相关论文: Deep Finite Volume Method for Partial Differential…

Physics-informed neural networks (PINNs) are effective in solving integer-order partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly…

Physics-informed neural networks (PINNs) have attracted significant attention for solving partial differential equations (PDEs) in recent years because they alleviate the curse of dimensionality that appears in traditional methods. However,…

This paper focuses on proposing a deep learning initialized iterative method (Int-Deep) for low-dimensional nonlinear partial differential equations (PDEs). The corresponding framework consists of two phases. In the first phase, an…

In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational…

The wave equation is an important physical partial differential equation, and in recent years, deep learning has shown promise in accelerating or replacing traditional numerical methods for solving it. However, existing deep learning…

We present the variational multiscale (VMS) method for partial differential equations (PDEs) with stochastic coefficients and source terms. We use it as a method for generating accurate coarse-scale solutions while accounting for the effect…

The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, algorithmically, the standard adaptive finite…

Accurately depicting multiphysics interactions in interfacial systems requires computational frameworks capable of reconciling geometric adaptability with strict conservation fidelity. However, traditional spatiotemporal discretisation…

We review sharpening methods for finite volume schemes, with an emphasis on the basic structure of sharpening methods. It covers high order methods and non linear techniques for linear advection, Glimm's method, anti-diffusion techniques,…

In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep…

Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by…

Recent developments in mechanical, aerospace, and structural engineering have driven a growing need for efficient ways to model and analyse structures at much larger and more complex scales than before. While established numerical methods…

We show that the DDR method can be interpreted as defining a computable consistent discrete $\mathrm{L}^2$ product on a conforming FES defined by PDEs. Without modifying the numerical method itself, this point of view provides an…

Enforcing exact macroscopic conservation laws, such as mass and energy, in neural partial differential equation (PDE) solvers is computationally challenging in high dimensions. Traditional discrete projections rely on deterministic…

Variational Physics-Informed Neural Networks (VPINNs) utilize a variational loss function to solve partial differential equations, mirroring Finite Element Analysis techniques. Traditional hp-VPINNs, while effective for high-frequency…

This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the…

Utilizing physics-informed neural networks (PINN) to solve partial differential equations (PDEs) becomes a hot issue and also shows its great powers, but still suffers from the dilemmas of limited predicted accuracy in the sampling domain…

Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we…

Physics-informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering…

Despite rapid improvements in the performance of central processing unit (CPU), the calculation cost of simulating chemically reacting flow using CFD remains infeasible in many cases. The application of the convolutional neural networks…