PFT: Phonon Fine-tuning for Machine Learned Interatomic Potentials

TL;DR

PFT addresses the challenge that standard MLIPs, trained on energies, forces, and stresses, underconstrain PES curvature and thus degrade phonon properties. It directly optimizes second-order force constants by aligning MLIP Hessians with DFT-derived force constants using Hessian-vector products and column sampling to scale to large supercells, complemented by a co-training strategy to mitigate catastrophic forgetting. The approach yields a 55% average improvement in MDR Phonon metrics and achieves state-of-the-art performance among Materials Project–trained models, while also enhancing third-order derivative–dependent properties such as thermal conductivity (κ_SRME) and preserving upstream performance through co-training. These results demonstrate that Hessian-aware fine-tuning significantly improves vibrational and anharmonic predictions with practical computational costs, supporting broader adoption and extension to other higher-order derivatives and datasets.

Abstract

Many materials properties depend on higher-order derivatives of the potential energy surface, yet machine learned interatomic potentials (MLIPs) trained with standard a standard loss on energy, force, and stress errors can exhibit error in curvature, degrading the prediction of vibrational properties. We introduce phonon fine-tuning (PFT), which directly supervises second-order force constants of materials by matching MLIP energy Hessians to DFT-computed force constants from finite displacement phonon calculations. To scale to large supercells, PFT stochastically samples Hessian columns and computes the loss with a single Hessian-vector product. We also use a simple co-training scheme to incorporate upstream data to mitigate catastrophic forgetting. On the MDR Phonon benchmark, PFT improves Nequix MP (trained on Materials Project) by 55% on average across phonon thermodynamic properties and achieves state-of-the-art performance among models trained on Materials Project trajectories. PFT also generalizes to improve properties beyond second-derivatives, improving thermal conductivity predictions that rely on third-order derivatives of the potential energy.

Figures (8)

• Figure 1: Overview of PFT framework. a Finite-difference calculations rely on the construction of a supercell to obtain force constants from interactions beyond the unitcell. b MLIPs are pre-trained on standard unitcell DFT calculations. c-i Up to $O(3N)$ atomic displacements are applied to the supercell, with the number reduced by crystal symmetries; forces are computed with DFT, and numerical derivatives yield the force constant matrix. c-ii The same workflow can be used for MLIPs, replacing DFT force calculations with force predictions from the model. c-iii Alternatively, the force constants can also be computed as the analytical Hessian of the predicted energy directly. c-iv In this work, we use the Hessian-vector product to efficiently compute columns of the Hessian, and align with sampled DFT force constant columns during training. Note the shown force constants are downsampled for clarity.
• Figure 2: Hessian error vs. phonon properties. Error in the Hessians on the test subset of the MDR Phonon data are plotted against heat capacity errors for several foundation models trained on MPtrj. Hessian errors correlate with improved property prediction.
• Figure 3: Training ablation. The top row shows energy, force, and stress errors on the MPtrj validation set, while the bottom row shows energy, force, stress, and hessian errors on the MDR Phonon validation set during finetuning of Nequix MP. We compare phonon fine-tuning with and without co-training on MPtrj, as well directly finetuning energy, force, and stress on the phonon displacement calculations. Co-training mostly mitigates degradation of MPtrj performance at only a slight increase in hessian error, and that training directly on displacements worsens hessian performance over the base model. *We note that larger energy errors on MDR Phonon data are likely due to a mismatch in energies between MPtrj and MDR Phonon, see Sec. \ref{['energy_discrepancy']}.
• Figure 4: Phonon band structures. For ease of visualization, we display the phonon band structure for the three materials in the test split of MDR Phonon with the fewest number of atoms. In the case of Carbon structure (mp-990448), it results in a dynamically stable structure similar to DFT while the non-PFT model shows imaginary mode around the $q$-point $A$. Overall, we find that that PFT generally produces bands with closer alignment to those of the DFT ground truth. More phonon bandstructures are provided in Fig. \ref{['fig:pft_band_structure_extra']}.
• Figure 5: Thermal conductivity vs. Hessian error. Scatter plot of several models Hessian MAE on the MDR phonon test set vs. symmetric relative mean error in predicted phonon mode contributions to thermal conductivity $\kappa_\text{SRME}$. We find there is still a strong trend in between the two despite thermal conductivity using third-order force constants.