Neuroevolution Potential (NEP)

• Neuroevolution Potential (NEP) is a machine-learned framework that decomposes total potential energy into a sum of local atomic energies predicted via shallow neural networks.
• It combines Chebyshev/Legendre-based descriptors with evolutionary optimization and GPU acceleration to achieve high accuracy and scalability in molecular dynamics simulations.
• Recent advancements include multi-component descriptors, active-learning dataset curation, and physical augmentations (e.g., dispersion and ZBL corrections) to extend its application range.

Neuroevolution Potential (NEP) is a family of machine-learned interatomic potentials in which the total potential energy is written as a sum of local atomic energies and each atomic energy is predicted from an atom-centered descriptor by a shallow neural network. In its original formulation, NEP combined a Chebyshev- and Legendre-based descriptor, a one-hidden-layer feedforward network, training by separable natural evolution strategy (SNES), and a GPU-native implementation in GPUMD, with the explicit aim of combining high accuracy and low cost in large-scale molecular dynamics, especially for heat transport and strongly anharmonic or disordered materials (Fan et al., 2021). Subsequent work extended NEP from material-specific models to multi-component descriptors, dispersion-corrected and ZBL-hybrid variants, unified metallic and 89-element foundation models, quantum-nuclear simulations, coarse-grained and mixed-resolution models, and active-learning and dataset-curation frameworks (Fan, 2021, Liang et al., 30 Apr 2025, Fan et al., 1 Mar 2026, Wang et al., 15 Apr 2026).

1. Formal definition and mathematical structure

In NEP, the total potential energy is decomposed as

$U=\sum_{i=1}^{N} U_i,$

where $U_i$ is the local energy of atom $i$ and depends on a descriptor of its local environment within finite cutoffs (Fan et al., 2021). A standard one-hidden-layer form used throughout the literature is

$$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$

with descriptor components $q^i_\nu$, hidden width $N_{\rm neu}$, descriptor dimension $N_{\rm des}$, and trainable weights and biases (Fan, 2021).

The original NEP descriptor consists of radial and angular components. For the radial channel,

$q_n^i = \sum_{j\neq i} g_n(r_{ij}),$

and for the angular channel,

$$q_{nl}^i = \sum_{j\neq i}\sum_{k\neq i} g_n(r_{ij}) g_n(r_{ik}) P_l(\cos\theta_{ijk}),$$

where $P_l$ is a Legendre polynomial and $U_i$0 is built from Chebyshev polynomials multiplied by a smooth cosine cutoff (Fan et al., 2021, Fan, 2021). The descriptor is constructed to be invariant under translation, rotation, inversion, and permutation of atoms of the same species (Fan et al., 2021). Later expositions also write the angular part through spherical-harmonic moments $U_i$1 and higher-body contractions, making explicit the 3-body, 4-body, and 5-body structure already implicit in the NEP family (Xu et al., 25 May 2025, Xu et al., 2024, Wu et al., 17 Jun 2026).

A recurrent misconception is that NEP is a graph neural network or message-passing model. In the cited work it is instead a descriptor-based, atom-centered potential with a deliberately shallow network; much of its expressivity is placed in the structured descriptor rather than in architectural depth (Zhou et al., 2024, Liang et al., 30 Apr 2025).

2. Optimization and training paradigms

The original NEP framework trains the parameter vector with SNES rather than standard backpropagation (Fan et al., 2021). Its training objective combines errors in energies, forces, and virials, together with $U_i$2 and $U_i$3 regularization. In the original paper, the loss is written as a weighted sum of $U_i$4, $U_i$5, $U_i$6, $U_i$7, and $U_i$8, with practical default regularization values $U_i$9 reported to work well across the benchmark systems (Fan et al., 2021).

This optimizer choice had a methodological consequence: descriptor parameters could be made trainable with little additional machinery, because evolutionary search did not require explicit derivatives of the loss with respect to those descriptor parameters. That convenience was exploited directly in the multi-component NEP2 descriptor, where newly introduced species-dependent coefficients $i$0 were optimized together with neural-network weights and biases (Fan, 2021).

