Neuroevolution Potential (NEP) Method

• Neuroevolution Potential (NEP) is a machine learning interatomic potential that uses local atomic environment descriptors and neural network regression for quantum-accurate energy predictions.
• It combines separable natural evolution strategy with active learning to deliver high accuracy in energies, forces, and virials at computational costs similar to empirical force fields.
• NEP extensions, such as NEP-D3 and TNEP, enable modeling of long-range dispersion and tensorial properties, broadening its application in complex materials and simulations.

The Neuroevolution Potential (NEP) method is a machine learning interatomic potential framework that combines local atomic environment descriptors with neural network regression, trained via a separable natural evolution strategy. NEP delivers quantum-mechanical accuracy for energies, forces, and virials at computational costs comparable to empirical force fields, enabling large-scale, long-timescale molecular dynamics simulations for a wide spectrum of materials systems, including those with significant anharmonicity, disorder, or chemical complexity.

1. Model Formulation and Theoretical Foundations

The NEP potential models the total energy of a system as a sum over atom-centered site energies, with each site energy expressed as a neural network function of local structural descriptors:

$U = \sum_i U_i(\mathbf{q}^i)$

$$U_i = \sum_{\mu=1}^{N_\mathrm{neu}} w^{(1)}_\mu \tanh\left( \sum_{\nu=1}^{N_\mathrm{des}} w^{(0)}_{\mu\nu} q^i_\nu - b^{(0)}_\mu \right) - b^{(1)}$$

where:

• $q^i_\nu$ are atomic environment descriptors for atom $i$,
• $w^{(0)}, w^{(1)}$, and $b^{(0)}, b^{(1)}$ are trainable neural network parameters,
• $N_\mathrm{des}$ is the descriptor dimension, $N_\mathrm{neu}$ the number of hidden neurons,
• The activation function is typically hyperbolic tangent.

The descriptors combine radial and angular components. Radial descriptors use Chebyshev polynomials; angular descriptors incorporate Legendre polynomials over pairs and triples of neighbor atoms:

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

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

$$g_n(r_{ij}) = \frac{T_n(x_{ij})+1}{2} f_c(r_{ij}), \quad x_{ij} = 2(r_{ij}/r_c-1)^2-1$$

These formulations achieve symmetry with respect to translation, rotation, and permutation of like atoms.

The network parameters and descriptor expansions are optimized using a separable natural evolution strategy (SNES), a population-based, derivative-free optimizer well-suited for the high-dimensional non-convex loss landscape encountered in neural network regression of atomistic data.

2. Descriptor Construction and Chemistry Encoding

The NEP formalism supports modeling of both single-component and multi-component systems. For multiple species, radial functions are modified with trainable type-dependent prefactors for each descriptor and species pair, significantly improving regression accuracy for alloys and compounds. In NEP2 and subsequent versions, each radial function is weighted by a trainable parameter $c_{nij}$:

$g_n(r_{ij}) = T_n(\ldots) f_c(r_{ij}) c_{nij}$

where $n$ indexes the basis, $i, j$ the atom types, and $c_{nij}$ is optimized during training and constrained to avoid near-zero values.

The dimension of the descriptor vector scales as:

$N_\text{des} = (n^\text{R}_{\max} + 1) + (n^\text{A}_{\max} + 1)l_\max$

where $n^\text{R}_{\max}$ and $n^\text{A}_{\max}$ are the maximum orders of the radial and angular expansions, and $l_\max$ the maximum angular momentum.

3. Data Efficiency and Training Algorithms

NEP employs an active-learning-style iterative approach for dataset construction:

1. Generate initial structures covering polymorphs, amorphous forms, surfaces, and defects.
2. Compute DFT energies, forces, and virials.
3. Train a provisional NEP.
4. Use this NEP to run MD simulations across the relevant phase space.
5. Use descriptor-space farthest-point sampling to select new, maximally informative configurations.
6. Perform DFT calculations for these, augment the dataset, and retrain NEP.
7. Iterate until property convergence is achieved.

This methodology yields high-accuracy potentials with training sets (hundreds to a few thousand configurations) that are smaller and less redundant than in typical deep learning potentials, without sacrificing transferability for large, unanticipated atomic configurations.

4. Extensions and Specialized Frameworks

Several NEP variants extend the method:

• NEP-D3: Incorporates long-range dispersion via additive D3 correction, enabling accurate modeling of van der Waals interactions in layered and porous materials. The total energy is:

$U^{\rm{NEP-D3}} = U^{\rm{NEP}} + U^{\rm{D3}}$

where $U^{\rm{D3}}$ is environment-dependent, not merely pairwise.

