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We analyze a nonlocal PDE model describing the dynamics of adaptation of a phenotypically structured population, under the effects of mutation and selection, in a changing environment. Previous studies have analyzed the large-time behavior…
We propose exact results for the full counting statistics, or the scaled cumulant generating function, pertaining to the transfer of arbitrary conserved quantities across an interface in homogeneous integrable models out of equilibrium. We…
The co-evolution of structure and dynamics, known as adaptivity, is a fundamental property in various systems and drives diverse emergent behaviors. However, the adaptivity in previous works is primarily stemmed from pairwise situations,…
We propose and illustrate an approach to coarse-graining the dynamics of evolving networks (networks whose connectivity changes dynamically). The approach is based on the equation-free framework: short bursts of detailed network evolution…
In this article, we study a non-Newtonian Stokes-Transport system. This set of PDEs was introduced as a model for describing the behavior of a cloud of particles in suspension in a Stokes fluid, and is a nonlinear coupling between a…
This paper concerns periodic solutions for a 1D-model with nonlocal velocity given by the periodic Hilbert transform. There is a rich literature showing that this model presents singular behavior of solutions via numerics and mathematical…
We have previously shown an analysis of our dimer model in the over-damped regime to show directed transport in equilibrium. Here we analyze the full model with inertial terms present to establish the same result. First we derive the…
In this paper, we prove that a particular nondegenerate, nonlinear, autonomous parabolic partial differential equation with a nonlocal mass transfer admits the local existence of classical solutions. The equation was developed to…
The key to generalization is controlling the complexity of the network. However, there is no obvious control of complexity -- such as an explicit regularization term -- in the training of deep networks for classification. We will show that…
We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in…
Stability of Wardrop equilibria is analyzed for dynamical transportation networks in which the drivers' route choices are influenced by information at multiple temporal and spatial scales. The considered model involves a continuum of…
Recent studies have shown that high disparities in effective learning rates (ELRs) across layers in deep neural networks can negatively affect trainability. We formalize how these disparities evolve over time by modeling weight dynamics…
In this paper we study the phase of self-similar solutions to general Nonlinear Schr\"odinger equations. From this analysis we gain insight on the dynamics of nontrivial solutions and a deeper understanding of the way collective coordinate…
The time evolution problem for non-self adjoint second order differential operators is studied by means of the path integral formulation. Explicit computation of the path integral via the use of certain underlying stochastic differential…
This paper introduces neuroevolution for solving differential equations. The solution is obtained through optimizing a deep neural network whose loss function is defined by the residual terms from the differential equations. Recent studies…
The asymptotic analysis of a linear high-field Wigner-BGK equation is developped by a modified Chapman-Enskog procedure. By an expansion of the unknown Wigner function in powers of the Knudsen number $\epsilon$, evolution equations are…
We propose a novel approach for learning the evolution that employs differentiable neural networks to approximate the full GENERIC structure. Instead of manually choosing the fitted parameters, we learn the whole model together with the…
In this paper, starting from the microscopic dynamics of isolated dislocations, we explain how to derive formally mean field models for the dynamics of dislocation densities. Essentially these models are tranport equations, coupled with the…
We present the first numerically stable nonlinear evolution for the leading-order gravitational effective field theory (Quadratic Gravity) in the spherically-symmetric sector. The formulation relies on (i) harmonic gauge to cast the…
We prove gradient estimates for solutions of the oblique derivative problem for a large class of elliptic and parabolic quasilinear PDEs. In particular, we expand on previous work of the author using a maximum principle argument. In…