Related papers: Forward-Backward Evolution Equations and Applicati…
We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…
Inverse problems are often ill-posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is…
We review the concept of well-posedness in the context of evolutionary problems from mathematical physics for a particular subclass of problems from elasticity theory. The complexity of physical phenomena appears as encoded in so called…
In the field of evolutionary multi-objective optimization, the approximation of the Pareto front (PF) is achieved by utilizing a collection of representative candidate solutions that exhibit desirable convergence and diversity. Although…
We study well-posedness and long time behavior of the nonlinear Vlasov-Poisson- Fokker-Planck system with an external confining potential. The system describes the time evolution of particles (e.g.$\,\,$in a plasma) undergoing diffusion,…
In this work, we concern with the high order numerical methods for coupled forward-backward stochastic differential equations (FBSDEs). Based on the FBSDEs theory, we derive two reference ordinary differential equations (ODEs) from the…
We investigate finite-time Lyapunov exponents (FTLEs), a measure for exponential separation of input perturbations, of deep neural networks within the framework of continuous-depth neural ODEs. We demonstrate that FTLEs are powerful…
This article introduces and solves a general class of fully coupled forward-backward stochastic dynamics by investigating the associated system of functional differential equations. As a consequence, we are able to solve many different…
We investigate degenerate saddle point problems, which can be viewed as limit cases of standard mixed formulations of symmetric problems with large jumps in coefficients. We prove that they are well-posed in a standard norm despite the…
Although real-coded differential evolution (DE) algorithms can perform well on continuous optimization problems (CoOPs), it is still a challenging task to design an efficient binary-coded DE algorithm. Inspired by the learning mechanism of…
We study inverse boundary problems for evolutionary PDEs using only a single passive boundary observation, where data from an unknown internal source propagate through an unknown medium without active inputs. The goal is the simultaneous…
Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant…
Estimating the stability boundary is a fundamental and challenging problem in transient stability studies. It is known that a proper level set of a Lyapunov function or an energy function can provide an inner approximation of the stability…
We propose some numerical schemes for forward-backward stochastic differential equations (FBSDEs) based on a new fundamental concept of transposition solutions. These schemes exploit time-splitting methods for the variation of constants…
Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281--354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert…
Decomposition has been the mainstream approach in the classic mathematical programming for multi-objective optimization and multi-criterion decision-making. However, it was not properly studied in the context of evolutionary multi-objective…
The Forward-Forward algorithm is an alternative learning method which consists of two forward passes rather than a forward and backward pass employed by backpropagation. Forward-Forward networks employ layer local loss functions which are…
In this paper we consider evolution equations in the abstract Hilbert space under the special conditions imposed on the operator at the right-hand side of the equation. We establish the method that allows us to formulate the existence and…
We present an application of recent well-posedness results in the theory of delay differential equations for ordinary differential equations arXiv:2308.04730 to a generalized population model for stem cell maturation. The weak approach…
The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the…