  Machine learning approaches have recently emerged as powerful tools to probe
structure-property relationships in crystals and molecules. Specifically,
Machine learning interatomic potentials (MLIP) can accurately reproduce
first-principles data at a cost similar to that of conventional interatomic
potential approaches. While MLIP have been extensively tested across various
classes of materials and molecules, a clear characterization of the anharmonic
terms encoded in the MLIP is lacking. Here, we benchmark popular MLIP using the
anharmonic vibrational Hamiltonian of ThO$_2$ in the fluorite crystal
structure. This anharmonic Hamiltonian was constructed from density functional
theory (DFT) using our highly accurate and efficient irreducible derivative
methods, and then used to generate molecular dynamics trajectories. This data
set was used to train three classes of MLIP: Gaussian Approximation Potentials,
Artificial Neural Networks (ANN), and Graph Neural Networks (GNN). The results
were assessed by directly comparing phonons and their interactions, as well as
phonon linewidths, phonon lineshifts, and thermal conductivity. The models were
also trained on a DFT molecular dynamics dataset, demonstrating good agreement
up to fifth-order for the ANN and GNN. Our analysis demonstrates that MLIP have
great potential for accurately characterizing anharmonicity in materials
systems at a fraction of the cost of conventional first principles-based
approaches.
