Related papers: Solving High-Dimensional PDEs Using Linearized Neu…

In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give an algorithm to train a ReLU DNN with one hidden layer to *global optimality* with runtime…

While deep learning has achieved remarkable success in solving partial differential equations (PDEs), it still faces significant challenges, particularly when the PDE solutions have low regularity or singularities. To address these issues,…

Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when…

This paper establishes an approximation theorem for randomized neural networks (RaNNs) whose hidden-layer parameters are uniformly sampled from a prescribed bounded domain. Our analysis shows that, for RaNNs of the form $\mathop{\sum}_i W_i…

We consider the use of extreme learning machines (ELM) for computational partial differential equations (PDE). In ELM the hidden-layer coefficients in the neural network are assigned to random values generated on $[-R_m,R_m]$ and fixed,…

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…

This work addresses the accurate and efficient simulation of physical phenomena governed by parametric Partial Differential Equations (PDEs) characterized by varying boundary conditions, where parametric instances modify not only the…

We present a probabilistic framework for nonlinearities, based on doubly truncated Gaussian distributions. By setting the truncation points appropriately, we are able to generate various types of nonlinearities within a unified framework,…

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…

Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing…

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…

Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which…

This article discusses the uncertainty quantification (UQ) for time-independent linear and nonlinear partial differential equation (PDE)-based systems with random model parameters carried out using sampling-free intrusive stochastic…

Explicit, unconditionally stable, high-order schemes for the approximation of some first- andsecond-order linear, time-dependent partial differential equations (PDEs) are proposed.The schemes are based on a weak formulation of a…

We propose an abstract discontinuous Galerkin neural network (DGNN) framework for analyzing the convergence of least-squares methods based on the residual minimization when feasible solutions are neural networks. Within this framework, we…

This work presents a deep learning-based framework for the solution of partial differential equations on complex computational domains described with computer-aided design tools. To account for the underlying distribution of the training…

Random Feature Methods (RFMs) and their variants such as extreme learning machine finite-basis physics-informed neural networks (ELM-FBPINNs) offer a scalable approach for solving partial differential equations (PDEs) by using localized,…

Partial-differential-equation (PDE)-constrained optimization is a well-worn technique for acquiring optimal parameters of systems governed by PDEs. However, this approach is limited to providing a single set of optimal parameters per…

Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically…

Classifiers that are linear in their parameters, and trained by optimizing a convex loss function, have predictable behavior with respect to changes in the training data, initial conditions, and optimization. Such desirable properties are…