Constrained by the fixed mathematical form of most empirical potentials used in classical molecular dynamics (MD) simulations, many properties of materials cannot be captured within experimental accuracy. On the other hand, accurate electronic structure calculations based on quantum theory, most notably density functional theory (DFT), are limited to several hundred atoms within a picosecond, which makes the method inadequate for modeling systems beyond the nanoscale. A combination of speed from classical MD and fidelity from DFT can be achieved through machine learning methods. Herewith, we developed an approach named spatial density neural network force fields (SDNNFFs) by training neural networks to “learn” and predict DFT-level forces. Our model focuses on the usage of a three-dimensional mesh of density functions, which together act as a mapping of the atomic environment and provides a physical representation of the forces acting on the central atom. Several notable advantages arise from the SDNNFF, including (1) the avoidance of the chain rule on the total energy and other variables by direct calculation of the forces from the neural network, (2) the ever large $N\times t$ scaling of the training data, where $N$ is the number of atoms in a supercell and $t$ is the number of evaluated structures by first-principles, and (3) the significant reduction in parameters and human effort needed to successfully train a force- and/or property-converged neural network force field. Overall, we focus on modeling DFT-level forces with minimal computational cost and parametrization for rapid prediction of phonon-based properties and future molecular dynamics of large-scale systems. To demonstrate the SDNNFF, we trained several models on diamond structures, including bulk silicon (Si), diamond, silicon carbide (SiC), and boron arsenide (BAs), and predicted their phonon dispersions and lattice thermal conductivities using the direct solution to the phonon Boltzmann transport equation. For phonon properties, we utilized a fitting method for obtaining the second- and third-order force constants, which outperforms the highly force-sensitive finite displacement method when employing neural network force fields. In comparison to DFT lattice thermal conductivity, we obtained high precision results from our SDNNFF within 0.7% for Si, 6.2% for diamond, 2.76% for SiC, and 7.46% for BAs, with further agreement with experiments. The phonon dispersions from the SDNNFF also matched those from direct DFT and experiments. The developed approach for accurately predicting phonon transport properties of crystalline materials would largely benefit the design of advanced materials with improved performance, such as complex thermoelectric devices and low thermal resistance interfaces for nanoelectronics. Future applications of our SDNNFF model could be extended toward including atomic energy into the algorithm and simulating large-scale heterogeneous systems for quasielectronic representations for various properties.

• Received 11 February 2020
• Revised 15 May 2020
• Accepted 24 June 2020

DOI:https://doi.org/10.1103/PhysRevB.102.035203