BENO：用于椭圆型偏微分方程的边界嵌入神经算子
BENO: Boundary-embedded Neural Operators for Elliptic PDEs
AI总结：
本文提出边界嵌入神经算子 BENO，通过双分支图神经网络与 Transformer 编码器将复杂几何和非齐次边界条件嵌入椭圆型偏微分方程求解，在多种边界条件测试中平均超越现有方法 60.96%。
摘要：
椭圆型偏微分方程（PDEs）是一类重要的时间无关偏微分方程，在流体动力学、等离子体物理和固体力学等许多科学与工程领域中发挥着关键作用。近年来，神经算子已成为一种有前景的技术，通过直接将输入映射到解来更高效地求解椭圆型偏微分方程。然而，现有网络通常无法处理现实世界中存在的复杂几何形状和非齐次边界值。在此，我们提出边界嵌入神经算子（BENO），这是一种新颖的神经算子架构，将复杂几何形状和非齐次边界值嵌入到椭圆型偏微分方程的求解过程中。受经典格林函数启发，BENO 由两个图神经网络（GNNs）分支组成，分别用于内部源项和边界值。此外，一个 Transformer 编码器将全局边界几何映射为潜在向量，该向量影响图神经网络每个消息传递层。我们在具有各种边界条件的椭圆型偏微分方程中广泛测试了我们的模型。我们表明，所有现有基线方法都未能学习到解算子。相比之下，我们的模型凭借边界嵌入架构，平均优于最先进的神经算子和强基线 60.96%。我们的源代码可在 https://github.com/AI4Science-WestlakeU/beno.git 找到。
英文摘要：
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 https://github.com/AI4Science-WestlakeU/beno.git.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
