We prove that the gradient descent training of a two-layer neural network on empirical or population risk may not decrease population risk at an order faster than t−4/(d−2) under mean field scaling. The loss functiona...
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Many applications of computer vision rely on the alignment of similar but non-identical images. We present a fast algorithm for aligning heterogeneous images based on optimal transport. Our approach combines the speed...
The equations of motion describing inviscid fluid flow are solved numerically in two dimensions for the case where the flow can be described by patches of constant vorticity. The case where the vorticity is described ...
The equations of motion describing inviscid fluid flow are solved numerically in two dimensions for the case where the flow can be described by patches of constant vorticity. The case where the vorticity is described initially by two circular patches is studied in detail. The numerical evidence indicates that when the minimum distance between the two patches is initially less than the radius of the patches a singularity forms in finite time on the boundary curves of the patches. The singularity appears to be a jump discontinuity in the tangent vector of the boundary curve.
The correspondence principle postulated for the description of hydrodynamic turbulence [Phys. Rev. Lett. 57, 1722 (1986)] combined with the theory of thermal boundary layer [B. Castaing et al. (private communication)]...
The correspondence principle postulated for the description of hydrodynamic turbulence [Phys. Rev. Lett. 57, 1722 (1986)] combined with the theory of thermal boundary layer [B. Castaing et al. (private communication)] is applied to high Rayleigh number convection in a Bénard cell. Quantitative interpretation of recent experimental data [B. Castaing et al. (private communication)] is presented. The predicted intermittency exponent following from comparison of the theory with experiment is 0.175<μ<0.275. A crucial experimental test of the renormalization group theory of turbulence is proposed.
Deviations from classical scaling behavior are shown to result in flattened energy and dissipation–fluctuation inertial‐range spectra in fully developed turbulence.
Deviations from classical scaling behavior are shown to result in flattened energy and dissipation–fluctuation inertial‐range spectra in fully developed turbulence.
A new dimensional analysis for high Rayleigh number thermal convection is proposed to give an alternative interpretation of the scaling laws observed recently by Castaing et al. [J. Fluid Mech. (in press)]. The key as...
A new dimensional analysis for high Rayleigh number thermal convection is proposed to give an alternative interpretation of the scaling laws observed recently by Castaing et al. [J. Fluid Mech. (in press)]. The key assumption in the present approach is that the central fluctuating temperature field actively interacts with the turbulent velocity field, and this interaction leads to a velocity inertial subrange that deviates significantly from Kolmogorov’s freely cascading inertial range.
A new algorithm based on spectral element discretization and non-oscillatory ideas is developed for the solution of hyperbolic partial differential equations. A conservative formulation is proposed based on cell avera...
A new algorithm based on spectral element discretization and non-oscillatory ideas is developed for the solution of hyperbolic partial differential equations. A conservative formulation is proposed based on cell averaging and reconstruction procedures, that employs a staggered grid of Gauss-Chebyshev and Gauss-Lobatto Chebyshev discretizations. The non-oscillatory reconstruction procedure is based on ideas similar to those proposed by Cai et al. (Math. Comput. 52, 389 (1989)) but employs a modified technique which is more robust and simpler in terms of determining the location and strength of a discontinuity. It is demonstrated through model problems of linear advection, inviscid Burgers equation, and one-dimensional Euler system that the proposed algorithm leads to stable, non-oscillatory accurate results. Exponential accuracy away from the discontinuity is realized for the inviscid Burgers equation example.
A time-independent field theoretical framework for turbulence is suggested, based upon a variational principle for a stationary solution of the Fokker-Planck equation. We obtain a functional equation for the effective...
A time-independent field theoretical framework for turbulence is suggested, based upon a variational principle for a stationary solution of the Fokker-Planck equation. We obtain a functional equation for the effective Action of this spatial field theory and investigate its general properties and some numerical solutions. The equation is completely universal, and allows for the scale invariant solutions in the inertial range. The critical indices are not fixed at the kinematical level, but rather should be found from certain eigenvalue conditions, as in the field theory of critical phenomena. Unlike the Wyld field theory, there are no divergences in our Feynman integrals, due to some magic cancellations. The simplest possible Gaussian approximation yields crude but still reasonable results (there are deviations from Kolmogorov scaling in 3 dimensions, but at 2.7544 dimensions it would be exact). Our approach allows us to study some new problems, such as spontaneous parity breaking in 3d turbulence. It turns out that with the appropriate helicity term added to the velocity correlation function, logarithmic infrared divergences arise in our field theory which effectively eliminates these terms. In order to build a quantitative theory of turbulence, one should consider more sophisticated Ansatz for the effective Action, which would require serious numerical work.
The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss...
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