Related papers: Optimised finite difference computation from symbo…
Differential Equations are among the most important Mathematical tools used in creating models in the science, engineering, economics, mathematics, physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine, environmental…
This paper deals with numerical solutions to a partial differential equation of fractional order. Generally this type of equation describes a transition from anomalous diffusion to transport processes. From a phenomenological point of view,…
The idea of neural Ordinary Differential Equations (ODE) is to approximate the derivative of a function (data model) instead of the function itself. In residual networks, instead of having a discrete sequence of hidden layers, the…
Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
The physics-informed neural network (PINN) is effective in solving the partial differential equation (PDE) by capturing the physics constraints as a part of the training loss function through the Automatic Differentiation (AD). This study…
The goal of this paper is to investigate an approach for derivative-free optimization that has not received sufficient attention in the literature and is yet one of the simplest to implement and parallelize. It consists of computing…
Traditional numerical techniques for solving time-dependent partial-differential-equation (PDE) initial-value problems (IVPs) store a truncated representation of the function values and some number of their time derivatives at each time…
We present a new finite element method, called $\phi$-FEM, to solve numerically elliptic partial differential equations with natural (Neumann or Robin) boundary conditions using simple computational grids, not fitted to the boundary of the…
Modeling real processes often results in several suitable models. In order to be able to distinguish, or discriminate, which model best represents a phenomenon, one is interested, e.g., in so-called T-optimal designs. These consist of the…
We propose UPOQA, a derivative-free optimization algorithm for partially separable unconstrained problems, leveraging quadratic interpolation and a structured trust-region framework. By decomposing the objective into element functions,…
Finite differences, as a subclass of direct methods in the calculus of variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
In this paper, we propose a novel Hermite weighted essentially non-oscillatory (HWENO) fast sweeping method to solve the static Hamilton-Jacobi equations efficiently. During the HWENO reconstruction procedure, the proposed method is built…
We consider the optimal control problem of a general nonlinear spatio-temporal system described by Partial Differential Equations (PDEs). Theory and algorithms for control of spatio-temporal systems are of rising interest among the…
The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of…
In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the $d$-dimensional advection-diffusion equation, our proposal is a two-level algorithm…
Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a…
In this article, we introduce an original hybrid quantum-classical algorithm based on a variational quantum algorithm for solving systems of differential equations. The algorithm relies on a spectral decomposition of the trial functions…
The take-home message of this paper is that solving optimal control problems can be computationally straightforward, provided that differentiable partial differential equation (PDE) solvers are available. Although this might seem to be a…