Direction-aware topological descriptors for elastic stiffness tensor prediction in porous materials
Classical topological descriptors used in topological data analysis (TDA) are invariant under permutations of spatial axes and therefore cannot represent the loading direction, which is essential for modeling anisotropic mechanical response. Here, this limitation is addressed by introducing a direction-aware TDA framework in which the loading axis is explicitly embedded into filtration functions used to compute both persistent homology and Euler characteristic profile descriptors. We apply this framework to predict the full elastic stiffness tensor of porous microstructures using non-directional as well as direction-aware descriptors of the structures as well as convolutional neural networks trained directly on the voxelized structure. We show that the performance of all those are comparable on the diagonal uniaxial, Poisson, and shear components. However for the twelve off-diagonal, normal shear coupling components - which govern elastic anisotropy and are the hardest to predict - only direction-aware topology retains meaningful predictive power, with all baselines, non-directional descriptors and the CNN collapsing to near-chance accuracy. When used as inputs to gradient-boosted tree models, the proposed descriptors match or exceed the accuracy of the convolutional neural network specifically on these hardest-to-predict coupling terms, despite relying on a compact, physically interpretable representation that is orders of magnitude smaller than the raw voxel grid. Overall, the results establish direction-aware TDA as a practical route for linking porous microstructure to the full anisotropic elastic response, capturing coupling terms that conventional descriptors and end-to-end deep learning models fail to resolve.