PFT: Phonon Fine-tuning for Machine Learned Interatomic Potentials

Abhijeet Gangan; Jaime Marian; Mit Kotak; Teddy Koker; Tess Smidt

REVIEW 1 major objections 6 minor 2 cited by

Fine-tuning an interatomic potential by matching its energy Hessian to DFT force constants reduces average vibrational-property error by 55 percent.

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2026-08-03 10:59 UTC pith:QNUQMYMZ

load-bearing objection PFT is a genuinely useful method for supervising PES curvature in universal MLIPs, with strong empirical gains; the main caveat is unvalidated zero-force labels on phonon supercells. the 1 major comments →

arxiv 2601.07742 v4 pith:QNUQMYMZ submitted 2026-01-12 cond-mat.mtrl-sci cs.LG

PFT: Phonon Fine-tuning for Machine Learned Interatomic Potentials

Teddy Koker , Abhijeet Gangan , Mit Kotak , Jaime Marian , Tess Smidt This is my paper

classification cond-mat.mtrl-sci cs.LG

keywords phonon fine-tuningmachine learned interatomic potentialsenergy Hessianforce constantsvibrational propertiesthermal conductivitycatastrophic forgettingequivariant neural networks

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper argues that the standard way of training machine-learned interatomic potentials—matching energies, forces, and stresses—leaves the curvature of the potential energy surface under-constrained, and that phonon-derived properties suffer as a result. It proposes phonon fine-tuning (PFT), a training objective that directly matches the model's energy Hessian to DFT-computed second-order force constants. On a benchmark of 1,000 held-out materials, PFT reduces average error across maximum phonon frequency, vibrational entropy, Helmholtz free energy, and heat capacity by 55% relative to the base model, and also improves thermal conductivity predictions that depend on third-order derivatives. The method scales to large supercells by stochastically sampling a single Hessian column per structure and computing the loss with one Hessian-vector product, and a co-training schedule keeps the model from forgetting its original training data.

Core claim

The central claim is that curvature, not just energy and forces, must be explicitly supervised to get vibrational properties right. PFT does this by adding a loss term that compares the analytical Hessian of the predicted energy—the second derivative with respect to pairs of atomic coordinates—with the force constants obtained from finite-displacement DFT phonon calculations. Because only one column of the Hessian is needed per structure in expectation, the extra cost is linear in the number of atoms rather than quadratic. The paper shows that Hessian error tracks phonon property error across existing models, that PFT lowers that error, and that the improvement transfers to thermal conductiv

What carries the argument

The mechanism is the Hessian of the predicted potential energy, which for a crystal is the same object as the second-order force constant matrix. PFT forms its gradient by sampling one atomic coordinate (one column of the Hessian) per structure in a batch and computing the column via a Hessian-vector product: one forward-mode Jacobian-vector product through a reverse-mode gradient. This makes the fine-tuning loss a weighted sum of the usual energy/force/stress errors plus a force-constant error; in expectation the sampled columns train the full Hessian. A co-training loop interleaves the original upstream energy/force/stress data after each phonon step to mitigate catastrophic forgetting.

Load-bearing premise

The load-bearing assumption is that the phonon supercells are relaxed tightly enough that force and stress labels can be set to zero; if residual relaxation forces or stresses are sizable, the fine-tuning objective is fitted to biased labels and the Hessian matching is distorted.

What would settle it

Measure the residual DFT forces on displaced phonon supercells before the zero-force assumption is applied; if those residuals are large and coincide with the materials where PFT fails, the assumption is unsafe. Alternatively, run a controlled experiment where the DFT force constants in the PFT loss are replaced by random values while keeping everything else identical: if phonon property error still drops, the improvement is not caused by Hessian matching.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

