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.

翻译：本文提出GIT-Net，一种受积分变换算子启发、用于逼近偏微分方程（PDE）算子的深度神经网络架构。GIT-Net利用了常用于定义PDE的微分算子，在专用函数基（如傅里叶基）中表达时通常具有稀疏表示这一特性。与刚性积分变换不同，GIT-Net通过深度神经网络对自适应广义积分变换进行参数化。与近期提出的多种替代方法相比，GIT-Net的计算和内存需求随网格离散化程度呈优雅扩展，便于将其应用于复杂几何区域的PDE问题。数值实验表明，GIT-Net是一种具有竞争力的神经网络算子，在多种PDE问题中兼具小测试误差与低评估成本。这区别于现有神经网络算子——它们通常仅在某一方面表现突出。