Density functional theory based neural network force fields from energy decompositions

In order to develop force fields (FF) for molecular dynamics simulations that retain the accuracy of ab initio density functional theory (DFT), we developed a machine learning protocol based on an energy decomposition scheme that extracts atomic energies from DFT calculations. Our DFT to FF (DFT2FF) approach provides almost hundreds of times more data for the DFT energies, which dramatically improves accuracy with less DFT calculations. In addition, we use piecewise cosine basis functions to systematically construct symmetry invariant features into the neural network model. We illustrate this DFT2FF approach for amorphous silicon where only 800 DFT configurations are sufficient to achieve an accuracy of 1 meV/atom for energy and 0.1 eV/A for forces. We then use the resulting FF model to calculate the thermal conductivity of amorphous Si based on long molecular dynamics simulations. The dramatic speedup in training in our DFT2FF protocol allows the adoption of a simulation paradigm where an accurate and problem specific FF for a given physics phenomenon is trained on-the-spot through a quick DFT precalculation and FF training.

Figure 1

Schematics of the neural network models. (a) High dimensional neural network potential (HDNNP), (b) single atom neural network potential (SANNP). The leftmost box is the simulation system, the second column represents the atomic descriptors including interatomic distances, and three-body angles, the third column represents the symmetry functions, the fourth column represents the neural network for the model, and the fifth column represents the energy term(s) to be trained for.

Figure 2

Piecewise cosine functions as basis functions to construct the symmetry functions. (a) graphical representation of the piecewise cosine functions when $M=10$ basis functions are used. (b) comparison of the radial distribution and the normalized values of the two-body piecewise cosine features with their respective nodes with $M=100$.

Figure 3

Training curves for SANNP and HDNNP. The errors reported are calculated from the test set. Similar results are obtained for the validation set. After 250 iterations, the errors for SANNP and HDNNP are 0.094 and 0.168 eV/A, respectively.

Figure 4

Comparison between the fitted SANNP model and DFT for the test set with 100 configurations. (a) comparison of SANNP atomic energies and DFT atomic energies in the test set. (b) comparison of the SANNP total energies and DFT total energies in the test set. (c) comparison of the SANNP forces and DFT forces in the test set. Note, none of the test set data is used in training the SANNP model.

Figure 5

Errors in (a) total energies and (b) forces as functions of the size $M$ of the piecewise cosine functions. Inner cutoff of $R_{\mathrm{inner}}=0.0\mathrm{A}$ (or no cutoff) and $R_{\mathrm{inner}}=1.9\mathrm{A}$ are also compared.

Figure 6

Comparison of DFT, neural network potential, and empirical force fields along a sequence of AIMD trajectories. (a) total energies between DFT and NNP (b) projected forces between DFT and neural network forces (c) scaled dot product between DFT and neural network forces (d) total energies between DFT, NNP, and classical force fields (with constant shifts for force field energies); (e) total energies between the neural network potential and various levels of DFT over 100 fs (f) total energies between the neural network potential and various levels of DFT over 10 fs.

Figure 7

Comparison of (a) radial distributions and (b) angular distributions between DFT (blue) and single atom neural network potential, SANNP (orange).

Figure 8

(a) The time evolution of the heat current autocorrection function (HCACF), $\langle J_x\left(t\right)J_x\left(0\right)\rangle$, for amorphous silicon at 300 K; (b) The time integration of the HCACF using Eq. (9) at 300 K; (c) The temperature dependence of the thermal conductivity from 150 to 600 K.