PFT: Phonon Fine-tuning for Machine Learned Interatomic Potentials

Many materials properties depend on higher-order derivatives of the potential energy surface, yet machine learned interatomic potentials (MLIPs) trained with 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 by 55% on average across phonon thermodynamic properties and achieves state-of-the-art accuracy 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.

💡 Research Summary

The paper addresses a critical shortcoming of current machine‑learned interatomic potentials (MLIPs): while they are typically trained to reproduce DFT energies, forces, and stresses, this indirect supervision does not guarantee that the curvature of the potential energy surface (PES) – the second‑order derivatives or force‑constant matrix – is accurate. Since phonon spectra, vibrational thermodynamics, and many derived properties (e.g., thermal conductivity) depend directly on these second‑order derivatives, errors in curvature lead to degraded predictions of phonon frequencies, entropy, free energy, heat capacity, and related quantities.

To remedy this, the authors introduce Phonon Fine‑tuning (PFT), a fine‑tuning procedure that directly aligns the MLIP’s Hessian (energy second derivatives) with DFT‑computed force‑constant matrices obtained from finite‑displacement phonon calculations. The core loss function, L_PFT, augments the standard energy‑force‑stress (EFS) loss with an explicit Hessian term L_Φ that measures the mean‑absolute error between sampled columns of the MLIP Hessian and the corresponding DFT force‑constant columns.

A major computational obstacle is that phonon calculations require large supercells (hundreds to thousands of atoms) to avoid spurious interactions and capture long‑range forces. The full Hessian scales as O(N²) in memory and compute, which is prohibitive. PFT circumvents this by stochastic column sampling: for each structure in a training batch, a random atom‑direction pair (a, i) is selected, defining a unit vector v that picks out the (a,i) column of the Hessian. The column is obtained with a single Hessian‑vector product (HVP), ∇²E·v, which can be computed efficiently via a forward‑mode Jacobian‑vector product through a reverse‑mode gradient (Pearlmutter’s trick). This reduces the per‑step cost to O(N) while still providing an unbiased estimator of the full Hessian loss across the batch.

Because phonon data consist only of equilibrium configurations, fine‑tuning solely on them can cause catastrophic forgetting of the diverse non‑equilibrium structures present in the original Materials Project (MP) pre‑training set. To preserve the model’s performance on the upstream data, the authors interleave PFT steps with conventional EFS steps on the MP dataset. The co‑training ratio K (e.g., K=5) determines how many upstream batches are processed for each phonon batch. Empirically, this simple schedule mitigates performance loss on the MP validation set (energy, force, stress errors increase by <1 %) while still achieving a substantial reduction in Hessian error.

The methodology is evaluated on the MDR Phonon benchmark (PBE‑based) and on Matbench Discovery tasks (thermal conductivity and stability classification). Key findings include:

1. Correlation between Hessian error and phonon property error – Lower Hessian MAE correlates strongly with reduced errors in maximum phonon frequency, vibrational entropy, Helmholtz free energy, and heat capacity.
2. 55 % average reduction in phonon thermodynamic errors – When PFT is applied to the Nequix‑MP foundation model, the mean absolute errors across the four thermodynamic metrics drop by roughly half compared with the baseline.
3. State‑of‑the‑art thermal conductivity prediction – κ_SRME improves from 0.446 to 0.306 W m⁻¹ K⁻¹, surpassing all other MP‑trained models.
4. Preservation of upstream performance – Co‑training keeps the original MP validation errors essentially unchanged, whereas fine‑tuning without co‑training degrades them markedly.
5. Scalability to larger base models – Applying PFT to a model pre‑trained on the OMat24 dataset yields consistent gains, demonstrating that the approach is not limited to a specific architecture or dataset size.

The paper’s contributions can be summarized as three technical advances: (i) a direct Hessian‑matching loss that forces the MLIP to reproduce the true PES curvature, (ii) an efficient linear‑scaling HVP‑based training pipeline that makes large‑supercell phonon fine‑tuning feasible, and (iii) a straightforward co‑training recipe that prevents catastrophic forgetting. Together, these innovations close the gap between MLIP‑based phonon calculations and high‑fidelity DFT results, enabling reliable high‑throughput vibrational property predictions and opening avenues for further extensions such as higher‑order derivative training, non‑equilibrium phonon data, and multi‑property joint optimization.