Multi-element Kernels (simple)¶

Multi-element 2-, 3-, and 2+3-body kernels that restrict all signal variance hyperparameters to a single value.

flare.kernels.mc_simple.many_body_mc(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

many-body multi-element kernel between two force components.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 3-body kernel.
Return type:	float

flare.kernels.mc_simple.many_body_mc_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

many-body multi-element kernel between two local energies.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – One-element array containing the 2-body cutoff. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2-body force/energy kernel.
Return type:	float

flare.kernels.mc_simple.many_body_mc_en_jit(q_array_1, q_array_2, c1, c2, species1, species2, sig, ls)¶

many-body many-element kernel between energy components accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – many-body bond array of the first local environment. bond_array_2 (np.ndarray) – many-body bond array of the second local environment. c1 (int) – atomic species of the central atom in env 1 c2 (int) – atomic species of the central atom in env 2 etypes1 (np.ndarray) – atomic species of atoms in env 1 etypes2 (np.ndarray) – atomic species of atoms in env 2 species1 (np.ndarray) – all the atomic species present in trajectory 1 species2 (np.ndarray) – all the atomic species present in trajectory 2 sig (float) – many-body signal variance hyperparameter. ls (float) – many-body length scale hyperparameter. r_cut (float) – many-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the many-body kernel.
Return type:	float

flare.kernels.mc_simple.many_body_mc_force_en(env1, env2, d1, hyps, cutoffs, cutoff_func=<function quadratic_cutoff>)¶

many-body single-element kernel between two local energies.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – Two-element array containing the 2-, 3-, and many-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the many-body force/energy kernel.
Return type:	float

flare.kernels.mc_simple.many_body_mc_force_en_jit(q_array_1, q_array_2, q_neigh_array_1, q_neigh_grads_1, c1, c2, etypes1, species1, species2, d1, sig, ls)¶

many-body many-element kernel between force and energy components accelerated with Numba.

Parameters:	c1 (int) – atomic species of the central atom in env 1 c2 (int) – atomic species of the central atom in env 2 etypes1 (np.ndarray) – atomic species of atoms in env 1 species1 (np.ndarray) – all the atomic species present in trajectory 1 species2 (np.ndarray) – all the atomic species present in trajectory 2 d1 (int) – Force component of the first environment. sig (float) – many-body signal variance hyperparameter. ls (float) – many-body length scale hyperparameter.
Returns:	Value of the many-body kernel.
Return type:	float

flare.kernels.mc_simple.many_body_mc_grad(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶: gradient manybody-body multi-element kernel between two force components.

flare.kernels.mc_simple.many_body_mc_grad_jit(q_array_1, q_array_2, q_neigh_array_1, q_neigh_array_2, q_neigh_grads_1, q_neigh_grads_2, c1, c2, etypes1, etypes2, species1, species2, d1, d2, sig, ls)¶

gradient of many-body multi-element kernel between two force components w.r.t. the hyperparameters, accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – many-body bond array of the first local environment. bond_array_2 (np.ndarray) – many-body bond array of the second local environment. neigh_dists_1 (np.ndarray) – matrix padded with zero values of distances of neighbours for the atoms in the first local environment. neigh_dists_2 (np.ndarray) – matrix padded with zero values of distances of neighbours for the atoms in the second local environment. num_neigh_1 (np.ndarray) – number of neighbours of each atom in the first local environment num_neigh_2 (np.ndarray) – number of neighbours of each atom in the second local environment c1 (int) – atomic species of the central atom in env 1 c2 (int) – atomic species of the central atom in env 2 etypes1 (np.ndarray) – atomic species of atoms in env 1 etypes2 (np.ndarray) – atomic species of atoms in env 2 etypes_neigh_1 (np.ndarray) – atomic species of atoms in the neighbourhoods of atoms in env 1 etypes_neigh_2 (np.ndarray) – atomic species of atoms in the neighbourhoods of atoms in env 2 species1 (np.ndarray) – all the atomic species present in trajectory 1 species2 (np.ndarray) – all the atomic species present in trajectory 2 d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. sig (float) – many-body signal variance hyperparameter. ls (float) – many-body length scale hyperparameter. r_cut (float) – many-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the many-body kernel and its gradient w.r.t. sig and ls
Return type:	array

