PreFerred Potential (PFP)

• PreFerred Potential (PFP) is a universal machine-learning interatomic potential that predicts energies, forces, and charges using deep neural network architectures.
• It leverages an equivariant message passing framework and heterogeneous DFT data to achieve near-DFT accuracy with significantly reduced computational cost.
• The adaptable model supports various electronic-structure functionals and transfer-learning applications, enabling efficient simulations for materials discovery and adsorption studies.

PreFerred Potential (PFP) is a universal neural-network potential, and in later usage a universal machine-learning interatomic potential, developed to predict energies, forces, and, in some formulations, atomic charges for atomic configurations spanning broad chemical and structural domains. Its central aim is to bridge highly accurate but costly quantum-chemical methods such as DFT and classical interatomic potentials by providing near-DFT accuracy at a fraction of the computational cost. The original formulation emphasized applicability to arbitrary combinations of 45 elements and to molecular, crystalline, surface, cluster, adsorption, and disordered systems; later versions and deployment modes broadened the covered element sets, the training references, and the downstream uses of PFP-derived latent features in transfer-learning pipelines (Takamoto et al., 2021, Shinagawa et al., 9 Mar 2026).

1. Historical development and domain coverage

PFP originated in work by Takamoto et al. on a universal neural network potential for materials discovery applicable to arbitrary combinations of 45 elements. In that formulation, the design targets were universality, generalization to unseen or unstable structures, and efficiency sufficient for MD, MC, NEB, and screening workflows (Takamoto et al., 2021). The training corpus combined a molecular dataset of approximately 6 million structures, a crystal dataset of approximately 3 million structures, and the OC20 adsorption dataset, with calculation-mode conditioning used to distinguish molecular and crystal DFT environments (Takamoto et al., 2021).

Subsequent summaries describe PFP as expanding beyond the original 45-element setting. One technical overview states that the original PreFerred release covered 45 elements and was expanded to 72 elements in PFP version 2023 (Mao et al., 2024). A later crystal-structure-prediction study describes PFP as trained on approximately 42 million DFT-relaxed structures so that it can be applied to arbitrary combinations of 72 elements (Shibayama et al., 27 Mar 2025). PFP/MM, by contrast, describes broad chemical coverage as spanning up to 96 elements in periodic and cluster data for the CRYSTAL_U0_PLUS_D3 mode (Miyazaki et al., 17 Mar 2026). In PFP v8, the reported data sources are mode-specific: PFP-r²SCAN uses approximately $3\times 10^6$ structures across 70 elements, whereas PFP-PBE/+U uses approximately $6\times 10^7$ structures across 96 elements (Shinagawa et al., 9 Mar 2026).

Milestone	Reported scope	Distinguishing feature
Original PFP	45 elements	Arbitrary combinations of 45 elements (Takamoto et al., 2021)
PFP version 2023	72 elements	Expanded elemental coverage (Mao et al., 2024)
CSP usage	72 elements	Approx. 42 million DFT-relaxed structures (Shibayama et al., 27 Mar 2025)
PFP v8 modes	70 elements in r²SCAN; 96 in PBE/+U	Multi-mode PES conditioned on functional flavor (Shinagawa et al., 9 Mar 2026)

These versioned descriptions indicate that “PFP” is not a single fixed parametrization but a family of related models and service modes. A plausible implication is that comparisons across papers must be interpreted with attention to the specific release, functional conditioning, and covered chemical domain.

2. Core mathematical formulation and equivariant architecture

PFP is built on the TeaNet graph-neural-network architecture. Atoms are represented as nodes, neighbors within a cutoff generate directed edges, and message passing propagates information through scalar, vector, and rank-2 tensor channels while preserving Euclidean symmetry (Mao et al., 2024, Shinagawa et al., 9 Mar 2026). In the original 45-element description, TeaNet propagates rank-0, rank-1, and rank-2 features under full $\mathrm{E}(3)$ equivariance using five message-passing layers with per-layer cutoffs $\{3,3,4,6,6\}\,\text{\AA}$, yielding an effective receptive field of $22\,\text{\AA}$ (Takamoto et al., 2021).

Across the summaries, the total energy is decomposed into atomic contributions. A generic PFP form is

$$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$

where $\mathbf X_i$ encodes the local environment of atom $i$ (Shibayama et al., 27 Mar 2025). The PFP v8 overview writes the same principle in learned-embedding form,

$$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$

with forces and stress obtained by analytic differentiation (Shinagawa et al., 9 Mar 2026). The original 45-element architecture also includes an explicit learned Morse-style two-body term,

$$E_{\rm tot}=\sum_i E_i^{\rm NN}+\sum_{i<j}V_{\rm 2B}(r_{ij}),$$

introduced to stabilize very close atom pairs (Takamoto et al., 2021).

