Moment Tensor Machine-Learning Potential: Development and Applications

doi:10.11900/0412.1961.2025.00289

Acta Metall Sin

2026, Vol. 62

Issue (5): 785-802 DOI: 10.11900/0412.1961.2025.00289

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Moment Tensor Machine-Learning Potential: Development and Applications

CHEN Xing-Qiu¹(

), WANG Jiantao¹^,², LIU Peitao¹(

)

1 Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China
2 School of Materials Science and Engineering, University of Science and Technology of China, Shenyang 110016, China

Cite this article:

CHEN Xing-Qiu, WANG Jiantao, LIU Peitao. Moment Tensor Machine-Learning Potential: Development and Applications. Acta Metall Sin, 2026, 62(5): 785-802.

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Abstract

In recent years, artificial intelligence-based computational materials modeling has advanced rapidly, with machine learning potentials (MLPs) emerging as a central research direction. By fitting ab initio reference data into continuous and differentiable functional forms, MLPs retain near-quantum-mechanical accuracy while substantially reducing computational cost. This capability alleviates the limitations of ab initio methods in simulations of large-scale systems and long timescales. Consequently, MLPs serve as a critical link between atomistic simulations and macroscopic material property predictions, enabling new possibilities in computational materials science. This review focuses on the moment tensor potential (MTP), which offers an excellent balance between accuracy and computational efficiency. This paper provides a systematic overview from three perspectives: theoretical framework, algorithmic optimization, and practical applications. First, the mathematical foundations and design principles of MTP are analyzed. Next, strategies for improving accuracy and accelerating computation are discussed. Finally, representative case studies on typical material systems are presented to demonstrate the performance of MTP, and future development directions are outlined.

Key words: interatomic potential machine learning moment tensor atomistic simulation

Received: 28 September 2025

ZTFLH:

O469

Fund: National Natural Science Foundation of China(52422112);National Natural Science Foundation of China(52188101);Advanced Materials-National Science and Technology Major Project(2025ZD0618901);Strategic Priority Research Program of Chinese Academy of Sciences(XDA041040402);Science and Technology Major Project of Liaoning Province(2024JH1/11700032)

Corresponding Authors: CHEN Xing-Qiu, professor, Tel: 15002491386, E-mail: xingqiu.chen@imr.ac.cn; LIU Peitao, professor, Tel: 17696608803, E-mail: ptliu@imr.ac.cn

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Fig.1 Workflow of machine-learned force field (MLFF) construction (HEA—high-entropy alloy; F(x^*)—atomic energy; x₁-x_n —reference local configurations; x^*—local configuration to be predicted; ω₁-ω_N —model weights; K—kernel function; RC1 and RC2—coordinates; G₁ and G₂—local structure descriptors; $γ j i$ and $a i j i j$ —neural network model parameters)

Fig.2 Evolution of typical MLFF models (NN—neural network, MPNN—message passing neural network, KRR—kernel ridge regression, ETN—equivariant tensor network, HDNNP—high dimensional neural network potential, MSA—monomial symmetrization algorithm, GAP—Gaussian approximation potential, SNAP—spectral neighbor analysis method, MTP—moment tensor potential, Amp—atomistic machine-learning package, GDML—gradient domain machine learning, PROPhet—PROPerty prophet, DTNN—deep tensor neural network, qSNAP—quaderatic spectral neighbor analysis method, DP—deep potential, ACE—atomic cluster expansion, FCHL—Fabe-Christensen-Huang-Lilienfeld, LASP—large-scale atomistic simulation with neural network potential, EANN—embedded atom neural network, AIMNet—atoms-in-molecules Net, PIP—permutationally invariant polynomial, PaiNN—polarizable atom interaction neural network, RANN—rapid artificial neural network, NequIP—neural equivariant interatomic potential, NEP—Neuro evolution potential, GemNet—geometric message passing neural network, PFP—PreFerred potential, ml-ACE—multi-layer atomic cluster expansion, BIGDML—Bravais-inspired gradient domain machine learning, REANN—recursively embedded atom neural network, HermNet—heterogeneous relational message passing network, BOTNet—body-ordered-tensor-network, ETNs—equivalent tensor networks, PESPIP—potential energy surface PIP, GNoME—graph networks for materials exploration, EDDP—ephemeral data derived potential, CHGNet—crystal Hamiltonian graph neural network, ALIGNN-FF—atomistic line graph neural network-based force field, AIMNet2—2nd generation of atoms-in-molecules neural network potential, ViSNet—vector-scalar interactive graph neural network, SEGNN—spindistance edge graph neural network, MB-PIPNet—ManyBody PIPNet, HotPP—high-order tensor message passing interatomic potential, GPTFF—graph-based pre-trained transformer force field, EquiREANN—equivalent REANN, CAMP—Cartesian atomic moment machine learning interatomic potential, CACE—Cartesian atomic cluster expansion)

