"""
Functionality for enforcing rotational sum rules
"""
from sklearn.linear_model import Ridge
import numpy as np
from scipy.sparse import coo_matrix
from .utilities import SparseMatrix
[docs]def enforce_rotational_sum_rules(cs, parameters, sum_rules, **kwargs):
""" Enforces rotational sum rules by projecting parameters.
Note
----
The interface to this function might change in future releases.
Parameters
----------
cs : ClusterSpace
the underlying cluster space
parameters : numpy.ndarray
parameters to be constrained
sum_rules : list(str)
type of sum rules to enforce; possible values: 'Huang', 'Born-Huang'
ridge_alpha : float
hyperparameter to the ridge regression algorithm; keyword argument
passed to the optimizer; larger values specify stronger regularization,
i.e. less correction but higher stability [default: 1e-6]
iterations : int
number of iterations to run the projection since each step projects
the solution down to each nullspace in serial; keyword argument
passed to the optimizer [default: 10]
Returns
-------
numpy.ndarray
constrained parameters
Examples
--------
The rotational sum rules can be enforced to the parameters before
constructing a force constant potential as illustrated by the following
snippet::
cs = ClusterSpace(reference_structure, cutoffs)
sc = StructureContainer(cs)
# add structures to structure container
opt = Optimizer(sc.get_fit_data())
opt.train()
new_params = enforce_rotational_sum_rules(cs, opt.parameters,
sum_rules=['Huang', 'Born-Huang'])
fcp = ForceConstantPotential(cs, new_params)
"""
parameters = parameters.copy()
# handle optional kwargs input
iterations = kwargs.pop('iterations', 10)
ridge_alpha = kwargs.pop('ridge_alpha', 1e-6)
if len(kwargs) != 0:
raise ValueError('Unrecognised kwargs: {}'.format(kwargs))
# make orbit-parameter index map
params = []
n = 0
for orbit_index, orbit in enumerate(cs.orbits):
number_params_in_orbit = len(orbit.eigentensors)
params.append(list(range(n, n + number_params_in_orbit)))
n += number_params_in_orbit
# create lookuptable for force constants
lookup = {}
for orbit_index, orbit in enumerate(cs.orbits):
for of in orbit.orientation_families:
for cluster_index, perm_index in zip(of.cluster_indices,
of.permutation_indices):
cluster = cs._cluster_list[cluster_index]
perm = cs._permutations[perm_index]
lookup[tuple(cluster)] = [et.transpose(perm) for et in
of.eigentensors], orbit_index
# append the sum rule matrices
Ms = []
args = (lookup, params, cs.atom_list, cs._prim)
for sum_rule in sum_rules:
if sum_rule == 'Huang':
Ms.append(_create_Huang_constraint(*args))
elif sum_rule == 'Born-Huang':
Ms.append(_create_Born_Huang_constraint(*args))
else:
raise ValueError('At least one unknown sum rule: {}'
.format(sum_rule))
cvs_trans = cs._cvs
for i, M in enumerate(Ms):
row, col, data = [], [], []
for r, c, v in M.row_list():
row.append(r)
col.append(c)
data.append(np.float64(v))
M = coo_matrix((data, (row, col)), shape=M.shape)
Ms[i] = M.dot(cvs_trans)
for _ in range(iterations):
for M in Ms:
clf = Ridge(alpha=ridge_alpha)
d = M.dot(parameters)
clf.fit(M, d)
parameters -= clf.coef_ # subtract the correction
return parameters
def _create_Huang_constraint(lookup, parameter_map, atom_list, prim):
m = SparseMatrix(3**4, parameter_map[-1][-1] + 1, 0)
def R(i, j):
pi = atom_list[i].pos(prim.basis, prim.cell)
pj = atom_list[j].pos(prim.basis, prim.cell)
return pi - pj
for i in range(len(prim)):
for j in range(len(atom_list)):
ets, orbit_index = lookup.get(tuple(sorted((i, j))), (None, None))
if ets is None:
continue
inv_perm = np.argsort(np.argsort((i, j)))
et_indices = parameter_map[orbit_index]
for et, et_index in zip(ets, et_indices):
et = et.transpose(inv_perm)
Rij = R(i, j)
Cij = np.einsum(et, [0, 1], Rij, [2], Rij, [3])
Cij -= Cij.transpose([2, 3, 0, 1])
for k in range(3**4):
m[k, et_index] += Cij.flat[k]
return m
def _create_Born_Huang_constraint(lookup, parameter_map, atom_list, prim):
constraints = []
for i in range(len(prim)):
m = SparseMatrix(3**3, parameter_map[-1][-1] + 1, 0)
for j in range(len(atom_list)):
ets, orbit_index = lookup.get(tuple(sorted((i, j))), (None, None))
if ets is None:
continue
inv_perm = np.argsort(np.argsort((i, j)))
et_indices = parameter_map[orbit_index]
R = atom_list[j].pos(prim.basis, prim.cell)
for et, et_index in zip(ets, et_indices):
et = et.transpose(inv_perm)
tmp = np.einsum(et, [0, 1], R, [2])
tmp -= tmp.transpose([0, 2, 1])
for k in range(3**3):
m[k, et_index] += tmp.flat[k]
constraints.append(m)
M = SparseMatrix.vstack(*constraints)
return M