Dynamic Gaussian Graph Operator: Learning parametric partial
  differential equations in arbitrary discrete mechanics problems
Chu Wang Jinhong Wu Yanzhi Wang Zhijian Zha Qi Zhou
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.