Deep learning methods have access to be employed for solving physical systems governed by parametric partial differential equations (PDEs) due to massive scientific data. It has been refined to operator learning that focuses on learning non-linear mapping between infinite-dimensional function spaces, offering interface from observations to solutions. However, state-of-the-art neural operators are limited to constant and uniform discretization, thereby leading to deficiency in generalization on arbitrary discretization schemes for computational domain. In this work, we propose a novel operator learning algorithm, referred to as Dynamic Gaussian Graph Operator (DGGO) that expands neural operators to learning parametric PDEs in arbitrary discrete mechanics problems. The Dynamic Gaussian Graph (DGG) kernel learns to map the observation vectors defined in general Euclidean space to metric vectors defined in high-dimensional uniform metric space. The DGG integral kernel is parameterized by Gaussian kernel weighted Riemann sum approximating and using dynamic message passing graph to depict the interrelation within the integral term. Fourier Neural Operator is selected to localize the metric vectors on spatial and frequency domains. Metric vectors are regarded as located on latent uniform domain, wherein spatial and spectral transformation offer highly regular constraints on solution space. The efficiency and robustness of DGGO are validated by applying it to solve numerical arbitrary discrete mechanics problems in comparison with mainstream neural operators. Ablation experiments are implemented to demonstrate the effectiveness of spatial transformation in the DGG kernel. The proposed method is utilized to forecast stress field of hyper-elastic material with geometrically variable void as engineering application.

翻译：深度学习方法因大规模科学数据的可用性，已被应用于求解由参数化偏微分方程支配的物理系统。该类方法已逐步发展为算子学习范式，专注于学习无限维函数空间之间的非线性映射，从而建立从观测数据到解的接口。然而，现有最先进的神经算子局限于常数均匀离散化，导致在计算域任意离散格式下的泛化能力不足。本文提出一种新型算子学习算法——动态高斯图算子，将神经算子扩展至学习任意离散力学问题中的参数化偏微分方程。动态高斯图核通过将定义在一般欧氏空间中的观测向量映射至定义在高维均匀度量空间中的度量向量来实现学习。该积分核采用高斯核加权黎曼和近似进行参数化，并利用动态消息传递图描述积分项内的相互关系。选取傅里叶神经算子在空间域和频率域中对度量向量进行定位，度量向量被视为位于潜在均匀域上，其中空间与谱变换为解空间提供了高度正则化约束。通过将DGGO应用于求解数值任意离散力学问题并与主流神经算子对比，验证了其效率与鲁棒性。消融实验证明了动态高斯图核中空间变换的有效性。将该方法应用于几何可变孔洞超弹性材料的应力场预测作为工程应用实例。