Moment Tensor Potentials: a class of systematically improvable interatomic potentials

Alexander V. Shapeev

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Moment Tensor Potentials: a class of systematically improvable interatomic potentials

Alexander V. Shapeev

arXiv:1512.06054v2Dec 18, 2015physics.comp-phcond-mat.mtrl-scimath.NA

TL;DR

The paper addresses how to approximate quantum-mechanical interaction models with interatomic potentials that retain accuracy without quantum-mechanical computational cost. It proposes and analyzes Moment Tensor Potentials, then tests them on an existing quantum-mechanical database, reporting systematic improvement supported by theory and numerical experiments.

Problem
Quantum-mechanical calculations are accurate but too expensive for large molecular systems, while existing potentials have limited accuracy, transferability, or systematic improvability.
Method
The paper constructs nonparametric Moment Tensor Potentials using invariant polynomials and linear regression over atomic-environment representations satisfying required symmetries.
Results
The proposed potentials provably approximate any regular function with the required symmetries, and algebraic fitting convergence is confirmed by numerical experiments.
Takeaways & Limitations
MTP provides a systematically improvable interatomic-potential class that combines symmetry compliance with computational efficiency and is evaluated against existing schemes.
Takeaways & Limitations
Neural-network potentials may perform well in practice, but whether their descriptors are complete remains an open question.

Abstract

from arXiv · show

Density functional theory offers a very accurate way of computing materials properties from first principles. However, it is too expensive for modelling large-scale molecular systems whose properties are, in contrast, computed using interatomic potentials. The present paper considers, from a mathematical point of view, the problem of constructing interatomic potentials that approximate a given quantum-mechanical interaction model. In particular, a new class of systematically improvable potentials is proposed, analyzed, and tested on an existing quantum-mechanical database.

1 Introduction

Molecular modelling needs force calculations that balance quantum-mechanical accuracy against the much lower cost of empirical potentials. The paper proposes MTP as a systematically improvable, symmetry-preserving nonparametric potential based on invariant polynomials and moment tensors.

Quantum-mechanical calculations are accurate but computationally demanding, whereas empirical interatomic potentials are much cheaper but have limited accuracy and transferability.
Interatomic potentials scale as O(N), compared with O(N 3) for standard Kohn-Sham density functional theory calculations.
Parametric potentials cannot be systematically improved because they use a fixed number of numerical or functional parameters.
Nonparametric potentials can improve accuracy in principle, but increasing accuracy raises computational complexity and remains limited by the fitted quantum-mechanical model.
Nonparametric potentials require environment representations that are invariant, smooth, and complete, making descriptor construction nontrivial.
MTP combines linear regression with invariant polynomials and moment tensors, and can provably approximate any regular function satisfying the required symmetries while remaining computationally efficient.

2 Interatomic Potentials

Interatomic potentials approximate total interaction energy by summing local environment contributions subject to permutation, rotation/reflection, and cutoff smoothness requirements. Existing empirical and non-empirical representations differ in completeness, regression strategy, and computational cost.

The model assumes that the quantum-mechanical energy can be approximated by a sum of energies associated with individual atomic environments.
An atomic environment contains vectors from a central atom to neighboring atoms within a cutoff radius, typically 5 to 10 Å in practical calculations.
The local potential is mathematically a family of functions with different numbers of arguments because atomic environments can have different sizes.
2.1 Empirical Interatomic Potentials: Pair and embedded-atom potentials use descriptors based on relative coordinates and radial functions, but distance-only descriptors are incomplete because they ignore bond angles.
2.2 Non-empirical Interatomic Potentials: Neural-network potentials augment radial descriptors with angular descriptors, whereas GAP uses spherical-harmonic expansions that can represent regular symmetric functions but may be computationally expensive.

3 Moment Tensor Potentials

The paper constructs Moment Tensor Potentials from invariant polynomials that span permutation- and rotation-invariant polynomials, with efficient evaluation and provable approximation of regular quantum-mechanical interactions. The approximation analysis establishes algebraic convergence under stated analyticity and bounded-neighborhood assumptions.