Fig.3 Assessment of computational accuracy and efficiency for various MLFFs^[60] (J_Max—maximum angular momentum channel in the SNAP model, MD—molecular dynamics)

Fig.4 Decomposition routines of scalar basis functions $B α$ and $B β$ ^[66] (B_η₁-B_η₅—intermediate tensors, M₁-M₆—moment tensors)

Fig.5 Workflow of developing a (MTP) model combined with active learning sampling method (γ(x^*)—extrapolation grade for configuration x^*, γ_thr—grade threshold)

Fig.6 Efficiency and property comparisons of active learning vs random sampling for training a MTP model on the QM9 dataset^[64] (a, b) root-mean-square (RMS) error (a) and maximal error (b) as a function of training set size (c) parity plot of MTP-predicted enthalpy (H_MT) vs reference enthalpy (H_ref) for a model trained with random selection (d) parity plot for a model trained with active learning

Fig.7 Applications of MTP model on Ni-Al alloys
(a) distributions of projected first-nearest-neighboring pairs for both the complex stacking fault (CSF) and bulk Ni₃Al at 1454 K^[96] (d—displacement)
(b) temperature-dependent critical stresses to unlock a Kear-Wilsdorf lock (KWL)^[97] (τ₁, τ₂—critical stresses for unlocking the KWL)
(c) migration of stacking fault tetrahedra (SFT) cluster in Ni^[66]

Fig.8 Applications of MTP model in HEAs
(a) RMS errors in predicted energies and forces for the MTP model evaluated on the test data for different in-distribution subsystems of Ta-V-Cr-W^[103]
(b) scale and plasticity mechanism-dependent strengths of MoNbTaW and W in polycrystal and single-crystalline nanopillar^[104] (d_c, d_c1-d_c3—critical values; MPEA—multi-principal element alloy)
(c) kink nucleation mechanism and the observation of cross-slip locking^[105] (d₁₁₀—interplanar spacing of the (110) plane)

Fig.9 Applications of MTP model on solid-solid phase transitions
(a) lego-like metastable phases in Ti₃O₅^[111] (c and a—lattice constants)
(b) free energy as a function of collective variables during

β

λ

phase transition in Ti₃O₅^[111] (The dashed line denotes the PES for a concerted β-λ phase transition, whereas the solid line represents the PES for the transition pathway via intermediate metastable phases)
(c) phase diagram of In₂Se₃^[112]
(d) domain wall configurations in

β'

-2H In₂Se₃ phase^[112] (t—time)
(e) vibrational, orientational, and total entropy of low-temperature monoclinic (M-I), intermediate-temperature monoclinic (M-II), fcc (C), and high-pressure rhombohedral (R) phases of KPF₆ as a function of temperature^[113] (S_ori—orientational contribution, S_vib—vibrational contribution, S_tot—total contribution)

Fig.10 Applications of MTP model on surface and interface systems
(a) CO adsorption/binding energies and adsorption configurations on Rh(0001) surface^[121] (ML—monolayer, PBE—Perdew-Burke-Ernzerhof functional theory, RPA—random phase approximation, vdW-DF—Van der Waals density functional theory, MTP-RPA^Δ—MTP with RPA accuracy generated by the

Δ

-learning approach)
(b) Arrhenius plot of Li areal density at the interfacial regions Li-S cathode interface^[125] (E_a—Li-Fe diffusion energy barrier, D—diffusivity)
(c) free energy profile for the protons along the intermolecular axes, and snapshots from the path integral molecular dynamics (PIMD) trajectory representing proton transfer process^[122] (δ—reaction coordinate, R

O a H

and R

O b H

—distances between H and O_a/O_b atom)

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