Later work identified original NEP training as a computational bottleneck rather than an inference bottleneck. TorchNEP kept the NEP model form but replaced SNES with analytically derived gradients, mini-batch training, Adam with AMSGrad, compiler-assisted acceleration, and a two-stage loss-weighting protocol. Across benchmark datasets, TorchNEP reported average training acceleration of about $i$1, with all tested cases falling from hundreds of hours to less than one hour on a single V100 GPU, while the main accuracy gains were traced primarily to the two-stage training protocol rather than to the optimizer change alone (Wu et al., 17 Jun 2026).

A second misconception addressed in that work is that lower training error or larger descriptor complexity necessarily implies better prediction. On the 16-element metallic benchmark, increasing model complexity monotonically reduced training RMSE but did not consistently improve test energy errors or downstream physical properties, and in some cases worsened them (Wu et al., 17 Jun 2026).

3. Descriptor evolution and physical augmentation

The first major extension of NEP targeted multi-component systems. In the original multi-species descriptor, all radial basis functions for a given ordered type pair shared a fixed coefficient $i$2, with $i$3 in NEP1. NEP2 replaced this by a trainable basis-index-dependent coefficient $i$4, so that each radial channel could learn its own chemical weighting: $i$5 The number of such coefficients is $i$6, and the training imposed $i$7 (Fan, 2021). This change left the neural architecture and descriptor dimensionality unchanged, so molecular-dynamics cost was essentially unaffected. Empirically, it reduced energy and force RMSE in PbTe by about 30%, and in Al-Cu-Mg it reduced testing energy RMSE from $i$8 to $i$9 meV/atom, force RMSE from $$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$0 to $$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$1 meV/\AA, and virial RMSE from $$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$2 to $$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$3 meV/atom (Fan, 2021).

A second line of development added missing long-range or short-range physics explicitly rather than trying to force it into a purely local descriptor. NEP-D3 defined

$$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$4

using a two-body D3(BJ) correction with environment-dependent $$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$5 coefficients. This improved bilayer-graphene binding and sliding energetics relative to pure NEP and led to an approximately 10% reduction in thermal conductivity for three metal-organic frameworks, showing that dispersion could materially affect transport observables (Ying et al., 2023). Short-range repulsion was treated analogously by adding ZBL terms: NEP89 includes a ZBL background term, and dedicated oxide and cement models adopted explicit NEP-ZBL hybrids to improve robustness under close-contact events or high-energy nonequilibrium sampling (Liang et al., 30 Apr 2025, Xu et al., 25 May 2025, Gu et al., 15 Jan 2026).

These developments indicate a general pattern in the NEP literature: when locality alone is insufficient, the preferred strategy is often not to deepen the network, but to augment the physical model around the same compact descriptor-based core.

4. Dataset construction, active learning, and model curation

A substantial part of the NEP ecosystem is devoted to dataset generation and screening. NepTrain and NepTrainKit automated perturbation, selection, labeling, retraining, and interactive dataset inspection for NEP workflows (Chen et al., 2 Jun 2025). NepTrain orchestrates modules for perturbation, NEP fitting, GPUMD sampling, VASP labeling, representative selection, and workflow control; NepTrainKit provides a GUI for descriptor-space visualization, outlier inspection, configuration-type filtering, and editing (Chen et al., 2 Jun 2025). Their bond-length filter marks a structure as nonphysical if

$$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$6

with $$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$7 and $$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$8 the covalent radii of an atomic pair under periodic boundary conditions (Chen et al., 2 Jun 2025). In the $$U_i = \sum_{\mu=1}^{N_{\rm neu}} w^{(2)}_{\mu}\tanh\left(\sum_{\nu=1}^{N_{\rm des}} w^{(1)}_{\mu\nu} q^i_{\nu} - b^{(1)}_{\mu}\right) - b^{(2)},$$9 case study, the workflow began from 5000 perturbed candidates, selected 200 representative structures, and produced a final NEP with RMSEs below 1.9 meV/atom for energies, 46.0 meV/\AA\ for forces, and 17.0 meV/atom for virials, while recovering orthorhombic-to-tetragonal and tetragonal-to-cubic transition temperatures near 309 K and 404 K (Chen et al., 2 Jun 2025).

