Machine Learning Interatomic Potentials

• Machine learning interatomic potentials are data-driven models that learn atomic energy surfaces and forces from quantum mechanical reference data.
• They employ advanced descriptors like symmetry functions, SOAP, and graph-based methods to capture invariant local atomic environments.
• Balancing high accuracy with computational efficiency, MLIPs enable scalable simulations for materials design and high-throughput discovery.

A machine learning interatomic potential (MLIP) is a data-driven mathematical model that accurately reproduces the potential energy surface (PES) and associated forces of atomic systems by interpolating a reference database of quantum mechanical calculations. MLIPs replace the traditional analytic forms of interatomic interactions—often parameterized from physical intuition and a limited set of materials data—with flexible, high-dimensional regression schemes, typically constructed using kernel methods or artificial neural networks, and trained on large datasets generated via density functional theory (DFT) or higher-level quantum chemistry. This approach enables computationally efficient simulations of materials properties and processes with accuracy approaching that of first-principles methods, making MLIPs essential tools in contemporary atomistic modeling for chemistry, condensed matter physics, and materials science.

1. Fundamental Principles and Motivation

The foundational principle of machine learning interatomic potentials is that the true PES—dictating atomic forces, energies, and static and dynamical behavior—can be accurately learned as a multivariate function mapping atomic configurations to energies and forces, provided sufficient high-quality reference data. Instead of relying on fixed analytic expressions (as in, for example, the Stillinger-Weber or Tersoff forms), MLIPs utilize numerical structures—such as symmetry functions, kernel-based descriptors, or message-passing neural networks—to encode atomic neighborhoods and fit the energy landscape flexibly. The total energy $E$ of a system is commonly decomposed into local (per-atom) contributions, enforcing chemical locality, permutation, rotational, and translational invariance by design.

Mathematically, the local energy for atom $i$ can be written as

$E = \sum_i E_i(\mathcal{G}_i),$

where the descriptor vector $\mathcal{G}_i$ encodes the local atomic environment of atom $i$. The learning algorithm minimizes a composite loss function including errors in energies, forces, and stress tensors against the reference quantum mechanical data (2102.06163). The assumption of locality is systematically tested and, for most condensed-phase materials, provides an effective balance between accuracy and computational efficiency (1611.03277).

2. Descriptor Construction and Representation of Atomic Environments

A pivotal aspect of MLIPs is the construction of descriptors that transform Cartesian atomic coordinates into fixed-length, invariant vectors characterizing local chemical environments. Several major descriptor frameworks are widely used:

• Behler–Parrinello Symmetry Functions: These include radial and angular atom-centered functions designed to be invariant to permutation, rotation, and translation, serving as standard inputs for high-dimensional neural network potentials (HDNNP) (1906.08888).
• Smooth Overlap of Atomic Positions (SOAP): SOAP encodes local atomic density as a sum of Gaussians projected onto spherical harmonics and radial basis functions. The similarity between two environments is computed via a kernel as a function of the overlap of these projections (1611.03277, 1805.01568, 1906.08888). For example, the local density

$$\rho_a(\vec{r}) = \sum_{n,l,m} c_{nlm}^{(a)} g_n(r) Y_{lm}(\hat{r})$$

is expanded to obtain a power spectrum, which is used as a descriptor for kernel calculations.

• Bispectrum and Bispectrum Quadratic Extensions: Used in Spectral Neighbor Analysis Potential (SNAP), the bispectrum features are computed from hyperspherical harmonics expansions, with quadratic forms capturing more complex angular correlations (2205.01209, 1906.08888).
• Moment Tensor Potentials (MTP): These employ systematically improvable polynomial expansions in moment tensors that are contracted to provide rotationally invariant basis functions (1906.08888).
• Group-Theoretical Rotational Invariants: High-order invariants derived from the tensor products of spherical harmonics and projected onto the identity irreducible representation provide a complete, physically meaningful set for MLIPs, as demonstrated up to sixth-order for elemental Al (1901.02118).

The choice and completeness of descriptors directly govern the transferability and accuracy of the MLIP, particularly when extrapolating out of the training data regime.

3. Regression Frameworks and Model Architectures

MLIPs utilize various regression frameworks to map descriptors to energies and forces:

• Gaussian Approximation Potentials (GAP): GAP employs Gaussian process regression to interpolate between reference data points using kernels based on descriptor similarity. The total energy is often partitioned into two-body, three-body, and many-body (SOAP) terms:

$$E = (\delta^{2b})^2 \sum_i \varepsilon^{2b}(q^{2b}_i) + (\delta^{3b})^2 \sum_j \varepsilon^{3b}(q^{3b}_j) + (\delta^{MB})^2 \sum_a \varepsilon^{MB}(q^{MB}_a),$$

with each $\varepsilon^{(d)}(q^{(d)})$ as a linear combination of kernel functions centered on representative training environments (1611.03277, 1805.01568).

• Feedforward Neural Networks and HDNNP: Atomic or element-specific subnetworks are trained on symmetry function inputs, outputting energy terms summed to yield the total system energy (2404.18393, 2506.08339).
• Graph Neural Networks (GNNs) and Message-Passing Networks: Sophisticated methods such as NequIP, MACE, and Allegro use message-passing architectures to learn from both geometric and chemical features, mapping local and semi-local environments into equivariant representations and achieving state-of-the-art accuracy and transferability (2505.02503, 2505.22397).
• Linear and Nonlinear Atomic Cluster Expansion (ACE): ACE constructs the PES as a linear or nonlinear expansion over complete atomic cluster functions, systematically increasing accuracy with more terms (2505.02503).

The regression framework may incorporate uncertainty quantification—such as kernel-based error bars or ensemble averaging—to detect when the MLIP is extrapolating beyond its reliable domain (1805.01568, 2503.14293).

