Related papers: Microlocal analysis and evolution equations
As mathematical model for the evolutionary equations of species the masterequation is choiced. Two formulations will be demonstrated to include the changes of parameters into the masterequation - that is, on the one hand, the formation of a…
In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form \begin{equation*} \partial_t u + \partial_x(\Lambda^s u + u\Lambda^r u^2) = 0, \end{equation*} where…
The long- and short-time behavior of solutions to dissipative evolution equations is studied by applying the concept of hypocoercivity. Aiming at partial differential equations that allow for a modal decomposition, we compute estimates that…
We discuss a new method to solve in a semianalytical way the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations at NLO order in the x-space. The method allows to construct an evolution operator expressed in form of a rapidly…
This short, self-contained article seeks to introduce and survey continuous-time deep learning approaches that are based on neural ordinary differential equations (neural ODEs). It primarily targets readers familiar with ordinary and…
We study a class of nonlinear eigenvalue problems which involves a convolution operator as well as a superlinear nonlinearity. Our variational existence proof is based on constrained optimization and provides a one-parameter family of…
Evolutionary multi-objective clustering (EMOC), a modern clustering technique, has been widely applied to extract patterns, allowing us to analyze different aspects of complex data by considering multiple criteria. In this article, we…
We show how strongly continuous semigroups can be associated with evolutionary equations. For doing so, we need to define the space of admissible history functions and initial states. Moreover, the initial value problem has to be formulated…
Neuroevolution is a promising area of research that combines evolutionary algorithms with neural networks. A popular subclass of neuroevolutionary methods, called evolution strategies, relies on dense noise perturbations to mutate networks,…
We present a novel framework for the study of a large class of non-linear stochastic PDEs, which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of…
Evolutionary algorithms have been widely applied for solving dynamic constrained optimization problems (DCOPs) as a common area of research in evolutionary optimization. Current benchmarks proposed for testing these problems in the…
The machine learning methods for data-driven identification of partial differential equations (PDEs) are typically defined for a given number of spatial dimensions and a choice of coordinates the data have been collected in. This dependence…
The purpose of this paper is to use semiclassical analysis to unify and generalize Lp estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on ellipticity and order, and apply to…
Molecular discovery, when formulated as an optimization problem, presents significant computational challenges because optimization objectives can be non-differentiable. Evolutionary Algorithms (EAs), often used to optimize black-box…
We introduce the control conditions for 0th order pseudodifferential operators $\mathbf{P}$ whose real parts satisfy the Morse--Smale dynamical condition. We obtain microlocal control estimates under the control conditions. As a result, we…
This is the second part of a series of papers dealing with an extensive class of analytic difference operators admitting reflectionless eigenfunctions. In the first part, the pertinent difference operators and their reflectionless…
Machine intelligence can develop either directly from experience or by inheriting experience through evolution. The bulk of current research efforts focus on algorithms which learn directly from experience. I argue that the alternative,…
Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
Existence, uniqueness and stability of the solutions of linear stochastic evolution equations are investigated. The results obtained are used to prove theorems on solvability of linear second order stochastic partial differential equations…