The equivariance guarantees are expressed explicitly in the PFP/TeaNet summaries. Under $6\times 10^7$0, scalar features are invariant, vector features transform as $6\times 10^7$1, and rank-2 tensor features transform as $6\times 10^7$2. Bias-free linear maps and gated nonlinearities acting only on invariant norms ensure that each layer commutes with the group action, so predicted energies are invariant and forces rotate with the structure (Mao et al., 2024). This exact symmetry handling is one of the defining architectural properties that differentiate PFP from strictly invariant descriptor models.

Several downstream papers treat PFP as a black-box potential and do not reproduce low-level architectural detail. This omission is itself documented: Uchiyama et al. do not reproduce the detailed internal functional form of the PFP used through Matlantis, and some later overviews describe exact hyperparameters as proprietary (Uchiyama et al., 3 Feb 2026, Shinagawa et al., 9 Mar 2026). For encyclopedia purposes, this means that the public characterization of PFP is strongest at the level of symmetry structure, energy decomposition, training targets, and benchmarked behavior, rather than at the level of a fully specified open architecture.

3. Training data, targets, and functional conditioning

PFP is trained against large DFT corpora with combined losses on energies, forces, and, in some versions, charges or stress. The original universal-PFP loss aggregates energy, force, and charge terms,

$6\times 10^7$3

with typical weights $6\times 10^7$4, $6\times 10^7$5, and $6\times 10^7$6 (Takamoto et al., 2021). The PFP v8 description extends this to a weighted sum on energies, forces, and optionally Bader charges, plus weight decay regularization (Shinagawa et al., 9 Mar 2026).

The training datasets are deliberately heterogeneous. In the original 45-element paper, the molecular dataset contains optimized geometries, normal-mode-sampled distortions, high-temperature MD snapshots, reactive species, and two-body potentials; the crystal dataset contains bulk phases, clusters, surfaces, adsorption complexes, disordered snapshots, and transition-state guess structures (Takamoto et al., 2021). The PFP v8 report describes four sources: PFP-r²SCAN, PFP-PBE/+U, PFP-$6\times 10^7$7B97X-D, and OC20 adsorption, covering molecules, bulk crystals, surfaces, high-temperature disorder, low-coordination clusters, and adsorption complexes (Shinagawa et al., 9 Mar 2026). One PFP architectural summary reports approximately 22 million DFT calculations across bulk, slabs, and clusters in the original release and notes later expansion to 72 elements (Mao et al., 2024).

A distinctive feature of PFP is explicit conditioning on the reference electronic-structure mode. The original paper uses a learned embedding of a DFT-mode one-hot label to model molecular and crystal datasets consistently (Takamoto et al., 2021). PFP v8 generalizes this idea by embedding a one-hot “r2SCAN” flag in every atomic node so that the same network parameters reproduce multiple PESs, specifically PBE, PBE+U, $6\times 10^7$8B97X-D, and r²SCAN (Shinagawa et al., 9 Mar 2026). The stated motivation of v8 is that better zero-shot predictions versus experiments should be an explicit design target for universal MLIPs, rather than merely reproduction of PBE-level references (Shinagawa et al., 9 Mar 2026).

The r²SCAN training data in v8 were computed with VASP using PAW, a $6\times 10^7$9 cutoff, $\mathrm{E}(3)$0, $\mathrm{E}(3)$1 vacuum for slabs, and Gaussian smearing $\mathrm{E}(3)$2 (Shinagawa et al., 9 Mar 2026). This level of specification matters because PFP’s “universality” is tied to a heterogeneous but still explicitly labeled reference hierarchy, not to a single monolithic ab initio target.

4. PFP latent representations as transferable descriptors

A major development in the PFP literature is the use of pretrained latent features as descriptors for downstream prediction tasks. Two distinct transfer paradigms are reported.

First, the dielectric-tensor work freezes PFP and extracts intermediate multi-rank node and edge features,

$\mathrm{E}(3)$3

which are then passed to a lightweight equivariant readout network with two stacked equivariant blocks (Mao et al., 2024). The final dielectric tensor is predicted by averaging atomwise equivariant contributions containing an isotropic term, a $\mathrm{E}(3)$4 term, and a rank-2 tensor term, preserving the covariance relation

$\mathrm{E}(3)$5

under rotations (Mao et al., 2024). Here PFP functions not merely as an energy model but as a source of structurally and electronically informed equivariant embeddings.

