Related papers: A mesh-free method for interface problems using th…
Introducing flexibility in the time-discretisation mesh can improve convergence and computational time when solving differential equations numerically, particularly when the solutions are discontinuous, as commonly found in control problems…
In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already…
We show that an optimal finite element mesh refinement algorithm for a prototypical elliptic PDE can be learned by a recurrent neural network with a fixed number of trainable parameters independent of the desired accuracy and the input…
Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods are best suited to regular rectangular grids, which leads to…
As an alternative to PINNs, a Deep Ritz framework is proposed to solve fully nonlinear PDEs. A least-squares algorithm is advocated to decouple the nonlinearities from the variational features of several fully nonlinear PDEs. A splitting…
For the optimal control problem governed by elliptic equations with interfaces, we present a numerical method based on the Hansbo's Nitsche-XFEM. We followed the Hinze's variational discretization concept to discretize the continuous…
We propose in this paper a multilevel correction method to solve optimal control problems constrained by elliptic equations with the finite element method. In this scheme, solving optimization problem on the finest finite element space is…
We propose a boundary-corrected weak Galerkin mixed finite element method for solving elliptic interface problems in 2D domains with curved interfaces. The method is formulated on body-fitted polygonal meshes, where interface edges are…
In this paper, we introduce the locally conservative enriched immersed finite element method (EIFEM) to tackle the elliptic problem with interface. The immersed finite element is useful for handling interface with mesh unfit with the…
This paper proposes a novel Machine Learning-based approach to solve a Poisson problem with mixed boundary conditions. Leveraging Graph Neural Networks, we develop a model able to process unstructured grids with the advantage of enforcing…
Deep learning has been widely applied to solve partial differential equations (PDEs) in computational fluid dynamics. Recent research proposed a PDE correction framework that leverages deep learning to correct the solution obtained by a PDE…
Solutions of partial differential equations (PDEs) on manifolds have provided important applications in different fields in science and engineering. Existing methods are majorly based on discretization of manifolds as implicit functions,…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
We demonstrate how meshfree finite difference methods can be applied to solve vector Poisson problems with electric boundary conditions. In these, the tangential velocity and the incompressibility of the vector field are prescribed at the…
Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…
This paper proposes a meshless deep learning algorithm, enriched physics-informed neural networks (EPINNs), to solve dynamic Poisson-Nernst-Planck (PNP) equations with strong coupling and nonlinear characteristics. The EPINNs takes the…
Partial differential equations (PDEs) on surfaces are fundamental to scientific computing and geometry processing. A popular approach to solving PDEs on surfaces is the finite element method (FEM), where the surface is divided into discrete…
Real-time simulation of elastic structures is essential in many applications, from computer-guided surgical interventions to interactive design in mechanical engineering. The Finite Element Method is often used as the numerical method of…
We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…
Meshfree particle methods, such as Smoothed Particle Hydrodynamics (SPH) and the Moving Particle Semi-Implicit (MPS) method, are widely used to simulate complex free-surface and multiphase flows. A key challenge in these methods is the…