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Gaussian Approximation Potentials: a brief tutorial introduction
Albert P. Bartók, Gábor Csányi
TL;DR
The paper addresses how atomic energies can be represented through local neighbour environments while accounting for longer-range interactions. It presents the GAP framework and reports rather good agreement between a fitted model and reference calculations in a simple example.
Problem
The central challenge is finding a local functional that describes an atom or bond’s energy from its neighbour environment.
Method
GAP represents energies as a sum of range-separated terms, including long-range electrostatic, polarizability, and van der Waals contributions.
Results
The fitted model shows rather good agreement with reference calculations in the demonstrated example, except at the edges.
Takeaways & Limitations
The example supports using GAP with a fitted model to reproduce the demonstrated reference behavior.
Takeaways & Limitations
The separation into local and long-range contributions is an uncontrolled approximation because the required separation is not determined by the model.
Abstract
from arXiv · showhide
We present a swift walk-through of our recent work that uses machine learning to fit interatomic potentials based on quantum mechanical data. We describe our Gaussian Approximation Potentials (GAP) framework, discussing a variety of descriptors, how to train the model on total energies and derivatives and the simultaneous use of multiple models. We also show a small example using QUIP, the software sandbox implementation of GAP that is available for non-commercial use.
INTRODUCTION
The introduction frames accurate materials simulation as requiring local interatomic energy functions that can approach the Born–Oppenheimer potential energy surface while accommodating changing neighbor environments. It presents machine-learning methods as a route to fitting such potentials from electronic-structure data and focuses on a concise formal exposition.
- Motivation: Accurate materials potentials should approach the Born–Oppenheimer potential energy surface rather than remain limited by simple empirical forms.This would enhance the power of atomistic simulations for interpreting experiments and predicting materials behavior.
- Problem: The materials challenge is to learn a unique local functional for an atom or bond from its neighbor environment, rather than fit the total energy of a fixed atom count.The local function must also support bond formation and breaking as the number and identity of neighboring atoms change.
- Related work: Machine learning offers computational frameworks for addressing this problem through regression, classification, density estimation, and suitable variable transformations.The introduction situates this effort alongside artificial neural networks, Gaussian processes, and regularised non-parametric optimisation.
- Contribution: This tutorial focuses on a particular exposition that succinctly presents a formalism for fitting materials potential energy surfaces from electronic-structure data.The introduction refers readers elsewhere for detailed derivations of the underlying machine-learning and statistical results.
METHODOLOGY
The methodology represents total energy as range-separated local and long-range contributions, with local terms built from descriptors of atomic environments. This decomposition is an uncontrolled approximation whose accuracy and applicability require numerical testing, and long-range interactions may be necessary for polar or charged materials.
- Energy decomposition: Total energy is written as a sum of range-separated local terms and long-range contributions.The local terms have compact support within a cutoff radius, while long-range contributions include electrostatics, polarizability, and van der Waals interactions.
- Approximation and validation: The range-separated form is an uncontrolled approximation because the Schrödinger equation does not establish this decomposition a priori.Its accuracy and applicability must therefore be tested by numerical experiments.
- Local descriptors: Local energy functionals can use descriptors such as atom-pair distances, bond angles, or complete atomic environments.The index i counts instances such as bonds, angles, or atoms in a configuration.
- Local descriptors: Descriptors transform the Cartesian coordinates of atoms in a neighborhood into inputs for local energy terms.
- Long-range interactions: Materials with significant partial charges or easily polarizable electrons require complementing the local contribution with electrostatic and dispersion interactions.Long-range terms may remain empirical or include parameters fitted to data using approaches similar to those for the local term.
Gaussian Process Regression
Gaussian process regression predicts local energies from prior quantum-mechanical observations through covariance functions between atomic environments. The framework incorporates total energies and derivatives, compositional models, compact-support cutoffs, observation noise, and sparse-point approximations.
- Gaussian process formulation: Gaussian process predictions use kernel covariances between atomic environments and previous observations, without explicitly constructing basis functions or weights.The prediction depends on the kernel function and observed data, including the covariance vector between a new value and prior observations.
- Training data: Because quantum calculations provide total energies and derivatives rather than local energies, the model predicts local energies from covariances with these available quantities.Covariance matrices use total energies or derivative quantities, while prediction covariances connect the desired local energy with available energy and derivative data.
- Many-body models: Many-body interaction models can be fitted with independent Gaussian processes for two-body and three-body contributions.The total energy is approximated by two- and three-body terms with separate descriptors and independent Gaussian-process priors.
- Cutoffs: Compact support is imposed through spatial cutoffs in descriptors or kernels so only neighbours within a predefined radial distance are considered.The cutoff should change the covariance smoothly and minimise spurious effects on derivatives; the implementation typically uses d = 1 ˚A.
- Observation noise: The model treats quantum-mechanical observations as noisy because locality, additivity, finite cutoffs, and incomplete convergence introduce discrepancies.Gaussian noise terms are added separately for total energies and derivative quantities, modifying their covariance expressions.
- Computational efficiency: Computational cost can scale linearly with the number of sparse points rather than the total number of points.The passage identifies covariance-matrix and covariance-vector computation as potentially expensive and motivates sparse-point scaling.
Descriptors
Descriptors are central to fitting potential-energy surfaces and must encode invariance to rotations, inversion, and permutations of identical atoms. The section develops pair and triplet descriptors, kernel-level permutation symmetry, and SOAP-based neighbourhood representations while noting the limitations of increasingly complex symmetrisation.
- Descriptors: Appropriate atomic-environment representations are crucial to successful machine-learning fits of potential-energy surfaces and should explicitly encode rotational, inversion, and identical-atom permutation invariance.These invariances concern transformations of atomic positions around an environment’s centre.
- Descriptors: Pair descriptors use interatomic distances, while triplet descriptors symmetrise distances and apply cutoff functions to pair and triplet terms.The triplet cutoff is the product fcut(rij)fcut(rik).
- Descriptors: SOAP represents each atomic neighbourhood with a density function, expands it using radial basis functions and spherical harmonics, and forms rotationally invariant power-spectrum descriptors.The resulting covariance is normalised, and raising the kernel to a power ζ > 1 can sharpen differences between atomic environments.
SOFTWARE
The QUIP package implements the methodology as an extensible molecular simulation sandbox with Python and multiple simulation-package interfaces. Its GAP implementation supports diverse descriptors and covariance functions, and a silicon example shows good recovery of the Stillinger–Weber potential where training data exist.
- QUIP implements the methodology as an object-oriented FORTRAN95/2003 molecular simulation sandbox for atomic configurations and interatomic potentials.
- Interfaces connect QUIP to Python, including ASE compatibility, and packages such as LAMMPS, CP2K, and CASTEP.
- The GAP implementation provides over 20 descriptors and supports squared exponential and polynomial covariance functions.
- Agreement between the fitted model and the original Stillinger–Weber pair and angle terms was rather good, except at range edges lacking input data.
- A 600-configuration silicon-cluster training set sampled at 2000 K was used to fit two- and three-body interactions with a 4.1 Å cutoff.