If PFT's claims hold, any energy-conserving interatomic potential with automatic differentiation can be improved on phonon properties at a fraction of its original training cost.
The Hessian-error correlation suggests that curvature error, not just energy or force error, is a key bottleneck for vibrational property prediction.
Because PFT also improves thermal conductivity, which depends on third-order force constants, curvature-aware fine-tuning can generalize to anharmonic properties without directly supervising those higher-order derivatives.
Co-training preserves the base model's performance on stability classification, so phonon fine-tuning does not force a trade-off against general-purpose use.
Stochastic Hessian-column sampling makes it feasible to train on supercells large enough for accurate finite-displacement phonon calculations, which were previously impractical for full-Hessian training.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be applied as a universal post-processing step: since the needed labels are force constants that standard DFT phonon workflows already produce, any existing universal interatomic potential could be fine-tuned this way, not just the two base models tested.
The documented energy shift between the phonon data and the pretraining data may be suppressing energy accuracy; explicitly aligning or correcting that shift could improve PFT further.
Uniform random Hessian-column sampling is simple, but symmetry makes many columns redundant; weighting samples by how much they contribute to phonon modes could improve sample efficiency.
Adding third-order force constants directly to the PFT loss is a testable extension that could give a more direct path to thermal conductivity than relying on second-order transfer.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

PFT is a genuinely useful method for supervising PES curvature in universal MLIPs, with strong empirical gains; the main caveat is unvalidated zero-force labels on phonon supercells.

read the letter

What you should know: this is a solid, practical methods paper. The authors fine-tune an energy-conserving equivariant MLIP by matching analytically computed Hessian columns to DFT force constants from phonon calculations, using stochastic Hessian-vector products and a simple co-training scheme. It works. On held-out MDR Phonon data they get a 55% average reduction in phonon property error, reach state-of-the-art among MPtrj-trained models, and improve thermal conductivity κSRME from 0.446 to 0.306. The gains are consistent across two different base models, and the ablation showing that direct fine-tuning on displacement data worsens Hessian error is a nice piece of evidence that curvature supervision is doing the real work.

The genuine novelty is the combination: analytical Hessians for periodic universal MLIPs, stochastic column sampling to make large supercells tractable, and co-training to prevent catastrophic forgetting. The appendix validates the HVP approximation against full-Hessian training on 64-atom systems, with 30× speedup and 90% memory reduction. Code and weights are released. That is reproducible, concrete work.

Now the soft spots. The zero force/stress labels on phonon supercells are an assumption, not a verified fact. The PFT loss injects target force and stress of zero on every phonon structure, and the force-constant labels themselves assume F0=0 in the finite-difference formula. If the MDR Phonon relaxation was loose, residual forces around 1e-3 eV/Å would introduce errors in the force constants on the order of 0.1 eV/Å² at 0.01 Å displacement, comparable to the Hessian MAE gaps in Fig. 2. I don't think this sinks the paper—standard phonon workflows relax tightly, and the test set is genuinely held-out DFT, so the headline improvements are probably real—but it should be checked. The authors should report the relaxation convergence criterion and residual force norms, or better, use the actual forces from the displaced phonon calculations to supervise the EFS terms instead of assuming zero. They have that data. A second, minor issue: all benchmark numbers are single runs with no error bars, and λ_Φ was set to 100 without tuning. The 31.6 meV/atom energy shift between MPtrj and MDR Phonon is acknowledged and reasonably handled, though it muddies the reported energy MAE.

This paper is for anyone using universal MLIPs for phonon screening or thermal property prediction. It deserves serious peer review—the method is important, the evidence is strong, and the reproducibility raises it above typical MLIP fine-tuning papers. A good referee will ask for residual-force documentation and a sensitivity check on the loss weights, but the central empirical claim is credible. I'd bring it to a reading group and would certainly cite it.

Referee Report

1 major / 6 minor

Summary. The paper proposes phonon fine-tuning (PFT), a loss that augments standard energy/force/stress (EFS) training by matching the analytical Hessian of an MLIP energy to DFT force constants from finite-displacement phonon calculations. To make this tractable, PFT samples one Hessian column per structure and evaluates the loss with a single Hessian-vector product, giving O(N) training cost. A co-training schedule interleaves upstream EFS batches to limit catastrophic forgetting. On the PBE MDR Phonon benchmark, PFT improves the Nequix MP model by about 55% average MAE across four phonon properties and claims state-of-the-art accuracy among MPtrj-trained models; it also improves thermal-conductivity SRME and preserves Matbench Discovery performance when co-trained. An appendix reports that stochastic HVP training matches full-Hessian training on 64-atom supercells.

