Related papers: Numerical approximation based on deep convolutiona…
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…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…
Within recent years, considerable progress has been made regarding high-performance solvers for Partial Differential Equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely…
We introduce the deep multi-FBSDE method for robust approximation of coupled forward-backward stochastic differential equations (FBSDEs), focusing on cases where the deep BSDE method of Han, Jentzen, and E (2018) fails to converge. To…
We develop a numerical framework, the Deep Tangent Bundle (DTB) method, that is suitable for computing solutions of evolutionary partial differential equations (PDEs) in high dimensions. The main idea is to use the tangent bundle of an…
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…
In this paper, we propose and study neural network based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two…
To sidestep the curse of dimensionality when computing solutions to Hamilton-Jacobi-Bellman partial differential equations (HJB PDE), we propose an algorithm that leverages a neural network to approximate the value function. We show that…
We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in \cite{cstv}, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of…
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
We present approximation results and numerical experiments for the use of randomized neural networks within physics-informed extreme learning machines to efficiently solve high-dimensional PDEs, demonstrating both high accuracy and low…
In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of dimensionality. Specifically, we introduce a concept which we…
We investigate the potential of applying (D)NN ((deep) neural networks) for approximating nonlinear mappings arising in the finite element discretization of nonlinear PDEs (partial differential equations). As an application, we apply the…
Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for…
We present a lightweighted neural PDE representation to discover the hidden structure and predict the solution of different nonlinear PDEs. Our key idea is to leverage the prior of ``translational similarity'' of numerical PDE differential…
On the forefront of scientific computing, Deep Learning (DL), i.e., machine learning with Deep Neural Networks (DNNs), has emerged a powerful new tool for solving Partial Differential Equations (PDEs). It has been observed that DNNs are…
Recent work on Path-Dependent Partial Differential Equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using…
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 paper introduces a very simple and fast computation method for high-dimensional integrals to solve high-dimensional Kolmogorov partial differential equations (PDEs). The new machine learning-based method is obtained by solving a…