Fast Neural Network Approach for Direct Covariant Forces Prediction in Complex Multi-Element Extended Systems

Fast Neural Network Approach for Direct Covariant Forces Prediction in Complex Multi-Element Extended Systems

Abstract Neural network force field (NNFF) is a method for performing regression on atomic structure – force relationships, bypassing expensive quantum mechanics calculation which prevents the execution of long ab-initio quality molecular dynamics simulations. However, most NNFF methods for complex multi- element atomic systems indirectly predict atomic force vectors by exploiting just atomic structure rotation-invariant features and the network–feature spatial derivatives which are computationally expensive. We develop a staggered NNFF architecture exploiting both rotation-invariant and covariant features separately to directly predict atomic force vectors without using spatial derivatives, thereby reducing expensive structural feature calculation by ~180–480×. This acceleration enables us to develop NNFF which directly predicts atomic forces in complex ternary and quaternary–element extended systems comprised of long polymer chains, amorphous oxide, and surface chemical reactions. The staggered rotation-invariant-covariant architecture described here can also directly predict complex covariant vector outputs from local physical structures in domains beyond computational material science.

Introduction Ab-initio molecular dynamics (AIMD) is an atomistic simulation method widely used to study the movement of atoms in a physical system, where the forces experienced by each atom in the system is calculated using a quantum mechanics method such as density functional theory (DFT). Quantum interatomic force calculations are produced by solving a many-body system including electrons (e.g. Schrödinger, Kohn-Sham equations). The computational cost of these methods make AIMD computationally challenging for realistic physical phenomena that can be explored only when the simulated system is sufficiently large (many atoms) and/or has run for a long time (many time steps). However, AIMD is still widely used (despite the limitations in size and time scale) because it requires no prior assumption on the potential energy surface, and it can be used to accurately simulate interesting phenomena such as chemical reactions, phase changes, ionic transport, surface interactions, etc in a wide variety of material systems. Classical molecular dynamics (MD) methods based on fast force calculations using pre-fitted empirical functions are 105–106 × faster than AIMD, but the limitations of simple empirical functions often mean that they cannot be used to study complex atomic interactions, e.g. chemical reactions. Machine learning methods have increasingly been used to perform atomistic computations of energies and atomic forces with greater accuracy than empirical functions.1–11 Some of these latest methods have shown high force prediction accuracy (error within 1 kcal/mol∙Å = 0.043 eV/Å) for single-molecule systems in vacuum,12,13 but may be unsuitable for extended solid-state atomistic systems containing large number of atoms. Other kernel-based methods such as the Gaussian process regression have been used for developing force fields for single-element nanocluster in vacuum (force error within 0.20 eV/Å for many- body kernels).14 The neural network force field (NNFF) utilizes flexible neural network (NN) functions at fixed computation cost (independent of training sample size), which in turn enables indirect atomic structure – force regression involving many-body interactions, and has been used for complex multi- element extended systems.15–17 Due to the vast number of possible atomic configurations in 3- dimensional space, NNFF models were difficult to train.18 The development of ‘atomic fingerprints’, by Behler and Parrinello (B–P) and others,15,19 has enabled a step-change improvement in the accuracy of NNFF. Similar to convolutional and graph neural network (CNN and GNN),20–22 the atomic fingerprints rely on features obtained from localized proximity in space. Unlike CNN and GNN which utilize fixed-size grid- based and graph-based feature space respectively, Behler-Parrinello style atomic fingerprints use a fixed- size symmetry-function-based feature space utilizing all atoms located at 𝑟𝑟⃗𝑗𝑗 within a specified cutoff radius Rc from the target central atom i (൛𝑅𝑅ሬ⃗𝑖𝑖ൟ=

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