Second, Uchiyama et al. extract a 256-dimensional descriptor $\mathrm{E}(3)$6 from the final hidden layer of the pretrained PFP immediately before energy prediction and inject it into a three-dimensional EGNN for molecular property prediction (Uchiyama et al., 3 Feb 2026). No additional pooling, radial-basis expansion, or PCA is applied; the raw 256-dimensional vector is used directly (Uchiyama et al., 3 Feb 2026). In their EGNN-PFP construction, the initial node feature concatenates the PFP descriptor, an atomic-number embedding, and a four-dimensional geometric feature vector, while edge construction incorporates interatomic distance, descriptor similarity, a PFP-difference term, and a similarity-weighted distance (Uchiyama et al., 3 Feb 2026).

The empirical effect is reported on two chemically distinct benchmarks. On QM9, EGNN-PFP shows superior accuracy to both the original EGNN models and baseline models without PFP-derived descriptors for 11 of the 12 molecular properties, with reductions such as $\mathrm{E}(3)$7 for dipole $\mathrm{E}(3)$8, $\mathrm{E}(3)$9 meV for $\{3,3,4,6,6\}\,\text{\AA}$0, and $\{3,3,4,6,6\}\,\text{\AA}$1 meV for $\{3,3,4,6,6\}\,\text{\AA}$2; $\{3,3,4,6,6\}\,\text{\AA}$3 and $\{3,3,4,6,6\}\,\text{\AA}$4 worsen (Uchiyama et al., 3 Feb 2026). On tmQM, performance improves across all five target properties, including dipole $\{3,3,4,6,6\}\,\text{\AA}$5 $\{3,3,4,6,6\}\,\text{\AA}$6, HOMO-LUMO gap $\{3,3,4,6,6\}\,\text{\AA}$7 eV, and metal partial charge $\{3,3,4,6,6\}\,\text{\AA}$8 (Uchiyama et al., 3 Feb 2026).

The stated qualitative interpretation is that pretrained PFP embeddings carry rich, element-general information about local electronic potential fields that cannot be deduced from geometry alone (Uchiyama et al., 3 Feb 2026). At the same time, the same study explicitly notes that PFP is local and cannot capture fully delocalized properties such as the electronic spatial extent $\{3,3,4,6,6\}\,\text{\AA}$9 (Uchiyama et al., 3 Feb 2026). This limitation is central to understanding what PFP latents represent: they are powerful local descriptors, not complete global wavefunction surrogates.

5. Applications in materials discovery, adsorption, and reactive simulation

PFP has been applied across a wide range of atomistic tasks. In the original universal-PFP paper, reported case studies include lithium diffusion in $22\,\text{\AA}$0, molecular adsorption in MOFs, a Cu–Au order–disorder transition, and Fischer–Tropsch catalyst screening (Takamoto et al., 2021). For $22\,\text{\AA}$1, CI-NEB barriers along $22\,\text{\AA}$2, $22\,\text{\AA}$3, and $22\,\text{\AA}$4 are reported as $22\,\text{\AA}$5, $22\,\text{\AA}$6, and $22\,\text{\AA}$7 eV for PFP versus $22\,\text{\AA}$8, $22\,\text{\AA}$9, and $$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$0 eV for DFT; on a Co stepped surface, vanadium is identified as reducing the CO dissociation barrier by approximately $$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$1 (Takamoto et al., 2021).

Nanoparticle validation extends PFP into realistic finite systems. The nanoparticle study reports a Ru nanocluster cohesive-energy average absolute error of $$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$2 per atom for sizes up to $$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$3, a PdRuCu alloy excess-energy RMSE of approximately $$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$4 with correlation coefficient $$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$5, and a NO/Rh adsorption-energy mean absolute deviation of approximately $$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$6 (Huerta et al., 2021). It also reports single-GPU timings of approximately $$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$7 per $$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$8 step for NO–Rh MD and approximately $$E_{\mathrm{PFP}}(\{\mathbf R\})=\sum_{i=1}^N E_i(\mathbf X_i),$$9 per step for geometry optimization of a $\mathbf X_i$0-atom $\mathbf X_i$1 system (Huerta et al., 2021).

In crystal structure prediction, PFP is used as the energy and relaxation engine inside a genetic algorithm designed to expand convex-hull volume while preserving structural diversity. The CSP study states that PFP-driven GA evaluation can handle $\mathbf X_i$2 trials because PFP is at least $\mathbf X_i$3–$\mathbf X_i$4 times faster than DFT, and it reports new low-energy candidates $\mathbf X_i$5–$\mathbf X_i$6 below random-search baselines in benchmark settings (Shibayama et al., 27 Mar 2025). This use of PFP is significant because it exploits both transferability across compositions and cheap gradient-based relaxation.

