相关论文: HNS: An Efficient Hermite Neural Solver for Solvin…

A novel H3N3-2$_\sigma$ interpolation approximation for the Caputo fractional derivative of order $\alpha\in(1,2)$ is derived in this paper, which improves the popular L2C formula with (3-$\alpha$)-order accuracy. By an interpolation…

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…

In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…

Numerical simulation of time-dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high-fidelity solvers are often prohibitively expensive for long-horizon or time-critical settings.…

Deep neural operators can learn nonlinear mappings between infinite-dimensional function spaces via deep neural networks. As promising surrogate solvers of partial differential equations (PDEs) for real-time prediction, deep neural…

Deep Neural Networks (DNNs) are capable of solving complex problems in domains related to embedded systems, such as image and natural language processing. To efficiently implement DNNs on a specific FPGA platform for a given cost criterion,…

We develop a fast solver for the spectral element method (SEM) applied to the two-sided fractional diffusion equation on uniform, geometric and graded meshes. By approximating the singular kernel with a degenerate kernel, we construct a…

The splitting method is a powerful method for solving partial differential equations. Various splitting methods have been designed to separate different physics, nonlinearities, and so on. Recently, a new splitting approach has been…

A high order finite difference method is proposed for unstructured meshes to simulate compressible inviscid/viscous flows with/without discontinuities. In this method, based on the strong form equation, the divergence of the flux on each…

A new penalty-free neural network method, PFNN-2, is presented for solving partial differential equations, which is a subsequent improvement of our previously proposed PFNN method [1]. PFNN-2 inherits all advantages of PFNN in handling the…

Graph Neural Networks (GNNs) have achieved state-of-the-art performance in recommender systems. Nevertheless, the process of searching and ranking from a large item corpus usually requires high latency, which limits the widespread…

We study a family of $H^m$-conforming piecewise polynomials based on artificial neural network, named as the finite neuron method (FNM), for numerical solution of $2m$-th order partial differential equations in $\mathbb{R}^d$ for any $m,d…

This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear…

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

Humans are able to recognize structured relations in observation, allowing us to decompose complex scenes into simpler parts and abstract the visual world in multiple levels. However, such hierarchical reasoning ability of human perception…

This paper presents new fast algorithms for Hermite interpolation and evaluation over finite fields of characteristic two. The algorithms reduce the Hermite problems to instances of the standard multipoint interpolation and evaluation…

Diabetic Peripheral Neuropathy (DPN) affects nearly half of diabetes patients, requiring early detection. Corneal Confocal Microscopy (CCM) enables non-invasive diagnosis, but automated methods suffer from inefficient feature extraction,…

Partial differential equations (PDEs) underpin quantitative descriptions across the physical sciences and engineering, yet high-fidelity simulation remains a major computational bottleneck for many-query, real-time, and design tasks.…

Multiscale modeling is an effective approach for investigating multiphysics systems with largely disparate size features, where models with different resolutions or heterogeneous descriptions are coupled together for predicting the system's…

High order accurate Hermite methods for the wave equation on curvilinear domains are presented. Boundaries are treated using centered compatibility conditions rather than more standard one-sided approximations. Both first-order-in-time…