Добірка наукової літератури з теми "Infinite-width limit"
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Статті в журналах з теми "Infinite-width limit":
1
Pastur, L. "Eigenvalue distribution of large random matrices arising in deep neural networks: Orthogonal case." Journal of Mathematical Physics 63, no. 6 (June 1, 2022): 063505. http://dx.doi.org/10.1063/5.0085204.
Анотація:
This paper deals with the distribution of singular values of the input–output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent weight matrices alternate with diagonal matrices whose entries depend on the corresponding column of the nearest neighbor weight matrix. The problem has been considered in the several recent studies of the field for the Gaussian weights and biases and also for the weights that are Haar distributed orthogonal matrices and Gaussian biases. Based on a free probability argument, it was claimed in those papers that, in the limit of infinite width (matrix size), the singular value distribution of the Jacobian coincides with that of the analog of the Jacobian with special random but weight independent diagonal matrices, the case well known in random matrix theory. In this paper, we justify the claim for random Haar distributed weight matrices and Gaussian biases. This, in particular, justifies the validity of the mean field approximation in the infinite width limit for the deep untrained neural networks and extends the macroscopic universality of random matrix theory to this new class of random matrices.
2
Pacelli, R., S. Ariosto, M. Pastore, F. Ginelli, M. Gherardi, and P. Rotondo. "A statistical mechanics framework for Bayesian deep neural networks beyond the infinite-width limit." Nature Machine Intelligence 5, no. 12 (December 18, 2023): 1497–507. http://dx.doi.org/10.1038/s42256-023-00767-6.
3
Thorkildsen, Gunnar, and Helge B. Larsen. "X-ray diffraction in perfect t × l crystals. Rocking curves." Acta Crystallographica Section A Foundations of Crystallography 55, no. 5 (September 1, 1999): 840–54. http://dx.doi.org/10.1107/s0108767399002986.
Анотація:
A general formalism, based on the Takagi–Taupin equations, for calculating rocking curves in perfect t\times l crystals is presented. It includes nonsymmetrical scattering, refraction, and ordinary and anomalous absorption. t and l may be varied independently. In the limit of a semi-infinite crystal, the standard results from the fundamental theory are retrieved. For crystal dimensions less than the extinction length, the theory converges to the kinematical limit. Simulations for germanium and silicon show significant influence of crystal finiteness. When dynamical effects are prominent, the curves exhibit various degrees of asymmetry and the full width at half-maximum is generally larger than the corresponding Darwin width. This is attributed to combined Laue and Bragg contributions which are shifted with respect to each other owing to refraction.
4
Karr, D. G., J. C. Watson, and M. HooFatt. "Three-Dimensional Analysis of Ice Sheet Indentation: Limit Analysis Solutions." Journal of Offshore Mechanics and Arctic Engineering 111, no. 1 (February 1, 1989): 63–69. http://dx.doi.org/10.1115/1.3257141.
Анотація:
A method is presented for determining the collapse pressures of an ice sheet subjected to a uniformly distributed edge load by applying the upper-bound theorem of limit analysis. The ice sheet is idealized as a semi-infinite layer of elastic-perfectly plastic material. A quadratic anisotropic yield criterion is used to calculate the indentation pressures. The ice sheet consists of columnar ice and is assumed isotropic in the plane of the ice sheet. Upper-bound solutions are found by optimizing a three-dimensional discontinuous velocity field representing an assumed collapse pattern of the ice sheet. Solutions are based on various ratios of indentor width to ice thickness, thereby providing an envelope of indentation pressures over a range of aspect ratios, from conditions of plane strain to plane stress. Solutions are then compared with corresponding two and three-dimensional lower-bound analyses.
5
Landa, Haggai, Cecilia Cormick, and Giovanna Morigi. "Static Kinks in Chains of Interacting Atoms." Condensed Matter 5, no. 2 (May 13, 2020): 35. http://dx.doi.org/10.3390/condmat5020035.
Анотація:
We theoretically analyse the equation of topological solitons in a chain of particles interacting via a repulsive power-law potential and confined by a periodic lattice. Starting from the discrete model, we perform a gradient expansion and obtain the kink equation in the continuum limit for a power-law exponent n ≥ 1 . The power-law interaction modifies the sine-Gordon equation, giving rise to a rescaling of the coefficient multiplying the second derivative (the kink width) and to an additional integral term. We argue that the integral term does not affect the local properties of the kink, but it governs the behaviour at the asymptotics. The kink behaviour at the center is dominated by a sine-Gordon equation and its width tends to increase with the power law exponent. When the interaction is the Coulomb repulsion, in particular, the kink width depends logarithmically on the chain size. We define an appropriate thermodynamic limit and compare our results with existing studies performed for infinite chains. Our formalism allows one to systematically take into account the finite-size effects and also slowly varying external potentials, such as for instance the curvature in an ion trap.
