We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is unconditionally stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we prove that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.

A recent line of work has studied the statistical properties of neural networks to great success from a {\it mean field theory} perspective, making and verifying very precise predictions of neural network behavior and test time performance. In this paper, we build upon these works to explore two methods for taming the behaviors of random residual networks (with only fully connected layers and no batchnorm). The first method is {\it width variation (WV)}, ie varying the widths of layers as a function of depth. We show that width decay reduces gradient explosion without affecting the mean forward dynamics of the random network. The second method is {\it variance variation (VV)}, ie changing the initialization variances of weights and biases over depth. We show VV, used appropriately, can reduce gradient explosion of tanh and ReLU resnets from \exp (\Theta (\sqrt L)) and \exp (\Theta (\sqrt L)) respectively to constant \exp (\Theta (\sqrt L)) . A complete phase-diagram is derived for how variance decay affects different dynamics, such as those of gradient and activation norms. In particular, we show the existence of many phase transitions where these dynamics switch between exponential, polynomial, logarithmic, and even constant behaviors. Using the obtained mean field theory, we are able to track surprisingly well how VV at initialization time affects training and test time performance on MNIST after a set number of epochs: the level sets of test/train set accuracies coincide with the level sets of the expectations of certain gradient norms or of metric expressivity (as defined in\cite {yang_meanfield_2017}), a measure of expansion in a random neural network. Based on insights from past works in

In this paper, we prove the uniqueness of topological multivortex solutions for the self-dual Maxwell–Chern–Simons U(1)U(1) model if the Chern–Simons coupling parameter is sufficiently large and the charge of electron is sufficiently small or large. On the other hand, we also establish the sharp region of the flux for non-topological solutions and provide the classification of radial solutions of all types in the case of one vortex point.

In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau’s estimate for weak solutions of the heat equation and prove a sharp Yau’s gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition RCD(K; N).

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