Significance. If the underlying label assumptions are validated, this is a practical and scalable way to supervise PES curvature. The stochastic HVP formulation is mathematically clean, and the appendix's full-Hessian comparison is a genuine validation of the approximation. The evaluation is external: held-out DFT phonon calculations, two base models (Nequix MP and OAM), and public benchmarks. The paper also promises code, trained weights, and preprocessed data, which strengthens reproducibility. The co-training scheme is a useful contribution, and the forgetting numbers are encouraging. The main weakness is that the zero force/stress label assumption for phonon structures is stated but not validated, and the dataset energy mismatch is only partially addressed. These issues affect the reliability of the headline numbers but are fixable with additional reporting and targeted experiments.

major comments (1)

[Sec. 4.1 / Eq. (8)] The zero force/stress label assumption is central and unvalidated. The authors state 'we assume the force and stress labels are zero due to the strict structural relaxation procedure done prior to the phonon calculation,' but they do not report residual force/stress norms or the relaxation convergence criterion used for the MDR Phonon data. With λ_F=20 and λ_σ=5, any residual force of order 1e-3 eV/Å becomes a systematic target error, and Eq. (7) propagates the same residual into the ground-truth force constants as F0/Δ (0.1 eV/Å² at Δ=0.01 Å), which is comparable to the Hessian MAE gaps in Fig. 2. Because the headline 55% and SOTA claims rest on these labels, please report residual statistics and relaxation settings, or recompute with the actual DFT forces from the displacement data, or set λ_F=λ_σ=0 on phonon structures and show the conclusions are unchanged.

minor comments (6)

[Table 1] The eSEN-MP row lists only three values for four metrics, leaving the SOTA comparison incomplete. Add the missing value and align the columns.
[Fig. 2 caption] The caption says 'heat capacity errors,' but the panel plots four different phonon properties. Update the caption to describe the multi-panel layout.
[Sec. 4.1] The text says the phonon data 'contain[s] the original energy/force/stress calculations at each displacement' and then says it 'does not contain force, or stress labels needed for the PFT loss function.' This is contradictory. Clarify which structures and which labels are actually used in Eq. (8).
[Sec. A.2] The 31.6 meV/atom energy shift between MDR Phonon and MPtrj is acknowledged. Since PFT sets λ_E=0 (Table A.2), state explicitly that the energy shift does not enter the PFT loss, and comment on whether the large outliers could affect co-training batches.
[Eq. (9)] The expectation notation with E_{b~U,j~U} inside a sum over a,i is easy to misread. Spell out that one column (b,j) is sampled per structure and that the sum over a,i covers the entries of that column.
[General] Consider reporting repeated-seed statistics for the main PFT results; the stochastic column sampling and the random train/test split make seed variability a relevant quantity.

Circularity Check

0 steps flagged

No significant circularity: PFT is externally benchmarked fine-tuning; the zero force/stress label assumption is a correctness risk, not a circular derivation.

full rationale

The paper's central claim is an empirical result: fine-tuning an MLIP by adding a Hessian-matching term (Eq. 8) to the EFS loss improves phonon properties and thermal conductivity. The targets in L_Phi (Eq. 9) are DFT-computed force constants from the MDR Phonon database, not quantities derived from the model or from the fitted parameters. The test set is material-disjoint: 'These 91 materials along with randomly selected calculations from the remaining data are used as a test set of 1,000 materials' (Sec. 4.1), so the reported reductions are genuine held-out predictions. External benchmarks (MDR Phonon, Matbench Discovery thermal conductivity, Matbench Discovery stability) provide independent grounding. No equation reduces a prediction to a fitted input; co-training choices and hyperparameters are not used to define the evaluation metric. The self-citations (Nequix MP, Koker et al. 2025) are normal base-model citations and are not load-bearing as evidence: PFT is also applied to Nequix OAM, and both are evaluated against DFT ground truth. One limitation deserves explicit flagging, though it is not circularity: Sec. 4.1 states 'we assume the force and stress labels are zero due to the strict structural relaxation procedure done prior to the phonon calculation,' and the paper does not report residual force/stress norms or relaxation convergence criteria. If residuals are non-negligible, Eq. 8's EFS terms inject biased labels into the fine-tuning objective, and Eq. 7's finite-displacement force constants inherit a baseline offset F0/Δ. This is a potential validity threat to the quantitative headline, but it concerns input-label quality and does not make the derivation self-referential. The energy-shift discrepancy acknowledged in Sec. A.2 (31.6 meV/atom MAE) is likewise a data-consistency concern, not a circularity.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 0 invented entities