flare.kernels.mc_simple.many_body_mc_jit(q_array_1, q_array_2, q_neigh_array_1, q_neigh_array_2, q_neigh_grads_1, q_neigh_grads_2, c1, c2, etypes1, etypes2, species1, species2, d1, d2, sig, ls)¶

many-body multi-element kernel between two force components accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – many-body bond array of the first local environment. bond_array_2 (np.ndarray) – many-body bond array of the second local environment. neigh_dists_1 (np.ndarray) – matrix padded with zero values of distances of neighbours for the atoms in the first local environment. neigh_dists_2 (np.ndarray) – matrix padded with zero values of distances of neighbours for the atoms in the second local environment. num_neigh_1 (np.ndarray) – number of neighbours of each atom in the first local environment num_neigh_2 (np.ndarray) – number of neighbours of each atom in the second local environment c1 (int) – atomic species of the central atom in env 1 c2 (int) – atomic species of the central atom in env 2 etypes1 (np.ndarray) – atomic species of atoms in env 1 etypes2 (np.ndarray) – atomic species of atoms in env 2 etypes_neigh_1 (np.ndarray) – atomic species of atoms in the neighbourhoods of atoms in env 1 etypes_neigh_2 (np.ndarray) – atomic species of atoms in the neighbourhoods of atoms in env 2 species1 (np.ndarray) – all the atomic species present in trajectory 1 species2 (np.ndarray) – all the atomic species present in trajectory 2 d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. sig (float) – many-body signal variance hyperparameter. ls (float) – many-body length scale hyperparameter. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the many-body kernel.
Return type:	float

flare.kernels.mc_simple.three_body_mc(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

3-body multi-element kernel between two force components.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 3-body kernel.
Return type:	float

flare.kernels.mc_simple.three_body_mc_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

3-body multi-element kernel between two local energies.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 3-body force/energy kernel.
Return type:	float

flare.kernels.mc_simple.three_body_mc_en_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, cross_bond_inds_1, cross_bond_inds_2, cross_bond_dists_1, cross_bond_dists_2, triplets_1, triplets_2, sig, ls, r_cut, cutoff_func)¶

3-body multi-element kernel between two local energies accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 3-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 3-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. cross_bond_inds_1 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_inds_2 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_1 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_2 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. triplets_1 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the first local environment that are within a distance r_cut of atom m. triplets_2 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the second local environment that are within a distance r_cut of atom m. sig (float) – 3-body signal variance hyperparameter. ls (float) – 3-body length scale hyperparameter. r_cut (float) – 3-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 3-body local energy kernel.
Return type:	float

flare.kernels.mc_simple.three_body_mc_force_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

3-body multi-element kernel between a force component and a local energy.

Parameters:	env1 (AtomicEnvironment) – Local environment associated with the force component. env2 (AtomicEnvironment) – Local environment associated with the local energy. d1 (int) – Force component of the first environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 3-body force/energy kernel.
Return type:	float

flare.kernels.mc_simple.three_body_mc_force_en_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, cross_bond_inds_1, cross_bond_inds_2, cross_bond_dists_1, cross_bond_dists_2, triplets_1, triplets_2, d1, sig, ls, r_cut, cutoff_func)¶

3-body multi-element kernel between a force component and a local energy accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 3-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 3-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. cross_bond_inds_1 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_inds_2 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_1 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_2 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. triplets_1 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the first local environment that are within a distance r_cut of atom m. triplets_2 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the second local environment that are within a distance r_cut of atom m. d1 (int) – Force component of the first environment (1=x, 2=y, 3=z). sig (float) – 3-body signal variance hyperparameter. ls (float) – 3-body length scale hyperparameter. r_cut (float) – 3-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 3-body force/energy kernel.
Return type:	float

flare.kernels.mc_simple.three_body_mc_grad(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) -> ('float', 'ndarray')¶

3-body multi-element kernel between two force components and its gradient with respect to the hyperparameters.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 3-body kernel and its gradient with respect to the hyperparameters.
Return type:	(float, np.ndarray)

flare.kernels.mc_simple.three_body_mc_grad_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, cross_bond_inds_1, cross_bond_inds_2, cross_bond_dists_1, cross_bond_dists_2, triplets_1, triplets_2, d1, d2, sig, ls, r_cut, cutoff_func)¶

3-body multi-element kernel between two force components and its gradient with respect to the hyperparameters.