6
AKHMEDIEV, N., J. M. SOTO-CRESPO, M. GRAPINET, and Ph GRELU. "DISSIPATIVE SOLITON PULSATIONS WITH PERIODS BEYOND THE LASER CAVITY ROUND TRIP TIME." Journal of Nonlinear Optical Physics & Materials 14, no. 02 (June 2005): 177–94. http://dx.doi.org/10.1142/s0218863505002645.
Анотація:
We review recent results on periodic pulsations of the soliton parameters in a passively mode-locked fiber laser. Solitons change their shape, amplitude, width and velocity periodically in time. These pulsations are limit cycles of a dissipative nonlinear system in an infinite-dimensional phase space. Pulsation periods can vary from a few to hundreds of round trips. We present a continuous model of a laser as well as a model with parameter management. The results of the modeling are supported with experimental results obtained using a fiber laser.
7
Zeng, Y., and S. Weinbaum. "Stokes flow through periodic orifices in a channel." Journal of Fluid Mechanics 263 (March 25, 1994): 207–26. http://dx.doi.org/10.1017/s0022112094004088.
Анотація:
This paper develops a three-dimensional infinite series solution for the Stokes flow through a parallel walled channel which is obstructed by a thin planar barrier with periodically spaced rectangular orifices of arbitrary aspect ratio B’/d’ and spacing D’. Here B’ is the half-height of the channel and d’ is the half-width of the orifice. The problem is motivated by recent electron microscopic studies of the intercellular channel between vascular endothelial cells which show a thin junction strand barrier with discontinuities or breaks whose spacing and width vary with the tissue. The solution for this flow is constructed as a superposition of Hasimoto's (1958) general solution for the two-dimensional flow through a periodic slit array in an infinite plane wall and a new three-dimensional solution which corrects for the top and bottom boundaries. In contrast to the well-known solutions of Sampson (1891) and Hasimoto (1958) for the flow through zero-thickness orifices of circular or elliptic cross-section or periodic slits in an infinite plane wall, which exhibit characteristic viscous velocity profiles, the present bounded solutions undergo a fascinating change in behaviour as the aspect ratio B’/d’ of the orifice opening is increased. For B’/d’ [Lt ] 1 and (D’ –- d’)/B’ of O(1) or greater, which represents a narrow channel, the velocity has a minimum at the orifice centreline, rises sharply near the orifice edges and then experiences a boundary-layer-like correction over a thickness of O(B’) to satisfy no-slip conditions. For B’/d’ of O(1) the profiles are similar to those in a rectangular duct with a maximum on the centreline, whereas for B’/d’ [Gt ] 1, which describes widely separated channel walls, the solution approaches Hasimoto's solution for the periodic infinite-slit array. In the limit (D’ –- d’)/B’ [Lt ] 1, where the width of the intervening barriers is small compared with the channel height, the solutions exhibit the same behaviour as Lee & Fung's (1969) solution for the flow past a single cylinder. The drag on the zero-thickness barriers in this case is nearly the same as for the cylinder for all aspect ratios.
8
DELEBECQUE, FANNY. "AN ASYMPTOTIC MODEL FOR THE TRANSPORT OF AN ELECTRON GAS IN A SLAB." Mathematical Models and Methods in Applied Sciences 21, no. 07 (July 2011): 1443–78. http://dx.doi.org/10.1142/s0218202511005453.
Анотація:
We study the limiting behavior of a Schrödinger–Poisson system describing a three-dimensional quantum gas that is confined along the vertical z-direction in a fine slab. The starting point is the three-dimensional Schrödinger–Poisson system with Dirichlet conditions on two horizontal planes z = 0 and z = ε, where the small parameter ε is the scale width of the slab. The limit ε → 0 appears to be an infinite system of two-dimensional nonlinear Schrödinger equations. Our strategy combines a refined analysis of the Poisson kernel acting on strongly confined densities and a time-averaging process that allows us to deal with the fast time oscillations.
