相关论文: NeuraLSP: An Efficient and Rigorous Neural Left Si…

We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of…

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…

There have been growing interests in leveraging experimental measurements to discover the underlying partial differential equations (PDEs) that govern complex physical phenomena. Although past research attempts have achieved great success…

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc.…

Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order…

In this work, a non-gradient descent learning (NGDL) scheme was proposed for deep feedforward neural networks (DNN). It is known that an autoencoder can be used as the building blocks of the multi-layer perceptron (MLP) DNN, the MLP is…

The 'exact subgraph' approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational…

Deep Neural Networks have achieved remarkable success relying on the developing high computation capability of GPUs and large-scale datasets with increasing network depth and width in image recognition, object detection and many other…

This work proposes a framework, embedded within the Performance Estimation framework (PEP), for obtaining worst-case performance guarantees on stochastic first-order methods. Given a first-order method, a function class, and a noise model…

We propose a preconditioner to accelerate the convergence of the GMRES iterative method for solving the system of linear equations obtained from discretize-then-optimize approach applied to optimal control problems constrained by a partial…

We address the problem of compressed sensing using a deep generative prior model and consider both linear and learned nonlinear sensing mechanisms, where the nonlinear one involves either a fully connected neural network or a convolutional…

Numerical discretisations of partial differential equations (PDEs) can be written as discrete convolutions, which, themselves, are a key tool in AI libraries and used in convolutional neural networks (CNNs). We therefore propose to…

Neural network-based approaches have recently shown significant promise in solving partial differential equations (PDEs) in science and engineering, especially in scenarios featuring complex domains or incorporation of empirical data. One…

We present a neural network-based method for solving linear and nonlinear partial differential equations, by combining the ideas of extreme learning machines (ELM), domain decomposition and local neural networks. The field solution on each…

We develop numerical methods for elliptic systems governed by partial segregation constraints, in which three nonnegative components are required to have a vanishing pointwise product throughout the domain. This constraint enforces that at…

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as…

Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…

Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs) by ensuring that the outputs and gradients of deep learning models adhere to the governing equations. However, constrained by…

A DualTPD method is proposed for solving nonlinear partial differential equations. The method is characterized by three main features. First, decoupling via Fenchel--Rockafellar duality is achieved, so that nonlinear terms are discretized…

At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly,…