Parameters:	bond_array_1 (np.ndarray) – 3-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 3-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. cross_bond_inds_1 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_inds_2 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_1 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_2 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. triplets_1 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the first local environment that are within a distance r_cut of atom m. triplets_2 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the second local environment that are within a distance r_cut of atom m. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. sig (float) – 3-body signal variance hyperparameter. ls (float) – 3-body length scale hyperparameter. r_cut (float) – 3-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 3-body kernel and its gradient with respect to the hyperparameters.
Return type:	(float, float)

flare.kernels.mc_simple.three_body_mc_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, cross_bond_inds_1, cross_bond_inds_2, cross_bond_dists_1, cross_bond_dists_2, triplets_1, triplets_2, d1, d2, sig, ls, r_cut, cutoff_func)¶

3-body multi-element kernel between two force components accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 3-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 3-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. cross_bond_inds_1 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_inds_2 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_1 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_2 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. triplets_1 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the first local environment that are within a distance r_cut of atom m. triplets_2 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the second local environment that are within a distance r_cut of atom m. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. sig (float) – 3-body signal variance hyperparameter. ls (float) – 3-body length scale hyperparameter. r_cut (float) – 3-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 3-body kernel.
Return type:	float

flare.kernels.mc_simple.three_body_se_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, cross_bond_inds_1, cross_bond_inds_2, cross_bond_dists_1, cross_bond_dists_2, triplets_1, triplets_2, sig, ls, r_cut, cutoff_func)¶

3-body multi-element kernel between a force component and a local energy accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 3-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 3-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. cross_bond_inds_1 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_inds_2 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_1 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_2 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. triplets_1 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the first local environment that are within a distance r_cut of atom m. triplets_2 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the second local environment that are within a distance r_cut of atom m. sig (float) – 3-body signal variance hyperparameter. ls (float) – 3-body length scale hyperparameter. r_cut (float) – 3-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 3-body force/energy kernel.
Return type:	float

flare.kernels.mc_simple.three_body_sf_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, cross_bond_inds_1, cross_bond_inds_2, cross_bond_dists_1, cross_bond_dists_2, triplets_1, triplets_2, sig, ls, r_cut, cutoff_func)¶

3-body multi-element kernel between two force components accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 3-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 3-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. cross_bond_inds_1 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_inds_2 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_1 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_2 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. triplets_1 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the first local environment that are within a distance r_cut of atom m. triplets_2 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the second local environment that are within a distance r_cut of atom m. sig (float) – 3-body signal variance hyperparameter. ls (float) – 3-body length scale hyperparameter. r_cut (float) – 3-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 3-body kernel.
Return type:	float

flare.kernels.mc_simple.three_body_ss_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, cross_bond_inds_1, cross_bond_inds_2, cross_bond_dists_1, cross_bond_dists_2, triplets_1, triplets_2, sig, ls, r_cut, cutoff_func)¶

3-body multi-element kernel between two force components accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 3-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 3-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. cross_bond_inds_1 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_inds_2 (np.ndarray) – Two dimensional array whose row m contains the indices of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_1 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the first local environment that are within a distance r_cut of both atom n and the central atom. cross_bond_dists_2 (np.ndarray) – Two dimensional array whose row m contains the distances from atom m of atoms n > m in the second local environment that are within a distance r_cut of both atom n and the central atom. triplets_1 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the first local environment that are within a distance r_cut of atom m. triplets_2 (np.ndarray) – One dimensional array of integers whose entry m is the number of atoms in the second local environment that are within a distance r_cut of atom m. sig (float) – 3-body signal variance hyperparameter. ls (float) – 3-body length scale hyperparameter. r_cut (float) – 3-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 3-body kernel.
Return type:	float

flare.kernels.mc_simple.two_body_mc(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

2-body multi-element kernel between two force components.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – One-element array containing the 2-body cutoff. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2-body kernel.
Return type:	float

flare.kernels.mc_simple.two_body_mc_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

2-body multi-element kernel between two local energies.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – One-element array containing the 2-body cutoff. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2-body force/energy kernel.
Return type:	float

flare.kernels.mc_simple.two_body_mc_en_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, sig, ls, r_cut, cutoff_func)¶

2-body multi-element kernel between two local energies accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 2-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 2-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. sig (float) – 2-body signal variance hyperparameter. ls (float) – 2-body length scale hyperparameter. r_cut (float) – 2-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 2-body local energy kernel.
Return type:	float

flare.kernels.mc_simple.two_body_mc_force_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

2-body multi-element kernel between a force component and a local energy.