9
VOJTA, MATTHIAS, YING ZHANG, and SUBIR SACHDEV. "RENORMALIZATION GROUP ANALYSIS OF QUANTUM CRITICAL POINTS IN d-WAVE SUPERCONDUCTORS." International Journal of Modern Physics B 14, no. 29n31 (December 20, 2000): 3719–34. http://dx.doi.org/10.1142/s0217979200004271.
Анотація:
We describe a search for renormalization group fixed points which control a second-order quantum phase transition between a dx2-y2-superconductor and some other superconducting ground state. Only a few candidate fixed points are found. In the finite temperature (T) quantum-critical region of some of these fixed points, the fermion quasiparticle lifetime is very short and the spectral function has an energy width of order kBT near the Fermi points. Under the same conditions, the thermal conductivity is infinite in the scaling limit. We thus provide simple, explicit, examples of quantum theories in two dimensions for which a purely fermionic quasiparticle description of transport is badly violated.
10
Jagannathan, Arjun, Kraig Winters, and Laurence Armi. "Stratified Flows over and around Long Dynamically Tall Mountain Ridges." Journal of the Atmospheric Sciences 76, no. 5 (May 1, 2019): 1265–87. http://dx.doi.org/10.1175/jas-d-18-0145.1.
Анотація:
Abstract Uniformly stratified flows approaching long and dynamically tall ridges develop two distinct flow components over disparate time scales. The fluid upstream and below a “blocking level” is stagnant in the limit of an infinite ridge and flows around the sides when the ridge extent is finite. The streamwise half-width of the obstacle at the blocking level arises as a natural inner length scale for the flow, while the excursion time over this half-width is an associated short time scale for the streamwise flow evolution. Over a longer time scale, low-level horizontal flow splitting leads to the establishment of an upstream layerwise potential flow beneath the blocking level. We demonstrate through numerical experiments that for sufficiently long ridges, crest control and streamwise asymmetry are seen on both the short and long time scales. On the short time scale, upstream blocking is established quickly and the flow is well described as a purely infinite-ridge overflow. Over the long time scale associated with flow splitting, low-level flow escapes around the sides, but the overflow continues to be hydraulically controlled and streamwise asymmetric in the neighborhood of the crest. We quantify this late-time overflow by estimating its volumetric transport and then briefly demonstrate how this approach can be extended to predict the overflow across nonuniform ridge shapes.
Дисертації з теми "Infinite-width limit":
1
Hajjar, Karl. "A dynamical analysis of infinitely wide neural networks." Electronic Thesis or Diss., université Paris-Saclay, 2024. http://www.theses.fr/2024UPASM001.
Анотація:
Durant la dernière décennie, les réseaux de neurones ont eu un succès retentissant dans de nombreuses tâches en pratique, cependant les arguments théoriques derrière ce succès restent insuffisants et une théorie mathématique appropriée pour étudier rigoureusement ces objets fait toujours défaut. Les limites des réseaux de neurones à largeur infinie sont récemment apparues comme une façon d'éclaircir certains aspects du problème. Dans cette thèse, nous étudions la limite des réseaux de neurones de largeur infinie avec une renormalisation particulière souvent dénommée ''champ moyen'' dans la littérature. La difficulté d'analyser les réseaux de neurones d'un point de vue théorique réside en partie dans la nature hautement non-linéaire de ces objets et dans l'énorme quantité de paramètres, ou poids (pouvant aller jusqu'à la centaine de milliards en pratique) qui interagissent lorsqu'ils sont mis à jour durant la descente de gradient. Nous examinons les trajectoires durant l'optimisation des réseaux de neurones de largeur infinie pendant la phase d'entraînement afin d'exhiber des propriétés de ces modèles dans certains cadres simples tels que les réseaux de neurones entièrement connectés avec une ou plusieurs couches cachées. Cette thèse traite de différents aspects de la dynamique d'optimisation des réseaux de neurones de largeur infinie: des méthodes pour rendre possible l'entraînement de ces modèles aux symétries qui peuvent émerger dans cette limite en passant par de nouveaux algorithmes d'optimisation qui adaptent le nombre de neurones à la volée durant la phase d'entraînementNeural networks have had tremendous success in many practical tasks over the last decade, yet the theoretical reasons behind their performance are poorly understood and we lack a proper mathematical theory to rigorously study the properties of those objects. Infinite-width limits of neural networks have recently emerged as a way to shed light on some of the aspects of the problem. In this thesis, we study the infinite-width limit of networks of different depths under a particular scaling often referred to as the ''mean-field'' scaling in the literature. Part of the reason why neural networks are difficult to analyze from a theoretical standpoint is because they are highly non-linear and involve a huge amount of parameters, or weights, (up to hundreds of billions in practice) which interact as they are updated during gradient descent. We investigate the optimization trajectories of the infinite-width limit of neural networks during training in order to exhibit properties of those models in simple settings such as fully-connected networks with one or more hidden layers. This thesis focuses on different aspects of the optimization dynamics of networks in the infinite-width limit: from methods to enable training those models at arbitrary depths to the symmetry properties that can emerge in that limit as well as novel optimization algorithms which adapt the number of neurons in an on-line fashion during training
Тези доповідей конференцій з теми "Infinite-width limit":
1
Osinski, Marek, Mohammad Mojahedie, and Michael W. Prairie. "Density of states in finite-barrier quantum wells." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/oam.1992.mz4.
