Explainable machine learning reveals a compact, low-dimensional geometric basis underlying sheet and skeletal TPMS architected materials

Triply periodic minimal surface (TPMS) architectures constitute one of the largest and most diverse families of architected materials, yet the geometric principles governing their mechanical and thermal behavior remain poorly understood. Here, we investigate fourteen TPMS topologies spanning both sheet- and skeletal-based realizations and characterize their geometry through a set of sixteen physically interpretable descriptors. Combining finite element simulations with explainable machine learning, we establish descriptor hierarchies for the effective mechanical and thermal properties across the TPMS design space. Surface area and compactness emerge as universal descriptors shared by both architecture classes, while descriptors of diminishing importance differ between sheet and skeletal networks, reflecting distinct manifestations of geometric heterogeneity. Despite these differences, a unified four-descriptor representation is identified that accurately predicts Young’s and shear moduli, thermal conductivity, as well as uniaxial and shear yield limits across all topologies for both network classes with coefficients of determination exceeding \(R^2=0.93\) . Interestingly, both principal component analysis (PCA) and the nonlinear Levina-Bickel intrinsic dimension estimator reveal that the geometric descriptor space and the associated performance space collapse onto a compact, nearly two-dimensional manifold, indicating that the apparent complexity of TPMS architectures arises from a remarkably low-dimensional organization. These findings provide a unified framework, applicable across sheet- and skeletal-based network classes, for interpretable modeling and materials informatics of TPMS architected materials, and establish a compact basis for future inverse-design methodologies.