notesum.ai

Published at November 5

Beyond Grid Data: Exploring Graph Neural Networks for Earth Observation

cs.LG

cs.AI

cs.CV

Released Date: November 5, 2024

Authors: Shan Zhao¹, Zhaiyu Chen¹, Zhitong Xiong¹, Yilei Shi¹, Sudipan Saha¹, Xiao Xiang Zhu¹

Aff.: ¹Not specified

Arxiv: http://arxiv.org/abs/2411.03223v1

Refer to caption

Mapping	Updating function	Physics examples
Node $\rightarrow$ Edge	$e_{i,j}=\phi^{e}(v_{i},v_{j})=(\nabla v)_{ij}$	$\nabla\phi=-E$ (Electric field)
Edge $\rightarrow$ Node	$v_{i}=\phi^{v}(e_{ij})=(\text{div}~{}e)_{i}$	$\nabla\cdot E=\rho/\epsilon_{0}$ (Maxwell’s equ.)
Node $\rightarrow$ Node	$v_{i}=\phi^{v}(v_{i},\{v_{j:(i,j)\in\mathcal{E}}\})=(\Delta v)_{i}$	$\Delta\phi=0$ (Laplace’s equ.)
Edge $\rightarrow$ 3-clique	$c_{ijk}=\phi^{c}(e_{ij},e_{jk},e_{ki})=(\text{curl}~{}e)_{ijk}$	$\nabla\times E=0$ (Irrotational E-field)
First order	$v^{\prime}_{i}=v_{i}+\alpha\phi^{v}(v_{i},\{v_{j:(i,j)\in\mathcal{E}}\})=v_{i}% +\alpha(\Delta v)_{i}$	$\dot{u}=\alpha\nabla^{2}u$ (Diffusion eqn.)
Second order	$v_{i}^{{}^{\prime\prime}}=2v^{\prime}_{i}-v_{i}+c^{2}\phi^{v}(v^{\prime}_{i},% \{v^{\prime}_{j:(i,j)\in\mathcal{E}}\})=2v^{\prime}_{i}-v_{i}+c^{2}(\Delta v^{% \prime})_{i}$	$\ddot{u}=c^{2}\Delta^{2}u$ (Wave eqn.)