Анотація:
Density of states in finite-barrier quantum wells is examined critically. In the infinite barrier limit, the two-dimensional (2D) density of states (DOS) has been shown to correspond to the bulk case.1 When finite wells are considered, this correspondence may no longer hold. In this paper, we propose a modification to the finite-well DOS, which retains the elegance of the infinite-well case while preserving the effects of the finite barrier. This is accomplished by either defining an effective infinite-well width that matches the parameters of the finite well or by defining a new effective mass. Both approaches are based on rigorous calculations of the quantized wave vectors. In the spirit of the effective mass concept, we concentrate on the latter case, particularly in the limits of very thin and very thick wells. We investigate the relationship between the DOS and the quantum-well effective mass, based on the allowed wave vectors. Examples will be given illustrating the conventional definition of 2D DOS, the modified DOS, and their comparison with the bulk case.
2
Gordon, J. L., and D. P. Jones. "Application of a Sixth Order Generalized Stress Function for Determining Limit Loads for Plates with Triangular Penetration Patterns." In ASME 2002 Pressure Vessels and Piping Conference. ASMEDC, 2002. http://dx.doi.org/10.1115/pvp2002-1298.
Анотація:
The capability to obtain limit load solutions of plates with triangular penetration patterns using fourth order functions to represent the collapse surface has been presented in previous papers. These papers describe how equivalent solid plate elastic-perfectly plastic finite element capabilities are generated and demonstrate how such capabilities can be used to great advantage in the analysis of tubesheets in large heat exchanger applications. However, these papers have pointed out that although the fourth order functions can produce sufficient accuracy for many practical applications, there are situations where improvements in the accuracy of in-plane and transverse shear are desirable. This paper investigates the use of a sixth order function to represent the collapse surface for improved accuracy of the in-plane response. Explicit elastic-perfectly plastic finite element solutions are obtained for unit cells representing an infinite array of circular penetrations arranged in an equilateral triangular array. These cells are used to create a numerical representation of the complete collapse surfaces for a number of ligament efficiencies (h/P where h is the minimum ligament width and P is the distance between hole centers). Each collapse surface is then fit to a sixth order function that satisfies the periodicity of the hole pattern. Sixth-order collapse functions were developed for h/P values between .05 and .50. Accuracy of the sixth order and the fourth order functions are compared. It was found that the sixth order function is indeed more accurate, reducing the error from 12.2% for the fourth order function to less than 3% for the sixth order function.
3
Mukoyama, Hiroshi, Shigeyuki Shimachi, and Yoshihide Hakozaki. "Contact Pressure Estimates of Tooth Surfaces of Gear Couplings." In ASME 2000 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2000. http://dx.doi.org/10.1115/detc2000/ptg-14452.
Анотація:
Abstract Recent demands for gear couplings are to reduce the backlash and to increase the shaft angle limit. On coping with these demands, the tooth contact pressure is recognized as the trade-off problem. In the traditional estimation of tooth contact pressure, the deflection of tooth is calculated by using the formula for spur gear that has long contact bearing in the face width direction, although gear coupling has it in the tooth depth direction. And, the Hertz depression of the tooth surface is estimated as that of the infinite plane. Additionally, the traditional methods don’t consider about the edge contact on the tip or end of tooth. A successive approximation method is established to find the load distribution on the mating teeth surfaces. As for the effect of the edge contact on the tip or end of tooth, it is cleared that the contact pressure distribution deforms itself severely, but the maximum pressure is almost constant. The expressions estimating the maximum pressure and the displacement of tooth base are constructed for 6 parameters as follows; total load coefficient, relative curvature of teeth surfaces, tooth module, ratio of tooth height to face width, angle of tip contact and deviation of end contact.