Related papers: Pseudo-time regularization for PDE with solution-d…
An adaptive regularization strategy for stabilizing Newton-like iterations on a coarse mesh is developed in the context of adaptive finite element methods for nonlinear PDE. Existence, uniqueness and approximation properties are known for…
Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Recently, mostly due to the high computational cost of traditional solution techniques, deep neural network based surrogates have gained…
This paper focuses on the regularization of backward time-fractional diffusion problem on unbounded domain. This problem is well-known to be ill-posed, whence the need of a regularization method in order to recover stable approximate…
In this paper we the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type. We carry out a convergence analysis in the sense of regularization methods and discuss…
In this paper, by employing the asymptotic expansion method, we prove the existence and uniqueness of a smoothing solution for a time-dependent nonlinear singularly perturbed partial differential equation (PDE) with a small-scale parameter.…
This article investigates the non-stationary reaction-diffusion-advection equation, emphasizing solutions with internal layers and the associated inverse problems. We examine a nonlinear singularly perturbed partial differential equation…
We propose a novel framework, Continuous_Time Attention, which infuses partial differential equations (PDEs) into the Transformer's attention mechanism to address the challenges of extremely long input sequences. Instead of relying solely…
We present an adaptive regularization algorithm that can be effectively applied to the optimization problem in deep learning framework. Our regularization algorithm aims to take into account the fitness of data to the current state of model…
This paper is concerned with backward problem for nonlinear space fractional diffusion with additive noise on the right-hand side and the final value. To regularize the instable solution, we develop some new regularized method for solving…
In this paper, a nonsmooth semilinear parabolic partial differential equation (PDE) is considered. For a reduced basis (RB) approach, a space-time formulation is used to develop a certified a-posteriori error estimator. This error estimator…
We introduce a novel formulation for curvature regularization by penalizing normal curvatures from multiple directions. This total normal curvature regularization is capable of producing solutions with sharp edges and precise isotropic…
We interpret steady linear statistical inverse problems as artificial dynamic systems with white noise and introduce a stochastic differential equation (SDE) system where the inverse of the ending time $T$ naturally plays the role of the…
Tensor decomposition methods allow us to learn the parameters of latent variable models through decomposition of low-order moments of data. A significant limitation of these algorithms is that there exists no general method to regularize…
In this paper, we develop an analytical framework for the partial differential equation underlying the consensus-based optimization model. The main challenge arises from the nonlinear, nonlocal nature of the consensus point, coupled with a…
We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity.…
PDE-constrained optimal control problems require regularisation to ensure well-posedness, introducing small perturbations that make the solutions challenging to approximate accurately. We propose a finite element approach that couples both…
This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain…
In this work we consider the regularization of a supervised learning problem by partial differential equations (PDEs) and derive error bounds for the obtained approximation in terms of a PDE error term and a data error term. Assuming that…
This paper is concerned with developing and analyzing two novel implicit temporal discretization methods for the stochastic semilinear wave equations with multiplicative noise. The proposed methods are natural extensions of well-known…
A method is developed within an adaptive framework to solve quasilinear diffusion problems with internal and possibly boundary layers starting from a coarse mesh. The solution process is assumed to start on a mesh where the problem is badly…