Parameters:	env1 (AtomicEnvironment) – Local environment associated with the force component. env2 (AtomicEnvironment) – Local environment associated with the local energy. d1 (int) – Force component of the first environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – One-element array containing the 2-body cutoff. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2-body force/energy kernel.
Return type:	float

flare.kernels.mc_simple.two_body_mc_force_en_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, d1, sig, ls, r_cut, cutoff_func)¶

2-body multi-element kernel between a force component and a local energy accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 2-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 2-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. d1 (int) – Force component of the first environment (1=x, 2=y, 3=z). sig (float) – 2-body signal variance hyperparameter. ls (float) – 2-body length scale hyperparameter. r_cut (float) – 2-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 2-body force/energy kernel.
Return type:	float

flare.kernels.mc_simple.two_body_mc_grad(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) -> (<class 'float'>, 'ndarray')¶

2-body multi-element kernel between two force components and its gradient with respect to the hyperparameters.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig, ls). cutoffs (np.ndarray) – One-element array containing the 2-body cutoff. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2-body kernel and its gradient with respect to the hyperparameters.
Return type:	(float, np.ndarray)

flare.kernels.mc_simple.two_body_mc_grad_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, d1, d2, sig, ls, r_cut, cutoff_func)¶

2-body multi-element kernel between two force components and its gradient with respect to the hyperparameters.

Parameters:	bond_array_1 (np.ndarray) – 2-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 2-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. d1 (int) – Force component of the first environment (1=x, 2=y, 3=z). d2 (int) – Force component of the second environment (1=x, 2=y, 3=z). sig (float) – 2-body signal variance hyperparameter. ls (float) – 2-body length scale hyperparameter. r_cut (float) – 2-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 2-body kernel and its gradient with respect to the hyperparameters.
Return type:	(float, float)

flare.kernels.mc_simple.two_body_mc_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, d1, d2, sig, ls, r_cut, cutoff_func)¶

2-body multi-element kernel between two force components accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 2-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 2-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. d1 (int) – Force component of the first environment (1=x, 2=y, 3=z). d2 (int) – Force component of the second environment (1=x, 2=y, 3=z). sig (float) – 2-body signal variance hyperparameter. ls (float) – 2-body length scale hyperparameter. r_cut (float) – 2-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 2-body kernel.
Return type:	float

flare.kernels.mc_simple.two_body_mc_stress_en_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, d1, d2, sig, ls, r_cut, cutoff_func)¶

2-body multi-element kernel between a partial stress component and a local energy accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 2-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 2-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. d1 (int) – First stress component of the first environment (1=x, 2=y, 3=z). d2 (int) – Second stress component of the first environment (1=x, 2=y, 3=z). sig (float) – 2-body signal variance hyperparameter. ls (float) – 2-body length scale hyperparameter. r_cut (float) – 2-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 2-body partial-stress/energy kernel.
Return type:	float

flare.kernels.mc_simple.two_body_mc_stress_force_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, d1, d2, d3, sig, ls, r_cut, cutoff_func)¶

2-body multi-element kernel between two force components accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 2-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 2-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. d1 (int) – First stress component of the first environment (1=x, 2=y, 3=z). d2 (int) – Second stress component of the first environment (1=x, 2=y, 3=z). d3 (int) – Force component of the second environment (1=x, 2=y, 3=z). sig (float) – 2-body signal variance hyperparameter. ls (float) – 2-body length scale hyperparameter. r_cut (float) – 2-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 2-body kernel.
Return type:	float

flare.kernels.mc_simple.two_body_mc_stress_stress_jit(bond_array_1, c1, etypes1, bond_array_2, c2, etypes2, d1, d2, d3, d4, sig, ls, r_cut, cutoff_func)¶

2-body multi-element kernel between two partial stress components: accelerated with Numba.

