Differentiable Gaussian Process Force Constants

The finite temperature properties of matter requires understanding thermal motion. In crystals this<br/>can be described by representing the collective excitation of the centre of mass motion as a phonon.<br/>Phonon frequencies (Energies) in the harmonic approximation come from the second order force con-<br/>stants (FC) of the potential energy surface (PES). The standard approach is to use a finite displacement<br/>method (FDM). Anharmonic contributions (required for finite thermal conductivity) require higher order<br/>force constants. FDM-based calculations scale poorly both with the size of the system, and the order of<br/>the force constants[1]. As an alternative approach, we can use a more sophisticated surrogate potential<br/>energy surface model [2].<br/>Gaussian processes (GPs) are a supervised machine learning method which can describe an arbitrary<br/>function. GPs are naturally Bayesian (probabilistic). Our reference data is the electronic structure from<br/>which conditions the model (training) [3, 4].<br/>Due to the underlying Gaussian form of a GP, the model is infinitely differentiable [5, 6]. This allows<br/>the model to be trained directly on forces (the derivative of energy), reducing the number of calculations<br/>required for a given accuracy of PES evaluation [7-9]. This differentiation can be extended to compute<br/>the second and the third derivative of PESs (harmonics and cubic anharmonic FCs) by using automatic<br/>differentiation (AD). By performing linear operations between arbitrary derivative orders of the GP, the<br/>covariance functions among PESs, forces and those FCs can be calculated.<br/>We implement this method in the Julia language, in our GPFC.jl package. We first use our tech-<br/>nique of anharmonic property calculations of Si, NaCl and PbTe which their atomic environments are<br/>represented on atomic Cartesian coordinates. To impose their space group symmetry, phonon coordinate<br/>representations are then used to describe the environment of these three materials resulting in faster con-<br/>vergence of anharmonic properties. To further impose a three-dimensional rotation symmetry, we aim to<br/>use an SO(3) representation (such as spherical harmonics) with our GP method to achieve accurate FCs<br/>with less data [3, 10]. We compare our results to standard approach of FDM (in Phono(3)py.py) [2]<br/>and cluster expansion (HiPhive.py) [11].<br/>References<br/>[1] A. Togo, L. Chaput and I. Tanaka, Distributions of phonon lifetimes in brillouin zones, Physical Review B 91 (2015) .<br/>[2] A. Togo et al., Implementation strategies in phonopy and phono3py, Journal of Physics: Condensed Matter 35 (2023) 353001.<br/>[3] A.P. Bartók et al., Gaussian approximation potentials: A brief tutorial introduction, International Journal of Quantum Chemistry 115 (2015) 1051.<br/>[4] H. Sugisawa et al., Gaussian process model of 51-dimensional potential energy surface for protonated imidazole dimer, The Journal of Chemical Physics 153 (2020) 114101.<br/>[5] E. Solak et al., Derivative observations in Gaussian Process Models of Dynamic Systems, NIPS’02: Proceedings of the 15th International Conference on Neural Information Processing Systems (2002) 8.<br/>[6] A. McHutchon, Differentiating Gaussian Processes.<br/>[7] E. Garijo del Río et al., Local Bayesian optimizer for atomic structures, Physical Review B 100 (2019) 104103.<br/>[8] S. Kaappa et al., Global optimization of atomic structures with gradient-enhanced gaussian process regression, Physical Review B 103 (2021) .<br/>[9] K. Asnaashari et al., Gradient domain machine learning with composite kernels: improving the accuracy of PES and force fields for large molecules, Machine Learning: Science and Technology 3 (2021) 015005.<br/>[10] V.L. Deringer et al., Gaussian process regression for materials and molecules, Chemical Reviews 121 (2021) 10073.<br/>[11] F. Eriksson et al., The hiphive package for the extraction of high-order force constants by machine learning, Advanced Theory and Simulations 2 (2019) .