  Lattices are architected metamaterials whose properties strongly depend on
their geometrical design. The analogy between lattices and graphs enables the
use of graph neural networks (GNNs) as a faster surrogate model compared to
traditional methods such as finite element modelling. In this work, we generate
a big dataset of structure-property relationships for strut-based lattices. The
dataset is made available to the community which can fuel the development of
methods anchored in physical principles for the fitting of fourth-order
tensors. In addition, we present a higher-order GNN model trained on this
dataset. The key features of the model are (i) SE(3) equivariance, and (ii)
consistency with the thermodynamic law of conservation of energy. We compare
the model to non-equivariant models based on a number of error metrics and
demonstrate its benefits in terms of predictive performance and reduced
training requirements. Finally, we demonstrate an example application of the
model to an architected material design task. The methods which we developed
are applicable to fourth-order tensors beyond elasticity such as piezo-optical
tensor etc.
