1. High Dimensional Atomic Neural Network
Construction of High-Dimensional Neural Network Potentials
Based on Atomic Pairs
K. V. Jovan Jose and Jörg Behler
AlgorithmAbstract
An accurate description of the interatomic potential is the crucial step in theoretical
simulations. Consequently, a large number of potentials of varying form and complexity
has been reported in the literature. Still, for some systems the accuracy that can be
achieved is not satisfying. Artificial Neural Networks (NN) have become a promising new
tool for the construction of efficient and accurate potentials due to their flexible functional
form. We present a new high-dimensional NN approach based on an expansion of the total
energy in terms of environment-dependent atom pairs. The advantages and drawbacks of
this approach are discussed and compared to the alternative approach employing a
summation of atomic energy contributions.
Methods
Artificial Neural Network Potential
PWSCF S. Baroni et al., www.quantum-espresso.org
FHI-aims Volker Blum et al., Comp. Phys. Comm. 180, 2175 (2009).
PBE XC functional J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
Conclusion and Outlook
Acknowledgements
We thank the DFG (Emmy Noether program), the FCI and the Academy of Sciences of NRW
for financial support.
Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, D-44780 Bochum, Germany
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+⋅+= ∑∑ ==
2
1
11
0
1
3
1
2
1
2
01
2
i
iijja
j
ja Gwwfwwf
T. B. Blank, S. D. Brown, A. W. Calhoun,and D. J. Doren, J. Chem. Phys. 103, 4129 (1995).
S. Lorenz, A. Groß, and M. Scheffler, Chem. Phys. Lett. 395, 210 (2004).
J. Behler, S. Lorenz, and K. Reuter, J. Chem. Phys. 127, 014705 (2007).
Advantages of Neural Networks
• Very flexible functional form
• No knowledge of the functional form required
• Valid for any electronic structure method.
• Now the NN-based method is applicable to large chemical systems
• NN reproduces total energies, forces and stress very accurately
• Fast and linear scaling with system size at any electronic structure method
Analytic Total Energy Expression:
E
Disadvantages of Neural Networks
• Inaccurate with extrapolation
• Many training points needed
• Non-physical energy functional form
Pair
Energies
Pair
Subnets
Ri={Xi,Yi,Zi}
Atomic
Positions
Symmetry
Functions
Atomic
Pairs
Results
Generate Training & Testing Sets & Initial Weights
Read Reference Structures, Energies & Forces
Yes
No
Evaluate Cost Function from Weights
Minimize the Cost Function & Update the Weights
Calculate the Cost Function RMSE
Optimum Cost Function
Nepoch < Nsp
Reference Calculations
Atomic
Subnets
Atomic
Positions
Symmetry
Functions
Atomic
Energies
Ri={Xi,Yi,Zi}
Total
Energy
∑=
=
Pairs
P
PEE
1
Input Hidden Output
High Dimensional Pair Neural Network
J. Behler, and M. Parrinello, Phys. Rev. Lett. 98, 146401 (2007).
J. Behler, R. Martonak, D. Donadio, and M. Parrinello, Phys. Stat. Sol. B245, 2618 (2008).
∑=
=
Atoms
A
AEE
1
Advantages
• Cost Function is independent of
translation and rotation
• Could handle periodic systems
Disadvantages
• Dimension α number of elements
• Non-physical energy functional
form
Input Hidden Output
Input Hidden Output
Ri={Xi,Yi,Zi}
No. Symmetry Functions: 15
No. Hidden layers & nodes: 2/15,15
No. of Epochs: 200
No. Training Points: 101
Final Energy & Force RMSE
0.00020 eV 0.00201 eV/Bohr
Advantages
Disadvantages
• Dimension α pair of elements
• More accurate representation
• Independent of translation &
rotation
• Could handle large systems
No. Symmetry Functions: 15
No. Hidden layers and nodes: 2/15,15
No. of Epochs: 200
No. Training Points: 101
Cu2Energy Fit Force Fit
Zn2O2Energy Fit Force Fit
ZnOEnergy Fit Force Fit
No. Symmetry Functions: 15
No. Hidden layers & nodes: 2/15,15
No. of Epochs: 200
No. Training Points: 101
Final Energy & Force RMSE
0.00055 eV 0.00140 eV/Bohr
Final Energy & Force RMSE
0.000855 eV 0.00450 eV/Bohr