In adsorption screening, Bonakala et al. combine UFF-based pre-screening with PFP refinement for ethylene capture in humid MOFs. For 88 MOF+guest configurations, PFP gives adsorption-energy MADs of $\mathbf X_i$7 for ethylene and $\mathbf X_i$8 for water versus DFT, while PFP relaxations are approximately $\mathbf X_i$9–$i$0 faster than CP2K/PBE-D3 geometry optimization (Bonakala et al., 8 Sep 2025). Full unit-cell relaxation can change $i$1 by up to $i$2, and the workflow ultimately identifies seven MOFs with optimal pore sizes, high ethylene affinity, and high $i$3 selectivity (Bonakala et al., 8 Sep 2025).

PFP/MM extends PFP to large condensed-phase reactive simulations by combining a PFP region with MM surroundings. For a $i$4-atom system with a 22-atom PFP region, reported throughput is $i$5 on a V100 and $i$6 on MN-Core 2, compared with $i$7 for a PFP-only treatment of all atoms; at the million-atom scale, throughput is approximately $i$8–$i$9 (Miyazaki et al., 17 Mar 2026). The framework is reported to reproduce a Ramachandran plot for alanine dipeptide, a solvent-stabilized intramolecular nucleophilic addition free-energy profile, and a cytochrome P450 Compound I hydroxylation landscape consistent with the accepted reaction mechanism (Miyazaki et al., 17 Mar 2026).

6. Benchmarks, limitations, and open methodological questions

PFP’s benchmark profile depends strongly on version and task. PFP v8 reports crystal formation-energy MAE $$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$0 versus experiment for 738 r²SCAN-supported compounds, matching DFT-r²SCAN and improving on PFP-PBE/+U $$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$1 and uncorrected PFP-PBE $$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$2 (Shinagawa et al., 9 Mar 2026). On GMTKN55, PFP-r²SCAN gives WTMAD-2 $$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$3, improved to $$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$4 with D3; for surface energies of selected fcc metals, PFP-r²SCAN gives MAE $$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$5, improved to $$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$6 with D3; for melting points across 11 listed materials, PFP-r²SCAN gives MAE approximately $$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$7, roughly halving the PFP-PBE error of approximately $$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$8 (Shinagawa et al., 9 Mar 2026).

The limitations are also explicitly catalogued. In the descriptor-transfer setting, Uchiyama et al. show that purely local PFP descriptors cannot recover long-range delocalized quantities such as $$E_\mathrm{tot}(\{\mathbf r\})=\sum_{i=1}^N E_i\bigl(\mathbf h_i^{(L)}\bigr),$$9 and that $$E_{\rm tot}=\sum_i E_i^{\rm NN}+\sum_{i<j}V_{\rm 2B}(r_{ij}),$$0 can worsen on QM9 (Uchiyama et al., 3 Feb 2026). In PFP v8, long-range many-body van der Waals interactions beyond D3 are not built in, truly nonlocal functionals such as rVV10 are absent, f-block elements are not yet supported in r²SCAN mode, and charged systems and isolated H atoms are excluded in $$E_{\rm tot}=\sum_i E_i^{\rm NN}+\sum_{i<j}V_{\rm 2B}(r_{ij}),$$1B97X-D mode (Shinagawa et al., 9 Mar 2026). Pt and Au melting points are underpredicted in r²SCAN mode, attributed to intrinsic functional softness and slight underfitting on those elements (Shinagawa et al., 9 Mar 2026).

For adsorption, MOF screening with PFP remains cutoff-based: true long-range electrostatics beyond $$E_{\rm tot}=\sum_i E_i^{\rm NN}+\sum_{i<j}V_{\rm 2B}(r_{ij}),$$2 are captured only implicitly through training, highly polar frameworks or charged MOFs may fall outside the training domain, multi-body van der Waals effects are not explicit, and electronic polarization is only whatever is embedded in the fitted potential (Bonakala et al., 8 Sep 2025). In PFP/MM, mechanical embedding means that the MM environment contributes classical electrostatics to the PFP region, so explicit solvent polarization of the reactive site is absent unless nearby solvent is included in the PFP region (Miyazaki et al., 17 Mar 2026).

Several methodological cautions follow from these reports. First, direct numerical comparisons across downstream models may be approximate when baselines use smaller subsets or higher-level DFT references, as noted explicitly for NatQG and QTAIM-GNN comparisons on tmQM (Uchiyama et al., 3 Feb 2026). Second, some internal architectural and hyperparameter details remain omitted or proprietary in application papers, which limits strict reproducibility of the underlying PFP instantiation (Uchiyama et al., 3 Feb 2026, Shinagawa et al., 9 Mar 2026). Third, “universality” in PFP denotes broad transfer across trained domains and element sets; it does not remove the dependence of accuracy on reference-functional choice, long-range physics treatment, or training coverage. This suggests that future advances are likely to combine broader reference data, more explicit long-range modeling, and richer downstream architectures rather than relying on locality alone.