Phonon interactions from lattice anharmonicity govern thermal properties and heat transport in materials. These interactions are described by $n$th order interatomic force constants ($n$IFCs), which can be viewed as high-dimensional tensors correlating the motion of $n$ atoms, or equivalently encoding $n$-phonon scattering processes in momentum space. Here, we introduce a tensor decomposition to efficiently compress $n$IFCs for arbitrary order $n$. Using tensor learning, we find optimal low-rank approximations of $n$IFCs by solving the resulting optimization problem. Our approach reveals the inherent low dimensionality of phonon-phonon interactions and allows compression of the three- and four-IFC tensors by factors of up to ${10}^3–{10}^4$ while retaining high accuracy in calculations of phonon scattering rates and thermal conductivity. Calculations of thermal conductivity using the compressed $n$IFCs achieve a speedup by nearly 3 orders of magnitude with $>98\%$ accuracy relative to the reference uncompressed solution. These calculations include both three- and four-phonon scattering and are shown for a diverse range of materials (Si, HgTe, MgO, TiNiSn, and ${\mathrm{ZrO}}_2$). In addition to accelerating state-of-the-art thermal transport calculations, the method shown here paves the way for modeling strongly anharmonic materials and higher-order phonon interactions.

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• Anharmonic lattice dynamics
• Phonons
• Thermal conductivity
• Density functional calculations
• Dimensionality reduction
• First-principles calculations
• Tensor network methods