Related papers: A Hybrid Kernel-Free Boundary Integral Method with…
This work addresses a novel version of the kernel-free boundary integral (KFBI) method for solving elliptic PDEs with implicitly defined irregular boundaries and interfaces. We focus on boundary value problems and interface problems, which…
The bidomain equations have been widely used to mathematically model the electrical activity of the cardiac tissue. In this work, we present a potential theory-based Cartesian grid method which is referred as the kernel-free boundary…
This work presents a generalized boundary integral method for elliptic equations on surfaces, encompassing both boundary value and interface problems. The method is kernel-free, implying that the explicit analytical expression of the kernel…
A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the…
We propose a method combining boundary integral equations and neural networks (BINet) to solve partial differential equations (PDEs) in both bounded and unbounded domains. Unlike existing solutions that directly operate over original PDEs,…
We develop a boundary integral equation solver for elliptic partial differential equations on complex \threed geometries. Our method is efficient, high-order accurate and robustly handles complex geometries. A key component is our singular…
A second-order accurate kernel-free boundary integral method is presented for Stokes and Navier boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose…
Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates…
This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations…
Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we…
In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…
This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we study the inverse problem of determining the…
Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis. Traditional numerical methods, like finite difference and finite element approaches, are widely used to solve governing…
Partial differential equations (PDEs) are widely used to describe relevant phenomena in dynamical systems. In real-world applications, we commonly need to combine formal PDE models with (potentially noisy) observations. This is especially…
In theory, diffusion curves promise complex color gradations for infinite-resolution vector graphics. In practice, existing realizations suffer from poor scaling, discretization artifacts, or insufficient support for rich boundary…
The singularities that arise in elliptic boundary value problems are treated locally by a singular function boundary integral method. This method extracts the leading singular coefficients from a series expansion that describes the local…
The boundary integral method is an efficient approach for solving time-harmonic acoustic obstacle scattering problems. The main computational task is the evaluation of an oscillatory boundary integral at each discretization point of the…
Conventionally, piecewise polynomials have been used in the boundary elements method (BEM) to approximate unknown boundary values. Since infinitely smooth radial basis functions (RBFs) are more stable and accurate than the polynomials for…
This article describes a numerical method based on the dual reciprocity boundary elements method (DRBEM) for solving some well-known nonlinear parabolic partial differential equations (PDEs). The equations include the classic and…
Edge detection is a cornerstone of image processing, yet existing methods often face critical limitations. Traditional deep learning edge detection methods require extensive training datasets and fine-tuning, while classical techniques…