This article introduces GIT-Net, a deep neural network architecture for approximating Partial Differential Equation (PDE) operators, inspired by integral transform operators. GIT-NET harnesses the fact that differential operators commonly used for defining PDEs can often be represented parsimoniously when expressed in specialized functional bases (e.g., Fourier basis). Unlike rigid integral transforms, GIT-Net parametrizes adaptive generalized integral transforms with deep neural networks. When compared to several recently proposed alternatives, GIT-Net's computational and memory requirements scale gracefully with mesh discretizations, facilitating its application to PDE problems on complex geometries. Numerical experiments demonstrate that GIT-Net is a competitive neural network operator, exhibiting small test errors and low evaluations across a range of PDE problems. This stands in contrast to existing neural network operators, which typically excel in just one of these areas.

这篇文章介绍了GIT-Net，一种深度神经网络架构，用于逼近偏微分方程算子，受到积分变换算子的启发。与刚性积分变换不同，GIT-Net使用深度神经网络参数化自适应广义积分变换。与其他最近提出的替代方法相比，GIT-Net的计算和内存需求随着网格离散化的扩展而平稳增长，促进其在复杂几何体上应用于PDE问题。数值实验表明，GIT-Net是一个具竞争力的神经网络算子，在一系列PDE问题中表现出较小的测试误差和低评估，与现有的神经网络算子形成对比，后者通常在这些领域中只有一个方面表现优异。

GIT-Net：运算符学习的广义积分变换