3.1 Representation with Invariant Polynomials: Invariant polynomials are symmetrized over permutations, rotations, and reflections to represent regular atomic-environment potentials.The construction begins by approximating a potential with a polynomial and then enforcing the required symmetries.
3.1 Representation with Invariant Polynomials: Generating invariant polynomials directly is difficult because the number of atom permutations grows as n!, motivating the more efficient moment-tensor construction.The proposed basis has computational complexity O(n) for fixed maximal tensor order and can additionally satisfy the R3 property.
3.1 Representation with Invariant Polynomials: Moment polynomials Mµ,ν are tensor-valued weighted moments of neighboring atoms and serve as the building blocks for the basis functions.For µ = 0, the first moments correspond to mass, center of mass, and second moments of inertia; higher µ introduce distance-based weights.
3.1 Representation with Invariant Polynomials: The basis functions Bα form a spanning set for all permutation- and rotation-invariant polynomials, although their linear representation is generally not unique.The indices encode tensor contractions, and permutations of the index matrix can yield equivalent basis functions.
3.2 Approximation Error Estimate: The approximation theorem provides permutation- and rotation-invariant polynomials of degree m for the tight-binding quantum-mechanical model under bounded-neighborhood and analytic-extension assumptions.The proof proceeds through complex-domain bounds, spectral estimates for the Hamiltonian, interpolation, and symmetrization.

4 Practical Implementation

The practical implementation replaces polynomial radial components as needed, precomputes efficient tensor contractions, and fits the resulting basis with regularized linear regression. Evaluation then computes moments and recursively assembles scalar basis functions before forming energies and forces.

4 Practical Implementation: The polynomial representation is modified to satisfy the R3 property required for practical implementation, while radial function spaces must represent regular functions of distance.Examples include cutoff-weighted inverse powers and exponentially decaying radial functions.
4 Practical Implementation: Precomputation recursively decomposes tensor-valued basis elements into products of smaller tensors and selects representations minimizing the combined tensor dimensions.Reordering indices provides additional decompositions that can reduce total computation cost.
4 Practical Implementation: Online evaluation computes the moment tensors, recursively evaluates remaining tensor elements, and retains the scalar Bα needed for the potential.The resulting scalar basis functions are linearly combined over atomic environments to obtain the interaction energy.
4 Practical Implementation: Regularized linear regression fits coefficients from reference energies and forces using an overdetermined system, with ℓ0, ℓ1, and ℓ2 regularization considered.The database contains atomic configurations together with their reference energies and forces.
4.2.1 ℓ2 regularization: The ℓ2 solution uses a diagonal penalty based on XTX, whose scale matches the database and basis-function covariance; γ is selected by 16-fold cross-validation.The diagonal scaling is chosen instead of the identity matrix for compatibility with XTX.
4.2.2 ℓ0 regularization: ℓ0 regularization produces significantly sparser coefficient vectors than ℓ2 and ℓ1 in the numerical experiments, but requires more precomputation time.The number of nonzero coefficients is varied until the fitting accuracy goal is reached.

5 Numerical Experiments

The experiments evaluate MTP convergence, accuracy, computational efficiency, and cross-validation against GAP on a tungsten Kohn–Sham DFT database. MTP shows algebraic error convergence and substantially lower computational cost while matching or improving GAP accuracy.

Convergence with the Number of Basis Functions: The RMS force-fitting error decreases algebraically with basis-set size, following (#A)^−0.227 for this database.The exponent is database-dependent, and smaller basis sets exhibit a faster preasymptotic convergence rate.
Performance Tests: MTP1 has about the same number of fitting parameters as GAP, with fitting accuracy about 1.5 times better and computation time 40 times smaller.The reported CPU times exclude initialization and atomistic-neighborhood construction.
Performance Tests: MTP2 matches GAP’s fitting accuracy while using fewer fitting parameters and achieving computation more than two orders of magnitude faster.MTP2 uses 760 extracted basis functions selected with ℓ0 regularization.
Performance Tests: Sixteen-fold cross-validation indicates that MTP2 is not overfitted, whereas MTP1 requires regularization to avoid overfitting.The regularization parameter is selected by minimizing the mean cross-validation RMS error across the 16 held-out splits.

6 Conclusion

The paper proposes systematically improvable nonparametric potentials with the required symmetries, proves algebraic fitting convergence in a simple setting, and confirms it numerically. The conclusion also identifies chemically equivalent atoms as the present scope boundary.

6 Conclusion: MTP is proposed as a systematically improvable nonparametric potential satisfying the required symmetries.The paper discusses accuracy and performance advantages over existing schemes.
6 Conclusion: Algebraic convergence of fitting is proved in a simple setting and confirmed through numerical experiments.
6 Conclusion: The work assumes that all atoms are chemically equivalent, while extension to multicomponent systems is left for future publications.The proposed extension would make radial functions depend on atomic types as well as positions.