Related papers: Solving the Poisson Equation with Dirichlet data b…
We discuss the expressive power of neural networks which use the non-smooth ReLU activation function $\varrho(x) = \max\{0,x\}$ by analyzing the approximation theoretic properties of such networks. The existing results mainly fall into two…
In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…
We establish in this work approximation results of deep neural networks for smooth functions measured in Sobolev norms, motivated by recent development of numerical solvers for partial differential equations using deep neural networks. {Our…
I provide an introduction to the application of deep learning and neural networks for solving partial differential equations (PDEs). The approach, known as physics-informed neural networks (PINNs), involves minimizing the residual of the…
Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to…
We provide a systematic investigation of using physics-informed neural networks to compute Lyapunov functions. We encode Lyapunov conditions as a partial differential equation (PDE) and use this for training neural network Lyapunov…
The purpose of this work is the study of solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the…
We consider approximation rates of sparsely connected deep rectified linear unit (ReLU) and rectified power unit (RePU) neural networks for functions in Besov spaces $B^\alpha_{q}(L^p)$ in arbitrary dimension $d$, on general domains. We…
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…
Solutions of evolution equation generally lies in certain Bochner-Sobolev spaces, in which the solution may has regularity and integrability properties for the time variable that can be different for the space variables. Therefore, in this…
Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations…
The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a…
This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semilinear partial differential equation (PDE) with low-regular solutions in two/three dimensions. The method is obtained by combining…
Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…
This study explores the number of neurons required for a Rectified Linear Unit (ReLU) neural network to approximate multivariate monomials. We establish an exponential lower bound on the complexity of any shallow network approximating the…
This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the…
In this work, we explore the numerical solution of geometric shape optimization problems using neural network-based approaches. This involves minimizing a numerical criterion that includes solving a partial differential equation with…
Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE). In PINNs, the residual form of the PDE of interest and its boundary conditions are lumped into a composite objective…
The traditional limitations of neural networks in reliably generalizing beyond the convex hulls of their training data present a significant problem for computational physics, in which one often wishes to solve PDEs in regimes far beyond…
In this work, we propose data-integrated neural networks (DataInNet) for solving partial differential equations (PDEs), offering a novel approach to leveraging data (e.g., source terms, initial conditions, and boundary conditions). The core…