The method introduces no new physical entities. The free parameters are training hyperparameters, not physical constants fitted to test data. The main axioms are domain assumptions about DFT force-constant quality, assumed-zero force/stress labels, and local-model expressivity for long-range force constants.

free parameters (6)

PFT force-constant loss weight λΦ = 100
Selected to be larger than other loss weights; described as 'not tuned' (Table A.2).
PFT force and stress loss weights λF, λσ = λF=20, λσ=5
Carried from Nequix MP; applied to phonon supercells with assumed-zero labels (Table A.2, Sec. 4.1).
Co-training ratio K = 4
Chosen as a reasonable trade-off between training time and overfitting to phonon data (Table A.2).
Number of fine-tuning epochs = 200
Not tuned; based on GPU budget, though validation metrics were close to converged (Table A.2).
Learning rate = 0.0001
Selected from {0.003, 0.001, 0.0003, 0.0001} based on early validation performance (Table A.2).
Co-training EFS loss weights = λE=500/750, λF=200, λσ=50
Increased by 10× from Nequix MP values to prevent co-train validation energy divergence (Table A.2).

axioms (7)

standard math Born-Oppenheimer PES is differentiable and energy-conserving MLIPs define forces and stresses as derivatives of a scalar energy (Eqs. 1–2).
Foundation of the MLIP framework; not questioned by the paper.
domain assumption Finite-displacement force constants (Eq. 7) with 0.01 Å displacement accurately approximate the true DFT Hessian.
Standard phonopy practice; accuracy depends on displacement size and supercell convergence.
domain assumption Force and stress labels for PFT phonon supercells are zero because structures are fully relaxed (Sec. 4.1).
Explicitly stated; residual relaxation forces/stresses could bias the EFS terms of the PFT loss.
standard math Sampling one Hessian column per structure trains the full Hessian in expectation (Eq. 9, Algorithm 1).
LΦ is a mean over Hessian columns, so uniform random sampling gives an unbiased estimator.
domain assumption A graph built on the DFT supercell can represent the force constants relevant to predicted phonon modes despite finite receptive field (Sec. 3.5).
Needed for analytical Hessians to match DFT force constants; empirically supported by the finite-displacement vs autodiff agreement in Table 1.
domain assumption PBE MDR Phonon force constants are consistent with the MPtrj PBE PES despite known energy shifts (Sec. A.2).
Energy MAE is 31.6 meV/atom with outliers up to 1.5 eV; force constants are assumed compatible even though energies are not.
domain assumption Triple-backward Hessian-vector-product training is numerically stable in JAX at the scales used.
Implementation requirement for the method; no formal or independent verification is provided.

pith-pipeline@v1.3.0-alltime-deepseek · 15528 in / 13006 out tokens · 123796 ms · 2026-08-03T10:59:43.519527+00:00 · methodology

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read the original abstract

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.

Figures

Figures reproduced from arXiv: 2601.07742 by Abhijeet Gangan, Jaime Marian, Mit Kotak, Teddy Koker, Tess Smidt.

**Figure 1.** 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 f… view at source ↗

**Figure 2.** 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. 3. Methodology 3.1. Hessian error and vibrational properties Because phonon spectra and derived vibrational properties are functions of the second-order force … view at source ↗

**Figure 3.** Figure 3: Training ablation. The left figure shows Hessian error on the MDR Phonon validation set, and the remaining figures show energy, force, and stress errors on the MPtrj validation set (top) and the MDR Phonon validation set (bottom). We compare phonon fine-tuning with and without co-training on MPtrj, as well as directly fine-tuning energy, force, and stress on the phonon displacement calculations. Co-trainin… view at source ↗

**Figure 4.** 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 PFT generally produces bands with clos… view at source ↗

**Figure 5.** 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 κSRME. We find there is still a strong trend in between the two despite thermal conductivity using third-order force constants. 200 400 600 Temperature (K) 0 1 2 5 10 20 (W m 1 K 1 ) AgI 200 400 600 Tem… view at source ↗

**Figure 6.** Figure 6: Macroscopic conductivity predictions. Macroscopic thermal conductivity calculations for zinc blend AgI (left), which exhibits low conductivity and zinc blend BeO (right), which exhibits high conductivity. DFT calculations are from (Togo, 2025). PFT produces conductivity predictions closer to the ground truth. We find that neither PFT models improve performance over the base model. This may be expected, es… view at source ↗

discussion (0)

Forward citations

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Reference graph

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