4. Accuracy, Efficiency, and Trade-offs

Benchmark studies show that when trained on sufficiently diverse and representative DFT data, MLIPs can achieve errors of 1–10 meV/atom for energies and 0.01–0.1 eV/Å for forces, closely matching DFT across a range of structural motifs and chemical environments (1901.02118, 1906.08888, 2505.02503). The specific accuracy depends critically on:

• Descriptor completeness and order (e.g., inclusion of high-order invariants or many-body SOAP terms).
• Regression model expressivity, regularization, and degree of freedom count.
• Training data quality and coverage of relevant configuration space.

Computationally, MLIPs enable simulations of systems orders of magnitude larger than direct DFT. GAP and HDNNP models are typically slower than more local polynomial expansions (MTP, linear ACE), but GPU-accelerated or message-passing GNN architectures (e.g., NequIP, MACE) bring MLIPs close to or surpassing the speed of non-accelerated empirical force fields, permitting nanosecond-scale and large-supercell molecular dynamics (2505.02503, 2505.22397).

Complex trade-offs arise: increasing model complexity or many-body order enhances accuracy but may introduce numerical instability, higher computational overhead, and overfitting risk. Pareto analyses reveal that nonlinear ACE and equivariant GNNs like NequIP and MACE often define the frontier of accuracy vs. computational efficiency (2505.02503).

5. Applications Across Chemistry and Materials Science

MLIPs have been successfully applied to diverse classes of materials and properties, including:

• Amorphous and Disordered Phases: The hierarchical GAP model enables realistic simulation of amorphous carbon, accurately describing liquid and surface states and phenomena such as high-temperature surface graphitization (1611.03277).
• Defects, Surfaces, and Mechanical Properties: MLIPs offer accurate descriptions of vacancy, interstitial, and defect cluster energetics (e.g., in tungsten (1908.07330), silicon (1805.01568), uranium mononitride (2411.14608)), matching DFT for migration barriers and defect formations.
• Thermodynamics and Phase Transitions: Capability for predicting melting curves, surface energies, and dynamical behavior under varying temperature and pressure, even extending to extreme conditions for carbon (2205.01209).
• Spectroscopy and Vibrational Properties: Specialized MLIPs, trained on harmonic force constants or by active learning, reproduce phonon spectra, vibrational free energies, and even infrared absorption features at a fraction of the DFT cost (2402.11383, 2506.13486).
• High-Throughput Materials Screening: MLIPs enable optimization of complex alloys (e.g., high-entropy alloys (2201.08906, 2408.06322)) over vast configuration spaces inaccessible to DFT, guiding synthesis and experimental verification.
• Complex Oxides, Battery Materials, and Multi-Component Systems: By leveraging GNN-based MLIPs and automated workflows, applications now encompass perovskites (2203.01117), La–Si–P phosphides (2506.08339), and multicomponent rock salt oxides (2408.06322).

6. Challenges, Current Limitations, and Future Directions

Despite their promise, MLIPs face several challenges:

• Transferability and Extrapolation: While MLIPs interpolate well within the trained configuration space, extrapolation outside this space—such as to novel coordination environments, high-pressure phases, or defect structures not represented in training—can lead to failure or unphysical predictions (2102.06163, 2505.02503). Hybrid or physically informed MLIPs that embed analytic models or physical constraints are being pursued to mitigate this (2102.06163).
• Data Generation and Active Learning: Automated, active learning–based data selection schemes—using uncertainty diagnostics from the regression model or committee variance—are increasingly adopted to minimize necessary DFT calculations while ensuring broad coverage of configuration space (2411.14608, 2506.13486).
• Efficient Model Design and Stability: The complexity of model architecture introduces risks of overfitting, instability, and high computational resource needs. Adaptive compositional strategies, guided by Fisher information analysis and multi-property error metrics, offer modular routes to model refinement (2504.19372). Smoothness of the PES and the accurate conservation of energy during MD simulations are critical for reliable long time-scale dynamics (2505.02503).
• Benchmarks and Usability: Community benchmarks now compare a wide range of MLIP types for accuracy, efficiency, and user-friendliness, with some models (e.g., ACE, MACE) delivering high accuracy and ease of integration in simulation codes, while others lag in user accessibility or require greater technical expertise for parameter tuning (2505.02503, 2505.22397).
• Industrial and High-Throughput Application: New libraries, such as the fully JAX-based MLIP library, facilitate industrial-grade, rapid molecular dynamics simulations using pre-trained or custom GNN-based models integrated with MD codes like ASE and JAX-MD, thus bridging the gap to industrial deployment (2505.22397).

Future directions include further integration of physical constraints, expansion to complex interfacial and multi-sublattice systems, systematic inclusion of long-range interactions, and scalable generative training protocols, all aimed at robust, automated deployment of MLIPs in predictive and discovery-oriented computational workflows.

7. Impact and Outlook

The emergence of MLIPs represents a paradigm shift in atomistic modeling, enabling DFT-accurate simulations of materials processes at scales and durations impossible with first-principles calculations. By leveraging advanced descriptors, regression techniques, and active learning, MLIPs regularly match or exceed the accuracy of conventional empirical potentials while extending transferability across structures, compositions, and property domains. Their integration into materials simulation pipelines, high-throughput discovery, advanced spectroscopy, and dynamical property prediction is rapidly accelerating, poised to become standard across many areas of physical science.

The field continues to evolve rapidly, with active research focused on automating model construction, quantifying and mitigating domain extrapolation risks, enhancing computational efficiency via hardware acceleration, and building accessible, reproducible workflows for both academic and industrial users. These advances position machine learning interatomic potentials as foundational tools for next-generation quantitative materials design and understanding.