  We present a SE(3)-equivariant graph neural network (GNN) approach that
directly predicting the formation factor and effective permeability from
micro-CT images. FFT solvers are established to compute both the formation
factor and effective permeability, while the topology and geometry of the pore
space are represented by a persistence-based Morse graph. Together, they
constitute the database for training, validating, and testing the neural
networks. While the graph and Euclidean convolutional approaches both employ
neural networks to generate low-dimensional latent space to represent the
features of the micro-structures for forward predictions, the SE(3) equivariant
neural network is found to generate more accurate predictions, especially when
the training data is limited. Numerical experiments have also shown that the
new SE(3) approach leads to predictions that fulfill the material frame
indifference whereas the predictions from classical convolutional neural
networks (CNN) may suffer from spurious dependence on the coordinate system of
the training data. Comparisons among predictions inferred from training the CNN
and those from graph convolutional neural networks (GNN) with and without the
equivariant constraint indicate that the equivariant graph neural network seems
to perform better than the CNN and GNN without enforcing equivariant
constraints.
