相关论文: Deep Hilbert--Galerkin Methods for Infinite-Dimens…

We consider the nonlinear Kolmogorov equation posed in a Hilbert space $H$, not necessarily of finite dimension. This model was recently studied by Cox et al. [24] in the framework of weak convergence rates of stochastic wave models. Here,…

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…

Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on…

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…

Neural networks are increasingly recognized as a powerful numerical solution technique for partial differential equations (PDEs) arising in diverse scientific computing domains, including quantum many-body physics. In the context of…

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…

We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…

In this paper, a highly parallel and derivative-free martingale neural network learning method is proposed to solve Hamilton-Jacobi-Bellman (HJB) equations arising from stochastic optimal control problems (SOCPs), as well as general…

This paper is concerned with developing accurate and efficient discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary…

Accurate approximation of scalar-valued functions from sample points is a key task in computational science. Recently, machine learning with Deep Neural Networks (DNNs) has emerged as a promising tool for scientific computing, with…

We study the derivative-informed learning of nonlinear operators between infinite-dimensional separable Hilbert spaces by neural networks. Such operators can arise from the solution of partial differential equations (PDEs), and are used in…

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…

Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton-Jacobi-Bellman (HJB) equations, which are notoriously difficult when the state dimension is large. Existing strategies for high-dimensional…

In this paper, we propose a random gradient-free method for optimization in infinite dimensional Hilbert spaces, applicable to functional optimization in diverse settings. Though such problems are often solved through finite-dimensional…

This paper presents a physics-informed machine learning approach for synthesizing optimal feedback control policy for infinite-horizon optimal control problems by solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation(PDE).…

A procedure for the numerical approximation of high-dimensional Hamilton-Jacobi-Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a…

We mathematically analyze and numerically study an actor-critic machine learning algorithm for solving high-dimensional Hamilton-Jacobi-Bellman (HJB) partial differential equations from stochastic control theory. The architecture of the…

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder…

This manuscript is devoted to the study of a class of nonlinear non-instantaneous impulsive first order abstract retarded type functional differential equations in an arbitrary separable Hilbert space H. A new set of sufficient conditions…

In this paper, we perform the convergence analysis of unsupervised Legendre--Galerkin neural networks (ULGNet), a deep-learning-based numerical method for solving partial differential equations (PDEs). Unlike existing deep learning-based…