• TNEP (Tensorial NEP): Provides the framework for modeling rank-1 (e.g., dipole moment) and rank-2 (e.g., polarizability, susceptibility) tensorial properties, extending NEP applicability to IR and Raman spectra:

$$\boldsymbol{\mu} = -\sum_{i}\sum_{j\neq i} r_{ij}^2 \frac{\partial U_i}{\partial \mathbf{r}_{ij}}$$

$$\alpha^{\upsilon\nu} = \sum_{i} U_i \delta^{\upsilon\nu} - \sum_{i}\sum_{j\neq i} r_{ij}^\upsilon \frac{\partial U_i}{\partial r_{ij}^\nu}$$

• ZBL Hybridization: For systems subject to extreme events (e.g., primary radiation damage), NEP includes a short-ranged Ziegler-Biersack-Littmark (ZBL) repulsion to prevent unphysical atom overlaps.
• Foundation/Universal NEPs: Using modular parameterization and active-learning-driven coverage, extended NEP models such as UNEP-v1 (16 elemental metals) and NEP89 (89 elements) have been developed, supporting high-fidelity atomistic simulation of virtually all classes of inorganic and organic materials.

5. Computational Implementation and Efficiency

The NEP method is fully implemented and optimized in the GPUMD package for both CPUs and GPUs. Notable features include:

• Highly parallel algorithms: Each atom is handled by an independent thread, allowing rapid computation for multi-million-atom systems.
• Empirical-potential-like speed: NEP achieves $\gtrsim 10^7$ atom-steps per second on a contemporary GPU (e.g., Nvidia RTX4090/V100), which is typically three to four orders of magnitude faster than state-of-the-art deep learning MLPs, and competitive with classical force fields.
• Minimal memory demand: NEP model parameter scaling is modest with respect to system complexity and number of species.
• Flexible input workflows: The NepTrain and NepTrainKit toolkits provide automated dataset construction, screening (e.g., bond-length filtering), descriptor-space coverage analysis, and graphical outlier detection, facilitating community adoption.

6. Applications and Impact

NEP has been applied to a broad range of materials and phenomena:

• Heat transport: NEP-enabled HNEMD and Green-Kubo MD give conductivity predictions matching BTE/DFT and experiment for crystals, alloys, MOFs, and 2D materials, including systems where perturbative BTE fails (e.g., strong anharmonicity, amorphous phases, nanostructured and defective samples).
• Phase stability and transitions: Universal NEP models allow robust phase diagram extrapolation under extreme P–T conditions (e.g., Al$_2$O$_3$ up to 200 GPa and 4000 K), including dynamic/hydrostatic transformation and amorphization.
• Alloy design and segregation: Unified NEPs, paired with highly efficient hybrid MC/MD algorithms, enable atomistically resolved, ab initio-accuracy studies of solute segregation, strengthening/embrittlement, and chemical ordering in polycrystals and high-entropy alloys.
• Mechanical properties and fracture: NEP provides crack propagation and dislocation dynamics insights at scales and statistics impossible with DFT or empirical methods.
• Tensorial response: The TNEP approach enables accurate vibrational spectroscopy (IR, Raman) simulations of complex molecules, liquids, and solids.
• Liquid water and biomolecular systems: NEP, fine-tuned to high-level quantum data (e.g., MB-pol), reproduces anomalous thermodynamic and transport properties of water (density, viscosity, diffusion, conductivity) across broad temperature/pressure ranges.
• Universality and extensibility: The NEP89 foundation model supports chemistry transfer for arbitrary materials and rapid fine-tuning on small data for specialized applications, bridging empirical efficiency with first-principles accuracy.

7. Limitations, Comparisons, and Future Prospects

Compared to other advanced MLPs:

• Accuracy: NEP approaches or matches the best-performing MLPs (e.g., MACE, Deep Potential, GAP) in energy and force generalization, but may be modestly surpassed in ultra-low-RMSE fitting in highly curated benchmarks.
• Efficiency: NEP provides the best efficiency/accuracy tradeoff for diverse, large-scale, and multi-chemical simulations.
• Long-range interactions and electronic effects: Standard NEP lacks explicit electronic polarization or charge transfer; ongoing developments address this via integration of long-range (e.g., D3, charge-equilibration) modules.
• Automated active learning and foundation expansion: The NepTrain and NepTrainKit tools address previous bottlenecks in dataset curation and error analysis, supporting widespread deployment and iteratively improved, universal NEP models.

Future directions include integration with more ab initio tensorial data for optical/ferroelectric phenomena, expanded periodic table coverage, improved modeling of electronic and quantum effects (via PIMD and hybrid methods), and seamless coupling to materials informatics and application-specific property pipelines.