The second behavior is at H = H.sub.c, and a magnetic phase transition occurs. The third behavior is when H > H.sub.c and the system becomes a partially polarized phase. Our results are consistent with the reported results obtained by the local density approximation, thermal Bethe ansatz method, mean-field approximation, and quantum transfer matrix method.
Data-driven computational mechanics Kirchdoerfer, T.; Ortiz, M.
Computer methods in applied mechanics and engineering,
06/2016, Letnik:
304
Journal Article
Recenzirano
Odprti dostop
We develop a new computing paradigm, which we refer to as data-driven computing, according to which calculations are carried out directly from experimental material data and pertinent constraints and ...conservation laws, such as compatibility and equilibrium, thus bypassing the empirical material modeling step of conventional computing altogether. Data-driven solvers seek to assign to each material point the state from a prespecified data set that is closest to satisfying the conservation laws. Equivalently, data-driven solvers aim to find the state satisfying the conservation laws that is closest to the data set. The resulting data-driven problem thus consists of the minimization of a distance function to the data set in phase space subject to constraints introduced by the conservation laws. We motivate the data-driven paradigm and investigate the performance of data-driven solvers by means of two examples of application, namely, the static equilibrium of nonlinear three-dimensional trusses and linear elasticity. In these tests, the data-driven solvers exhibit good convergence properties both with respect to the number of data points and with regard to local data assignment. The variational structure of the data-driven problem also renders it amenable to analysis. We show that, as the data set approximates increasingly closely a classical material law in phase space, the data-driven solutions converge to the classical solution. We also illustrate the robustness of data-driven solvers with respect to spatial discretization. In particular, we show that the data-driven solutions of finite-element discretizations of linear elasticity converge jointly with respect to mesh size and approximation by the data set.
We prove some new results concerning the approximation rate of neural networks with general activation functions. Our first result concerns the rate of approximation of a two layer neural network ...with a polynomially-decaying non-sigmoidal activation function. We extend the dimension independent approximation rates previously obtained to this new class of activation functions. Our second result gives a weaker, but still dimension independent, approximation rate for a larger class of activation functions, removing the polynomial decay assumption. This result applies to any bounded, integrable activation function. Finally, we show that a stratified sampling approach can be used to improve the approximation rate for polynomially decaying activation functions under mild additional assumptions.
In this paper, we investigate the approximation of a diffusion model problem with contrasted diffusivity for various nonconforming approximation methods. The essential difficulty is that the Sobolev ...smoothness index of the exact solution may be just barely larger than 1. The lack of smoothness is handled by giving a weak meaning to the normal derivative of the exact solution at the mesh faces. We derive robust and quasi-optimal error estimates. Quasi-optimality means that the approximation error is bounded, up to a generic constant, by the best approximation error in the discrete trial space, and robustness means that the generic constant is independent of the diffusivity contrast. The error estimates use a mesh-dependent norm that is equivalent, at the discrete level, to the energy norm and that remains bounded as long as the exact solution has a Sobolev index strictly larger than 1. Finally, we briefly show how the analysis can be extended to the Maxwell's equations.
The endpoint dilution assay's output, the 50% infectious dose (ID.sub.50 ), is calculated using the Reed-Muench or Spearman-Kärber mathematical approximations, which are biased and often ...miscalculated. We introduce a replacement for the ID.sub.50 that we call Specific INfection (SIN) along with a free and open-source web-application, midSIN (
Establishing a solid theoretical foundation for structured deep neural networks is greatly desired due to the successful applications of deep learning in various practical domains. This paper aims at ...an approximation theory of deep convolutional neural networks whose structures are induced by convolutions. To overcome the difficulty in theoretical analysis of the networks with linearly increasing widths arising from convolutions, we introduce a downsampling operator to reduce the widths. We prove that the downsampled deep convolutional neural networks can be used to approximate ridge functions nicely, which hints some advantages of these structured networks in terms of approximation or modeling. We also prove that the output of any multi-layer fully-connected neural network can be realized by that of a downsampled deep convolutional neural network with free parameters of the same order, which shows that in general, the approximation ability of deep convolutional neural networks is at least as good as that of fully-connected networks. Finally, a theorem for approximating functions on Riemannian manifolds is presented, which demonstrates that deep convolutional neural networks can be used to learn manifold features of data.
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