Related papers: PDGM: a Neural Network Approach to Solve Path-Depe…

Solving and optimising Partial Differential Equations (PDEs) in geometrically parameterised domains often requires iterative methods, leading to high computational and time complexities. One potential solution is to learn a direct mapping…

High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE…

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…

The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…

In this paper, we develop an adaptive Generalized Multiscale Discontinuous Galerkin Method (GMs-DGM) for a class of high-contrast flow problems, and derive a-priori and a-posteriori error estimates for the method. Based on the a-posteriori…

In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could…

The paper concerns classical solution of path-dependent partial differential equations (PPDEs) with coefficients depending on both variables of path and path-valued measure, which are crucial to understanding large-scale mean-field…

In this paper we present a new multiscale discontinuous Petrov--Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of Petrov--Galerkin version of…

We present the Deep Picard Iteration (DPI) method, a new deep learning approach for solving high-dimensional partial differential equations (PDEs). The core innovation of DPI lies in its use of Picard iteration to reformulate the typically…

The article proposes formulating and codifying a set of applied numerical methods, coined as Deep Learning Discrete Calculus (DLDC), that uses the knowledge from discrete numerical methods to interpret the deep learning algorithms through…

We propose conformable Adomian decomposition method (CADM) for fractional partial differential equations (FPDEs). This method is a new Adomian decomposition method (ADM) based on conformable derivative operator (CDO) to solve FPDEs. At the…

Probabilistic dependency graphs (PDGs) are a flexible class of probabilistic graphical models, subsuming Bayesian Networks and Factor Graphs. They can also capture inconsistent beliefs, and provide a way of measuring the degree of this…

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework…

We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of partial differential equations (PDEs). Our primary concern is…

This study proposes an approach based on a perturbation technique to construct global solutions to dynamic stochastic general equilibrium models (DSGE). The main idea is to expand a solution in a series of powers of a small parameter…

We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the…

Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately…

In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the…

Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus (see [15]), in the case of finite-dimensional…

In recent years, deep learning methods, exemplified by Physics-Informed Neural Networks (PINNs), have been widely applied to the numerical solution of differential equations. However, these methods may suffer from limited accuracy, high…