Parameters:	bond_array_1 (np.ndarray) – 2-body bond array of the first local environment. c1 (int) – Species of the central atom of the first local environment. etypes1 (np.ndarray) – Species of atoms in the first local environment. bond_array_2 (np.ndarray) – 2-body bond array of the second local environment. c2 (int) – Species of the central atom of the second local environment. etypes2 (np.ndarray) – Species of atoms in the second local environment. d1 (int) – First stress component of the first environment (1=x, 2=y, 3=z). d2 (int) – Second stress component of the first environment (1=x, 2=y, 3=z). d3 (int) – First stress component of the second environment (1=x, 2=y, 3=z). d4 (int) – Second stress component of the second environment (1=x, 2=y, 3=z). sig (float) – 2-body signal variance hyperparameter. ls (float) – 2-body length scale hyperparameter. r_cut (float) – 2-body cutoff radius. cutoff_func (Callable) – Cutoff function.
Returns:	Value of the 2-body kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_many_body_mc(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps, cutoffs, cutoff_func=<function quadratic_cutoff>)¶

2+many body kernel.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig2b, ls2b, sigmb, lsmb, sig_n). cutoffs (np.ndarray) – Two-element array containing the 2- and many-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3+many-body kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_many_body_mc_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, hyps, cutoffs, cutoff_func=<function quadratic_cutoff>)¶

2+3+many-body single-element energy kernel.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig2b ls2b, sigmb, lsmb, sig_n). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3+many-body kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_many_body_mc_force_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, hyps, cutoffs, cutoff_func=<function quadratic_cutoff>)¶

2+many-body multi-element kernel between two force and energy: components.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig1, ls1, sig2, ls2, sig3, ls3, sig_n). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3+many-body kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_many_body_mc_grad(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps, cutoffs, cutoff_func=<function quadratic_cutoff>)¶

2+many-body single-element kernel between two force components.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig1, ls1, sig2, ls2, sig3, ls3, sig_n). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3+many-body kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_three_body_mc(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

2+3-body multi-element kernel between two force components.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig1, ls1, sig2, ls2, sig_n). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3-body kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_three_body_mc_grad(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) -> ('float', 'ndarray')¶

2+3-body multi-element kernel between two force components and its gradient with respect to the hyperparameters.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig1, ls1, sig2, ls2, sig_n). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3-body kernel and its gradient with respect to the hyperparameters.
Return type:	(float, np.ndarray)

flare.kernels.mc_simple.two_plus_three_mc_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

2+3-body multi-element kernel between two local energies.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig1, ls1, sig2, ls2). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3-body energy/energy kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_three_mc_force_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, hyps: ndarray, cutoffs: ndarray, cutoff_func: Callable = <function quadratic_cutoff>) → float¶

2+3-body multi-element kernel between a force component and a local energy.

Parameters:	env1 (AtomicEnvironment) – Local environment associated with the force component. env2 (AtomicEnvironment) – Local environment associated with the local energy. d1 (int) – Force component of the first environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig1, ls1, sig2, ls2). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3-body force/energy kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_three_plus_many_body_mc(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps, cutoffs, cutoff_func=<function quadratic_cutoff>)¶

2+3-body single-element kernel between two force components.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig1, ls1, sig2, ls2, sig3, ls3, sig_n). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3+many-body kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_three_plus_many_body_mc_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, hyps, cutoffs, cutoff_func=<function quadratic_cutoff>)¶

2+3+many-body single-element energy kernel.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig1, ls1, sig2, ls2, sig3, ls3, sig_n). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3+many-body kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_three_plus_many_body_mc_force_en(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, hyps, cutoffs, cutoff_func=<function quadratic_cutoff>)¶

2+3+many-body single-element kernel between two force and energy: components.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig1, ls1, sig2, ls2, sig3, ls3, sig_n). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3+many-body kernel.
Return type:	float

flare.kernels.mc_simple.two_plus_three_plus_many_body_mc_grad(env1: flare.env.AtomicEnvironment, env2: flare.env.AtomicEnvironment, d1: int, d2: int, hyps, cutoffs, cutoff_func=<function quadratic_cutoff>)¶

2+3+many-body single-element kernel between two force components.

Parameters:	env1 (AtomicEnvironment) – First local environment. env2 (AtomicEnvironment) – Second local environment. d1 (int) – Force component of the first environment. d2 (int) – Force component of the second environment. hyps (np.ndarray) – Hyperparameters of the kernel function (sig1, ls1, sig2, ls2, sig3, ls3, sig_n). cutoffs (np.ndarray) – Two-element array containing the 2- and 3-body cutoffs. cutoff_func (Callable) – Cutoff function of the kernel.
Returns:	Value of the 2+3+many-body kernel.
Return type:	float