36 research outputs found
Uniform-in-time propagation of chaos for mean field Langevin dynamics
We study the mean field Langevin dynamics and the associated particle system.
By assuming the functional convexity of the energy, we obtain the
-convergence of the marginal distributions towards the unique invariant
measure for the mean field dynamics. Furthermore, we prove the uniform-in-time
propagation of chaos in both the -Wasserstein metric and relative entropy.Comment: 66 pages, 3 figures and 1 table. Contains corrections and
enhancements to arXiv:2212.03050v
Entropic fictitious play for mean field optimization problem
It is well known that the training of the neural network can be viewed as a
mean field optimization problem. In this paper we are inspired by the
fictitious play, a classical algorithm in the game theory for learning the Nash
equilibria, and propose a new algorithm, different from the conventional
gradient-descent ones, to solve the mean field optimization. We rigorously
prove its (exponential) convergence, and show some simple numerical examples
Three-Dimensional Surface Parameters and Multi-Fractal Spectrum of Corroded Steel.
To study multi-fractal behavior of corroded steel surface, a range of fractal surfaces of corroded surfaces of Q235 steel were constructed by using the Weierstrass-Mandelbrot method under a high total accuracy. The multi-fractal spectrum of fractal surface of corroded steel was calculated to study the multi-fractal characteristics of the W-M corroded surface. Based on the shape feature of the multi-fractal spectrum of corroded steel surface, the least squares method was applied to the quadratic fitting of the multi-fractal spectrum of corroded surface. The fitting function was quantitatively analyzed to simplify the calculation of multi-fractal characteristics of corroded surface. The results showed that the multi-fractal spectrum of corroded surface was fitted well with the method using quadratic curve fitting, and the evolution rules and trends were forecasted accurately. The findings can be applied to research on the mechanisms of corroded surface formation of steel and provide a new approach for the establishment of corrosion damage constitutive models of steel
Data from: Three-dimensional surface parameters and multi-fractal spectrum of corroded steel
To study multi-fractal behavior of corroded steel surface, a range of fractal surfaces of corroded surfaces of Q235 steel were constructed by using the Weierstrass-Mandelbrot method under a high total accuracy. The multi-fractal spectrum of fractal surface of corroded steel was calculated to study the multi-fractal characteristics of the W-M corroded surface. Based on the shape feature of the multi-fractal spectrum of corroded steel surface, the least squares method was applied to the quadratic fitting of the multi-fractal spectrum of corroded surface. The fitting function was quantitatively analyzed to simplify the calculation of multi-fractal characteristics of corroded surface. The results showed that the multi-fractal spectrum of corroded surface was fitted well with the method using quadratic curve fitting, and the evolution rules and trends were forecasted accurately. The findings can be applied to research on the mechanisms of corroded surface formation of steel and provide a new approach for the establishment of corrosion damage constitutive models of steel
Mean Field Optimization Problem Regularized by Fisher Information
Recently there is a rising interest in the research of mean field
optimization, in particular because of its role in analyzing the training of
neural networks. In this paper by adding the Fisher Information as the
regularizer, we relate the regularized mean field optimization problem to a
so-called mean field Schrodinger dynamics. We develop an energy-dissipation
method to show that the marginal distributions of the mean field Schrodinger
dynamics converge exponentially quickly towards the unique minimizer of the
regularized optimization problem. Remarkably, the mean field Schrodinger
dynamics is proved to be a gradient flow on the probability measure space with
respect to the relative entropy. Finally we propose a Monte Carlo method to
sample the marginal distributions of the mean field Schrodinger dynamics
Left and right parts of the fitting curve of the multi-fractal spectrum with <i>D</i> = 2.3 (Eq 10).
<p>ESS is the error sum of square; red indicates the left part of the fitting curve; blue indicates the right part of the fitting curve. D-value of the minimum and maximum value is 1.23.</p
Chemical composition of Q235 steel (wt. %).
<p>Chemical composition of Q235 steel (wt. %).</p