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High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial…
The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…
Deep learning has become a popular tool across many scientific fields, including the study of differential equations, particularly partial differential equations. This work introduces the basic principles of deep learning and the Deep…
In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal…
This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and…
Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A…
Deep neural networks are powerful tools for approximating functions, and they are applied to successfully solve various problems in many fields. In this paper, we propose a neural network-based numerical method to solve partial differential…
We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we…
At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly,…
This paper introduces a new method based on Deep Galerkin Methods (DGMs) for solving high-dimensional stochastic Mean Field Games (MFGs). We achieve this by using two neural networks to approximate the unknown solutions of the MFG system…
A framework for performing dynamic mesh adaptation with the discontinuous Galerkin method (DGM) is presented. Adaptations include modifications of the local mesh step size (h-adaptation) and the local degree of the approximating polynomials…
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…
We establish the convergence of the deep Galerkin method (DGM), a deep learning-based scheme for solving high-dimensional nonlinear PDEs, for Hamilton-Jacobi-Bellman (HJB) equations that arise from the study of mean field control problems…
In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method (DGM) uses the PDE residual in the least-squares sense as the loss function…
Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques…
We propose a deep learning based discontinuous Galerkin method (D2GM) to solve hyperbolic equations with discontinuous solutions and random uncertainties. The main computational challenges for such problems include discontinuities of the…
Randomized neural networks (RNN) are a variation of neural networks in which the hidden-layer parameters are fixed to randomly assigned values and the output-layer parameters are obtained by solving a linear system by least squares. This…
Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical…
This paper analyzes the convergence rate of a deep Galerkin method for the weak solution (DGMW) of second-order elliptic partial differential equations on $\mathbb{R}^d$ with Dirichlet, Neumann, and Robin boundary conditions, respectively.…
In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such…