Theoretical background

Structural features

The atomic energy is described by a function of polynomial invariants for the O(3) group. A pth-order polynomial invariant for a radial index n and a set of pairs composed of the angular number and the element unordered pair {(l1,t1),(l2,t2), …,(lp,tp)} is defined as a linear combination of products of p order parameters, expressed as

d_{nl_1l_2\cdots l_p,t_1t_2\cdots t_p,(\sigma)}^{(i)} =
\sum_{m_1,m_2,\cdots, m_p} c^{l_1l_2\cdots l_p,(\sigma)}_{m_1m_2\cdots m_p}
a_{nl_1m_1,t_1}^{(i)} a_{nl_2m_2,t_2}^{(i)} \cdots a_{nl_pm_p,t_p}^{(i)}.

Linearly independent coefficient sets are obtained using the group-theoretical projector operation method for a given set of angular numbers, ensuring that the linear combinations are invariant for arbitrary rotation. In terms of fourth- and higher-order polynomial invariants, multiple invariants are linearly independent for most of the angular number sets, which are distinguished by index \sigma if necessary. The order parameters of atom i and element pair (si,s) are approximately estimated from the neighboring atomic density of element s around atom i as

a_{nlm,\{s_i,s\}}^{(i)} = \sum_{\{j | r_{ij} \leq r_c,s_j = s\} }
f_n(r_{ij}) Y_{lm}^* (\theta_{ij}, \phi_{ij}),

where (r_{ij}, \theta_{ij}, \phi_{ij}) denotes the spherical coordinates of neighboring atom j centered at the position of atom i.

[1] A. Seko, A. Togo, and I. Tanaka, Group-theoretical high-order rotational invariants for structural representations: Application to linearized machine learning interatomic potential, Phys. Rev. B 99, 214108 (2019)

[2] A. Seko, Machine learning potentials for multicomponent systems: The Ti-Al binary system, Phys. Rev. B 102, 174104 (2020)

Potential energy models

In the repository, the atomic energy is measured from the energy of the isolated state of the atom. A potential energy model is identified with a combination of polynomial functions and structural features. Given a set of structural features D, polynomial functions are written as

F_1 \left(D\right) = \sum_i w_i d_i \\

F_{2,\rm pow} \left(D\right) = \sum_{i} w_{ii} d_i d_i \\

F_2 \left(D\right) = \sum_{\{i,j\}} w_{ij} d_i d_j \\

F_{3,\rm pow} \left(D\right) = \sum_{i} w_{iii} d_i d_i d_i \\

F_3 \left(D\right) = \sum_{\{i,j,k\}} w_{ijk} d_i d_j d_k

where w denotes a coefficient.

  • Pairwise structural features

D_{\rm pair}^{(i)} = \{d_{n0}^{(i)}\}

  • Polynomial invariants

D^{(i)} = D_{\rm pair}^{(i)} \cup D_2^{(i)} \cup D_3^{(i)} \cup D_4^{(i)} \cup \cdots

D_2^{(i)} = \{d_{nll}^{(i)}\}

D_3^{(i)} = \{d_{nl_1l_2l_3}^{(i)}\}

D_4^{(i)} = \{d_{nl_1l_2l_3l_4,(\sigma)}^{(i)}\}

Machine learning potentials in the repository are developed from the following potential energy models.

  • model type = 1, feature type = pair

E^{(i)} = F_1\left(D_{\rm pair}^{(i)} \right)
        + F_{2,\rm pow} \left(D_{\rm pair}^{(i)} \right)
        + F_{3,\rm pow} \left(D_{\rm pair}^{(i)} \right)
        + \cdots

  • model type = 2, feature type = pair (An extension of EAM potentials)

E^{(i)} = F_1\left(D_{\rm pair}^{(i)} \right)
        + F_{2} \left(D_{\rm pair}^{(i)} \right)
        + F_{3} \left(D_{\rm pair}^{(i)} \right)
        + \cdots

  • model type = 1, feature type = polynomial invariants (Linear polynomial form)

E^{(i)} = F_1 \left(D^{(i)} \right)

  • model type = 1, feature type = polynomial invariants

E^{(i)} = F_1 \left(D^{(i)} \right)
        + F_{2,\rm pow} \left(D^{(i)} \right)
        + F_{3,\rm pow} \left(D^{(i)} \right)
        + \cdots

  • model type = 2, feature type = polynomial invariants

E^{(i)} = F_1 \left(D^{(i)} \right)
        + F_{2} \left(D^{(i)} \right)
        + F_{3} \left(D^{(i)} \right)
        + \cdots

  • model type = 3, feature type = polynomial invariants

E^{(i)} = F_1 \left( D^{(i)} \right)
        + F_2 \left( D_{\rm pair}^{(i)} \right)
        + F_3 \left( D_{\rm pair}^{(i)} \right)
        + \cdots

  • model type = 4, feature type = polynomial invariants

E^{(i)} = F_1 \left( D^{(i)} \right)
        + F_2 \left( D_{\rm pair}^{(i)} \cup D_2^{(i)} \right)
        + \cdots

Parameters for developing MLPs

Parameters used for developing an MLP are found in log and input links at column “Files”.

  • Example of parameters

    # number of atom species
    n_type 2
    # use derivatives in training or not: True or False
    with_force True
    # regression method (ridge or lasso or normal)
    reg_method ridge
    # cutoff radius
    cutoff 10.0
    # model type with respect to structural features
    # 1 (power of features)
    # 2 (polynomial of all features)
    # 3 (polynomial of pair features + invariants)
    # 4 (polynomial of pair features and invariants(order=2) + invariants)
    model_type 1
    # degree of polynomials
    max_p 2
    # structural feature type (pair or gtinv)
    des_type gtinv
    # pairwise function type (gaussian or sph_bessel)
    pair_type gaussian
    # sequence for a in exp(-a(r-b)^2) [min, max, n]
    gaussian_params1 1.0 1.0 1
    # sequence for b in exp(-a(r-b)^2) [min, max, n]
    gaussian_params2 0 10.0 15
    # maximum order of group-theoretical invariants
    gtinv_order 4
    # maximum l values of group-theoretical invariants
    gtinv_maxl 7 7 2 0 0
    # use only symmetric invariants or not
    gtinv_sym False False False False False