Related papers: Solving Differential Equation with Constrained Mul…

A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface…

This work develops a computational framework that combines physics-informed neural networks with multi-patch isogeometric analysis to solve partial differential equations on complex computer-aided design geometries. The method utilizes…

We study existence and uniqueness of solutions to a class of nonlinear degenerate parabolic equations, in bounded domains. We show that there exists a unique solution which satisfies possibly inhomogeneous Dirichlet boundary conditions. To…

Approximating solutions of ordinary and partial differential equations constitutes a significant challenge. Based on functional expressions that inherently depend on neural networks, neural forms are specifically designed to precisely…

The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the…

We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An…

We propose deep neural network algorithms to calculate efficient frontier in some Mean-Variance and Mean-CVaR portfolio optimization problems. We show that we are able to deal with such problems when both the dimension of the state and the…

In this paper, feedforward neural networks are presented that have nonlinear weight functions based on look--up tables, that are specially smoothed in a regularization called the diffusion. The idea of such a type of networks is based on…

There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data…

In this paper, we present and compare four methods to enforce Dirichlet boundary conditions in Physics-Informed Neural Networks (PINNs) and Variational Physics-Informed Neural Networks (VPINNs). Such conditions are usually imposed by adding…

In theory, boundary and initial conditions are important for the wellposedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method…

This paper is concerned with the approximation of the solution of partial differential equations by means of artificial neural networks. Here a feedforward neural network is used to approximate the solution of the partial differential…

We present an approach for solving optimal Dirichlet boundary control problems of nonlinear optics by using deep learning. For computing high resolution approximations of the solution to the nonlinear wave model, we propose higher order…

We motivate the use of neural networks for the construction of numerical solutions to differential equations. We prove that there exists a feed-forward neural network that can arbitrarily minimise an objective function that is zero at the…

We introduce Differentiable Neural Radiosity, a novel method of representing the solution of the differential rendering equation using a neural network. Inspired by neural radiosity techniques, we minimize the norm of the residual of the…

This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations…

The challenge of applying learned knowledge from one domain to solve problems in another related but distinct domain, known as transfer learning, is fundamental in operator learning models that solve Partial Differential Equations (PDEs).…

We develop a mathematically rigorous framework for multilayer neural networks in the mean field regime. As the network's widths increase, the network's learning trajectory is shown to be well captured by a meaningful and dynamically…

Recently, innovative adaptations of the Ritz Method incorporating deep learning have been developed, known as the Deep Ritz Method. This approach employs a neural network as the test function for variational problems. However, the neural…

Solving partial differential equations (PDEs) is the canonical approach for understanding the behavior of physical systems. However, large scale solutions of PDEs using state of the art discretization techniques remains an expensive…