For large simulation cells, NEPMaker introduced a D-optimality-based active-learning framework implemented in GPUMD (Wang et al., 15 Apr 2026). Instead of labeling full large-cell configurations, NEPMaker detects extrapolative atomic environments on the fly, extracts local periodic structures, and optimizes their surrounding atoms and lattice with an uncertainty objective

$q^i_\nu$0

using $q^i_\nu$1, so that boundary environments remain close to the training distribution while the target local environment is preserved (Wang et al., 15 Apr 2026). The framework was demonstrated for Na melting, $q^i_\nu$2 phase transitions, and B4–B1 transformation in GaN, including metadynamics in supercells up to 27,648 atoms (Wang et al., 15 Apr 2026).

FAST extended the same logic to fine-tuning from a foundation model. For UO$q^i_\nu$3, it combined superionic-transition-targeted sampling, active-learning-enhanced exploration, and NEP89 fine-tuning to build a final dataset of only 500 configurations, explicitly centered on the difficult superionic regime (Zhuang et al., 19 Jun 2026). This suggests that, in NEP practice, data efficiency increasingly depends less on raw model compactness alone and more on physically targeted dataset design.

5. Unified, foundation, coarse-grained, and multiscale variants

NEP has moved from system-specific models to broader unified and multiscale forms. UNEP-v1 was deployed as a general-purpose unified NEP for 16 elemental metals and their alloys—Ag, Al, Au, Cr, Cu, Mg, Mo, Ni, Pb, Pd, Pt, Ta, Ti, V, W, and Zr—and enabled hybrid MCMD studies of solute segregation in polycrystalline Al. In that context, locality of site energies made each Monte Carlo swap almost free at large $q^i_\nu$4, and the same model family was cited as enabling 100-million-atom alloy simulations with ab initio accuracy on eight A100 GPUs (Song et al., 2024).

NEP89 generalized the framework further into a foundation model spanning 89 elements across inorganic and organic systems (Liang et al., 30 Apr 2025). Its trainable parameter count scales as

$q^i_\nu$5

with species dependence split between pair-specific expansion coefficients and per-species neural-network parameters (Liang et al., 30 Apr 2025). On a GeForce RTX 4090, NEP89 was reported to handle about 8,000,000 atoms at about $q^i_\nu$6 atom-steps/s, compared with roughly 1300 atoms at about $q^i_\nu$7 atom-steps/s for MACE-MP-0 in the cited benchmark, and it supported zero-shot or fine-tuned applications ranging from multicomponent alloys to methane combustion and protein–ligand dynamics (Liang et al., 30 Apr 2025). Fine-tuning examples included a 100-configuration adaptation for $q^i_\nu$8 and a 120-configuration active-learning refinement for crystalline benzene spectroscopy (Liang et al., 30 Apr 2025).

NEP-CG and NEP-AACG extended the same formalism to coarse-grained and mixed-resolution models (Fan et al., 1 Mar 2026). In NEP-CG, the coarse-grained free-energy surface is the potential of mean force

$q^i_\nu$9

and the exact coarse-grained force is the conditional mean force

$N_{\rm neu}$0

estimated by constrained atomistic simulations and time averaging (Fan et al., 1 Mar 2026). For liquid water, virial correction was essential for the equation of state; for an anisotropic C$N_{\rm neu}$1 monolayer, distinguishing crystallographically distinct bead types reduced stress errors by more than an order of magnitude; and NEP-AACG unified all-atom and CG descriptions in a gold nanowire fracture simulation at $N_{\rm neu}$2 (Fan et al., 1 Mar 2026).

