相关论文: HNS: An Efficient Hermite Neural Solver for Solvin…
Neural Stochastic Differential Equations (Neural SDEs) provide a principled framework for modeling continuous-time stochastic processes and have been widely adopted in fields ranging from physics to finance. Recent advances suggest that…
In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous…
In this paper, a meshless Hermite-HDMR finite difference method is proposed to solve high-dimensional Dirichlet problems. The approach is based on the local Hermite-HDMR expansion with an additional smoothing technique. First, we introduce…
The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the…
Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace.…
This paper presents a novel neural network design that learns the heuristic for Large Neighborhood Search (LNS). LNS consists of a destroy operator and a repair operator that specify a way to carry out the neighborhood search to solve the…
In this paper, we propose a novel $hr$-adaptive finite element method, enhanced by neural networks, for parabolic equations. The main challenge of the conventional $h$-adaptive finite element method is interpolating the finite element…
Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored…
Fast and accurate solutions of time-dependent partial differential equations (PDEs) are of pivotal interest to many research fields, including physics, engineering, and biology. Generally, implicit/semi-implicit schemes are preferred over…
Solving partial differential equations is difficult. Recently proposed neural resolution-invariant models, despite their effectiveness and efficiency, usually require equispaced spatial points of data. However, sampling in spatial domain is…
We introduce the Neural Hodge Corrective Solver (NHCS), a hybrid iterative-neural framework for partial differential equations that embeds learned corrective operators within the Discrete Exterior Calculus (DEC) formulation. The method…
A number of basic image processing tasks, such as any geometric transformation require interpolation at subpixel image values. In this work we utilize the multidimensional coordinate Hermite spline interpolation defined on non-equal spaced,…
Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…
This paper proposes two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (NPDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully…
Deep learning-based methods have shown remarkable effectiveness in solving PDEs, largely due to their ability to enable fast simulations once trained. However, despite the availability of high-performance computing infrastructure, many…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
We first consider the problem of approximating a few eigenvalues of a rational matrix-valued function closest to a prescribed target. It is assumed that the proper rational part of the rational matrix-valued function is expressed in the…
We extend the finite element interpolated neural network (FEINN) framework from partial differential equations (PDEs) with weak solutions in $H^1$ to PDEs with weak solutions in $H(\textbf{curl})$ or $H(\textbf{div})$. To this end, we…
Physics-informed neural networks (PINNs) show great advantages in solving partial differential equations. In this paper, we for the first time propose to study conformable time fractional diffusion equations by using PINNs. By solving the…
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…