Elliptic partial differential equations (PDEs) are a major class of
time-independent PDEs that play a key role in many scientific and engineering
domains such as fluid dynamics, plasma physics, and solid mechanics. Recently,
neural operators have emerged as a promising technique to solve elliptic PDEs
more efficiently by directly mapping the input to solutions. However, existing
networks typically cannot handle complex geometries and inhomogeneous boundary
values present in the real world. Here we introduce Boundary-Embedded Neural
Operators (BENO), a novel neural operator architecture that embeds the complex
geometries and inhomogeneous boundary values into the solving of elliptic PDEs.
Inspired by classical Green's function, BENO consists of two branches of Graph
Neural Networks (GNNs) for interior source term and boundary values,
respectively. Furthermore, a Transformer encoder maps the global boundary
geometry into a latent vector which influences each message passing layer of
the GNNs. We test our model extensively in elliptic PDEs with various boundary
conditions. We show that all existing baseline methods fail to learn the
solution operator. In contrast, our model, endowed with boundary-embedded
architecture, outperforms state-of-the-art neural operators and strong
baselines by an average of 60.96\%. Our source code can be found
this https URL

通过嵌入边界和复杂几何形状解决椭圆型偏微分方程的新型神经算子 (Boundary-Embedded Neural Operators)，在不均匀边界条件和复杂几何形状上的实验证明，该算子优于现有算法和基准算子，平均提升效果达 60.96%。

BENO: 嵌入边界的神经算子用于椭圆型偏微分方程

BENO: Boundary-embedded Neural Operators for Elliptic PDEs

In this paper, neural network approximation methods are developed for
elliptic partial differential equations with multi-frequency solutions. Neural
network work approximation methods have advantages over classical approaches in
that they can be applied without much concerns on the form of the differential
equations or the shape or dimension of the problem domain. When applied to
problems with multi-frequency solutions, the performance and accuracy of neural
network approximation methods are strongly affected by the contrast of the
high- and low-frequency parts in the solutions. To address this issue, domain
scaling and residual correction methods are proposed. The efficiency and
accuracy of the proposed methods are demonstrated for multi-frequency model
problems.

发展了神经网络逼近方法，用于具有多频解的椭圆偏微分方程。通过域缩放和残差校正方法解决了多频解中高频和低频部分的对比度问题，并展示了方法的效率和准确性。

利用领域缩放和残差校正方法增强的物理约束神经网络用于多频率椭圆问题

Enhanced physics-informed neural networks with domain scaling and  residual correction methods for multi-frequency elliptic problems

Machine learning has been successfully applied to various fields of
scientific computing in recent years. In this work, we propose a sparse radial
basis function neural network method to solve elliptic partial differential
equations (PDEs) with multiscale coefficients. Inspired by the deep mixed
residual method, we rewrite the second-order problem into a first-order system
and employ multiple radial basis function neural networks (RBFNNs) to
approximate unknown functions in the system. To aviod the overfitting due to
the simplicity of RBFNN, an additional regularization is introduced in the loss
function. Thus the loss function contains two parts: the $L_2$ loss for the
residual of the first-order system and boundary conditions, and the $\ell_1$
regularization term for the weights of radial basis functions (RBFs). An
algorithm for optimizing the specific loss function is introduced to accelerate
the training process. The accuracy and effectiveness of the proposed method are
demonstrated through a collection of multiscale problems with scale separation,
discontinuity and multiple scales from one to three dimensions. Notably, the
$\ell_1$ regularization can achieve the goal of representing the solution by
fewer RBFs. As a consequence, the total number of RBFs scales like
$\mathcal{O}(\varepsilon^{-n\tau})$, where $\varepsilon$ is the smallest scale,
$n$ is the dimensionality, and $\tau$ is typically smaller than $1$. It is
worth mentioning that the proposed method not only has the numerical
convergence and thus provides a reliable numerical solution in three dimensions
when a classical method is typically not affordable, but also outperforms most
other available machine learning methods in terms of accuracy and robustness.

提出了一种利用稀疏径向基函数神经网络的方法来解决具有多尺度系数的椭圆偏微分方程。该方法不仅在三维情况下具有数值收敛性并提供可靠的数值解，而且在准确性和鲁棒性方面优于大多数其他可用的机器学习方法。