相关论文: Cell-average based neural network method for hyper…

We introduce (U)NFV, a modular neural network architecture that generalizes classical finite volume (FV) methods for solving hyperbolic conservation laws. Hyperbolic partial differential equations (PDEs) are challenging to solve,…

The active flux (AF) method is a compact high-order finite volume method originally proposed for solving hyperbolic conservation laws, in which cell averages and point values at cell interfaces are evolved simultaneously. This paper…

Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…

We present a novel approach that integrates unfitted finite element methods and neural networks to approximate partial differential equations on complex geometries. Easy-to-generate background meshes (e.g., a simple Cartesian mesh) that cut…

We address the approximation of entropy solutions to initial-boundary value problems for nonlinear strictly hyperbolic conservation laws using neural networks. A general and systematic framework is introduced for the design of efficient and…

We propose an upwind finite volume method for a system of two kinetic equations in one dimension that are coupled through nonlocal interaction terms. These cross-interaction systems were recently obtained as the mean-field limit of a…

We present a subspace method based on neural networks (SNN) for solving the partial differential equation with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a…

A novel approach is presented for the solution of instantaneous chemical equilibrium problems. The chemical equilibrium can be considered, due to its intrinsically local character, as a mapping of the three-dimensional parameter space…

We compare the Finite Element Method (FEM) simulation of a standard Partial Differential Equation thermal problem of a plate with a hole with a Neural Network (NN) simulation. The largest deviation from the true solution obtained from FEM…

We propose a conservative energy method based on neural networks with subdomains for solving variational problems (CENN), where the admissible function satisfying the essential boundary condition without boundary penalty is constructed by…

In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of level set function, the…

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…

We develop a method to learn physical systems from data that employs feedforward neural networks and whose predictions comply with the first and second principles of thermodynamics. The method employs a minimum amount of data by enforcing…

Neural network-based methods for solving differential equations have been gaining traction. They work by improving the differential equation residuals of a neural network on a sample of points in each iteration. However, most of them employ…

Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FDNet), to learn…

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…

In this study, we analyzed the problem of accelerating the linear average consensus algorithm for complex networks. We propose a data-driven approach to tuning the weights of temporal (i.e., time-varying) networks using deep learning…

A neural network architecture is presented that exploits the multilevel properties of high-dimensional parameter-dependent partial differential equations, enabling an efficient approximation of parameter-to-solution maps, rivaling…

Various traditional numerical methods for solving initial value problems of differential equations often produce local solutions near the initial value point, despite the problems having larger interval solutions. Even current popular…

We introduce in this paper new and very effective numerical methods based on neural networks for the approximation of the mean curvature flow of either oriented or non-orientable surfaces. To learn the correct interface evolution law, our…