Related papers: Solving Nonlinear and High-Dimensional Partial Dif…
This study presents a meshfree two-dimensional fractional-order Element-Free Galerkin (2D f-EFG) method as a viable alternative to conventional mesh-based FEM for a numerical solution of (spatial) fractional-order differential equations…
Recent advances in deep learning makes solving parabolic partial differential equations (PDEs) in high dimensional spaces possible via forward-backward stochastic differential equation (FBSDE) formulations. The implementation of most…
We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence…
This paper proposes and analyzes two neural network methods to solve the master equation for finite-state mean field games (MFGs). Solving MFGs provides approximate Nash equilibria for stochastic, differential games with finite but large…
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem…
We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical…
Non-linear convection-reaction-diffusion (CRD) partial differential equations (PDEs) are crucial for modeling complex phenomena in fields such as biology, ecology, population dynamics, physics, and engineering. Numerical approximation of…
This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a new developed…
Machine learning, especially physics-informed neural networks (PINNs) and their neural network variants, has been widely used to solve problems involving partial differential equations (PDEs). The successful deployment of such methods…
In this thesis we develop techniques to efficiently solve numerical Partial Differential Equations (PDEs) using Graphical Processing Units (GPUs). Focus is put on both performance and re--usability of the methods developed, to this end a…
This paper aims to present a local discontinuous Galerkin (LDG) method for solving backward stochastic partial differential equations (BSPDEs) with Neumann boundary conditions. We establish the $L^2$-stability and optimal error estimates of…
In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep…
We introduce Discontinuous Galerkin Finite Element Operator Network (DG--FEONet), a data-free operator learning framework that combines the strengths of the discontinuous Galerkin (DG) method with neural networks to solve parametric partial…
This script offers an implementation-oriented introduction to deep learning methods for solving and estimating high-dimensional dynamic stochastic models in economics and finance. Its starting point is the curse of dimensionality:…
Probabilistic graphical modeling (PGM) provides a framework for formulating an interpretable generative process of data and expressing uncertainty about unknowns, but it lacks flexibility. Deep learning (DL) is an alternative framework for…
We consider a semilinear parabolic equation with a large class of nonlinearities without any growth conditions. We discretize the problem with a discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit Euler scheme)…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
This work develops a robust and efficient framework of the adjoint gradient-based aerodynamic shape optimization (ASO) using high-order discontinuous Galerkin methods (DGMs) as the CFD solver. The adjoint-enabled gradients based on…
Numerical solution of partial differential equations (PDEs) plays a vital role in various fields of science and engineering. In recent years, deep neural networks (DNNs) have emerged as a powerful tool for solving PDEs, leveraging their…
We use the ideas of goal-oriented error estimation and adaptivity to design and implement an efficient adaptive algorithm for approximating linear quantities of interest derived from solutions to elliptic partial differential equations…