6. Scientific applications, performance envelope, and limitations

NEP’s defining practical feature remains its inference efficiency. The original GPUMD implementation reported speed over $N_{\rm neu}$3 atom-step/s on one Tesla V100 and about $N_{\rm neu}$4 per atom per MD step, while retaining correct per-atom virial and heat-current formalisms for many-body potentials (Fan et al., 2021). That efficiency has since enabled million-atom heat-transport simulations in polycrystalline graphene on consumer GPUs, with cells up to 1,438,236 atoms and direct HNEMD analysis of in-plane and flexural phonon contributions (Zhou et al., 2024). It has also enabled fully GPU-resident PIMD/TRPMD with 32–64 beads and tens of thousands of atoms for LiH, MOFs, water, and aluminum, making nuclear quantum effects tractable at scales inaccessible to direct first-principles PIMD (Ying et al., 2024).

The application range is correspondingly broad. A universal alumina NEP trained on 3335 structures and 378,762 atoms reproduced equations of state, liquid and amorphous structure, and phase diagrams over wide $N_{\rm neu}$5 ranges, while also allowing moderate extrapolation to high-pressure phases (Zhang et al., 2024). A Ga$N_{\rm neu}$6O$N_{\rm neu}$7 NEP with energy-dependent weighting and process-oriented sampling outperformed tabGAP in both accuracy and speed, and supported 1,457,920-atom swift-heavy-ion irradiation simulations that reconciled competing experimental interpretations of track structure (Gu et al., 15 Jan 2026). A tobermorite/C-S-H NEP-ZBL model achieved energy RMSE 5.159 meV/atom with only 302 training structures and was used for elasticity, phonons, thermal conductivity, and 115,303-atom C-S-H simulations (Xu et al., 25 May 2025). For wurtzite GaN, a specialized NEP plus force-error extrapolation corrected room-temperature thermal conductivity to $N_{\rm neu}$8, highlighting how sensitive high-$N_{\rm neu}$9 transport is to residual force noise (Chen et al., 8 Feb 2025). For water, NEP-MB-pol combined MB-pol-quality reference data with PIMD/TRPMD and quantum-corrected transport analysis to reproduce structure, density, heat capacity, diffusion, viscosity, and thermal conductivity over 280–370 K (Xu et al., 2024).

The limitations are equally clear. Purely local NEP does not automatically recover long-range dispersion, long-range electrostatic non-analytic corrections, or all quantum-nuclear effects; these had to be supplied by D3, careful dataset design, PIMD/TRPMD, or quantum correction depending on the problem (Ying et al., 2023, Zhang et al., 2024, Ying et al., 2024). Transferability remains dataset dependent: NEP89 protein dynamics were described as qualitative rather than protein-force-field quality, Ga$N_{\rm des}$0O$N_{\rm des}$1 still showed some high-energy PES softening even after augmentation, and C-S-H predictions beyond strain $N_{\rm des}$2 were explicitly described as qualitative because those failure states were absent from training (Liang et al., 30 Apr 2025, Gu et al., 15 Jan 2026, Xu et al., 25 May 2025). Another misconception directly rejected by the recent literature is that broader models or larger descriptors automatically solve these problems; broad models frequently still benefit from targeted fine-tuning, and larger NEP models do not necessarily generalize better (Liang et al., 30 Apr 2025, Wu et al., 17 Jun 2026).

Taken together, the literature presents NEP not as a single fixed potential, but as a coherent modeling stack: a compact local-energy formalism; multiple training regimes, from SNES to analytical-gradient fine-tuning; physically targeted descriptor and energy augmentations; active-learning and curation infrastructure; and a rapidly expanding set of unified, quantum-aware, coarse-grained, and multiscale variants. Within that stack, the persistent theme is not maximal architectural complexity, but the controlled combination of structured local descriptors, efficient inference, and problem-specific data generation.