diff --git a/doc/src/compute_podfit.rst b/doc/src/compute_podfit.rst index 4bcb600415..fadfb0a092 100644 --- a/doc/src/compute_podfit.rst +++ b/doc/src/compute_podfit.rst @@ -21,16 +21,16 @@ Examples .. code-block:: LAMMPS - compute pod all podfit pod.txt data.txt + compute pod all podfit pod.txt data.txt Description """"""""""" -Fit a machine-learning interatomic potential (ML-IAP) based on proper orthogonal descriptors (POD). -Two input files are required for this command. The first input file describes -a POD potential, while the second input file specifies the DFT data. +Fit a machine-learning interatomic potential (ML-IAP) based on proper orthogonal descriptors (POD). +Two input files are required for this command. The first input file describes +a POD potential, while the second input file specifies the DFT data. -Below is a one-line description of all the keywords that can be assigned in the +Below is a one-line description of all the keywords that can be assigned in the first input file (pod.txt): * species (STRING): Chemical symbols for all elements in the system and have to match XYZ training files. @@ -42,11 +42,11 @@ first input file (pod.txt): * bessel_scaling_parameter1 0.0 (REAL): the 1st value of the Bessel scaling parameter * bessel_scaling_parameter2 2.0 (REAL): the 2nd value of the Bessel scaling parameter * bessel_scaling_parameter3 4.0 (REAL): the 3rd value of the Bessel scaling parameter -* onebody 1 (BOOL): turns on/off one-body potential -* twobody_number_radial_basis_functions 6 (INT): number of radial basis functions for two-body potential -* threebody_number_radial_basis_functions 5 (INT): number of radial basis functions for three-body potential -* threebody_number_angular_basis_functions 5 (INT): number of angular basis functions for three-body potential -* fourbody_snap_twojmax 0 (INT): band limit for SNAP bispectrum components (0,2,4,6,8... allowed) +* onebody 1 (BOOL): turns on/off one-body potential +* twobody_number_radial_basis_functions 6 (INT): number of radial basis functions for two-body potential +* threebody_number_radial_basis_functions 5 (INT): number of radial basis functions for three-body potential +* threebody_number_angular_basis_functions 5 (INT): number of angular basis functions for three-body potential +* fourbody_snap_twojmax 0 (INT): band limit for SNAP bispectrum components (0,2,4,6,8... allowed) * fourbody_snap_chemflag 0 (BOOL): turns on/off the explicit multi-element variant of the SNAP bispectrum components * quadratic22_number_twobody_basis_functions 0 (INT): number of two-body basis functions for the (2*2) quadratic potential * quadratic23_number_twobody_basis_functions 0 (INT): number of two-body basis functions for the (2*3) quadratic potential @@ -63,14 +63,14 @@ first input file (pod.txt): * cubic333_number_threebody_basis_functions 0 (INT): number of three-body basis functions for the (3*3*3) cubic potential * cubic444_number_fourbody_basis_functions 0 (INT): number of four-body basis functions for the (4*4*4) cubic potential -All keywords except species have default values. If keywords are not set in the input file, their defaults are used. +All keywords except species have default values. If keywords are not set in the input file, their defaults are used. Next, we describe all the keywords that can be assigned in the second input file (data.txt): -* file_format extxyz (STRING): only extended xyz format is currently supported -* file_extension xyz (STRING): extension of the data files +* file_format extxyz (STRING): only extended xyz format is currently supported +* file_extension xyz (STRING): extension of the data files * path_to_training_data_set (STRING): specifies the path to training data files in double quotes * path_to_test_data_set "" (STRING): specifies the path to test data files in double quotes -* percentage_training_data_set 1.0 (REAL): a real number (<= 1.0) specifies the percentage of the training set used to fit POD +* percentage_training_data_set 1.0 (REAL): a real number (<= 1.0) specifies the percentage of the training set used to fit POD * randomize_training_data_set 0 (BOOL): turns on/off randomization of the training set * fitting_weight_energy 100.0 (REAL): a real constant specifies the weight for energy in the least-squares fit * fitting_weight_force 1.0 (REAL): a real constant specifies the weight for force in the least-squares fit @@ -79,70 +79,70 @@ Next, we describe all the keywords that can be assigned in the second input file * energy_force_calculation_for_training_data_set 0 (BOOL): turns on/off energy and force calculation for the training data set * energy_force_calculation_for_test_data_set 0 (BOOL): turns on/off energy and force calculation for the test data set -All keywords except path_to_training_data_set have default values. If keywords are not set in the input file, their defaults are used. +All keywords except path_to_training_data_set have default values. If keywords are not set in the input file, their defaults are used. On successful training, it produces a number of output files: * training_errors.txt reports the errors in energy and forces for the training data set -* traning_analysis.txt reports detailed errors for all training configurations -* test_errors.txt reports errors for the test data set -* test_analysis.txt reports detailed errors for all test configurations -* coefficents.txt contains the coeffcients of the POD potential - -After training the POD potential, pod.txt and coefficents.txt are two files needed to use the +* traning_analysis.txt reports detailed errors for all training configurations +* test_errors.txt reports errors for the test data set +* test_analysis.txt reports detailed errors for all test configurations +* coefficents.txt contains the coeffcients of the POD potential + +After training the POD potential, pod.txt and coefficents.txt are two files needed to use the POD potential in LAMMPS. See :doc:`pair_style pod ` for using the POD potential. Examples about training and using POD potentials are found in the directory lammps/examples/pod. Parametrized Potential Energy Surface """"""""""""""""""""""""""""""""""""" -We consider a multi-element system of *N* atoms with :math:`N_{\rm e}` unique elements. -We denote by :math:`\boldsymbol r_n` and :math:`Z_n` position vector and type of an atom *n* in -the system, respectively. Note that we have :math:`Z_n \in \{1, \ldots, N_{\rm e} \}`, -:math:`\boldsymbol R = (\boldsymbol r_1, \boldsymbol r_2, \ldots, \boldsymbol r_N) \in \mathbb{R}^{3N}`, and -:math:`\boldsymbol Z = (Z_1, Z_2, \ldots, Z_N) \in \mathbb{N}^{N}`. The potential energy surface +We consider a multi-element system of *N* atoms with :math:`N_{\rm e}` unique elements. +We denote by :math:`\boldsymbol r_n` and :math:`Z_n` position vector and type of an atom *n* in +the system, respectively. Note that we have :math:`Z_n \in \{1, \ldots, N_{\rm e} \}`, +:math:`\boldsymbol R = (\boldsymbol r_1, \boldsymbol r_2, \ldots, \boldsymbol r_N) \in \mathbb{R}^{3N}`, and +:math:`\boldsymbol Z = (Z_1, Z_2, \ldots, Z_N) \in \mathbb{N}^{N}`. The potential energy surface (PES) of the system can be expressed as a many-body expansion of the form .. math:: E(\boldsymbol R, \boldsymbol Z, \boldsymbol{\eta}, \boldsymbol{\mu}) \ = \ & \sum_{i} V^{(1)}(\boldsymbol r_i, Z_i, \boldsymbol \mu^{(1)} ) + \frac12 \sum_{i,j} V^{(2)}(\boldsymbol r_i, \boldsymbol r_j, Z_i, Z_j, \boldsymbol \eta, \boldsymbol \mu^{(2)}) \\ - & + \frac16 \sum_{i,j,k} V^{(3)}(\boldsymbol r_i, \boldsymbol r_j, \boldsymbol r_k, Z_i, Z_j, Z_k, \boldsymbol \eta, \boldsymbol \mu^{(3)}) + \ldots + & + \frac16 \sum_{i,j,k} V^{(3)}(\boldsymbol r_i, \boldsymbol r_j, \boldsymbol r_k, Z_i, Z_j, Z_k, \boldsymbol \eta, \boldsymbol \mu^{(3)}) + \ldots -where :math:`V^{(1)}` is the one-body potential often used for representing external field -or energy of isolated elements, and the higher-body potentials :math:`V^{(2)}, V^{(3)}, \ldots` -are symmetric, uniquely defined, and zero if two or more indices take identical values. -The superscript on each potential denotes its body order. Each *q*-body potential :math:`V^{(q)}` -depends on :math:`\boldsymbol \mu^{(q)}` which are sets of parameters to fit the PES. Note -that :math:`\boldsymbol \mu` is a collection of all potential parameters -:math:`\boldsymbol \mu^{(1)}`, :math:`\boldsymbol \mu^{(2)}`, :math:`\boldsymbol \mu^{(3)}`, etc, -and that :math:`\boldsymbol \eta` is a set of hyperparameters such as inner cut-off radius -:math:`r_{\rm in}` and outer cut-off radius :math:`r_{\rm cut}`. +where :math:`V^{(1)}` is the one-body potential often used for representing external field +or energy of isolated elements, and the higher-body potentials :math:`V^{(2)}, V^{(3)}, \ldots` +are symmetric, uniquely defined, and zero if two or more indices take identical values. +The superscript on each potential denotes its body order. Each *q*-body potential :math:`V^{(q)}` +depends on :math:`\boldsymbol \mu^{(q)}` which are sets of parameters to fit the PES. Note +that :math:`\boldsymbol \mu` is a collection of all potential parameters +:math:`\boldsymbol \mu^{(1)}`, :math:`\boldsymbol \mu^{(2)}`, :math:`\boldsymbol \mu^{(3)}`, etc, +and that :math:`\boldsymbol \eta` is a set of hyperparameters such as inner cut-off radius +:math:`r_{\rm in}` and outer cut-off radius :math:`r_{\rm cut}`. -Interatomic potentials rely on parameters to learn relationship between atomic environments -and interactions. Since interatomic potentials are approximations by nature, their parameters -need to be set to some reference values or fitted against data by necessity. Typically, -potential fitting finds optimal parameters, :math:`\boldsymbol \mu^*`, to minimize a certain loss -function of the predicted quantities and data. Since the fitted potential depends on the data -set used to fit it, different data sets will yield different optimal parameters and thus different -fitted potentials. When fitting the same functional form on *Q* different data sets, we would -obtain *Q* different optimized potentials, :math:`E(\boldsymbol R,\boldsymbol Z, \boldsymbol \eta, \boldsymbol \mu_q^*), 1 \le q \le Q`. -Consequently, there exist many different sets of optimized parameters for empirical interatomic potentials. +Interatomic potentials rely on parameters to learn relationship between atomic environments +and interactions. Since interatomic potentials are approximations by nature, their parameters +need to be set to some reference values or fitted against data by necessity. Typically, +potential fitting finds optimal parameters, :math:`\boldsymbol \mu^*`, to minimize a certain loss +function of the predicted quantities and data. Since the fitted potential depends on the data +set used to fit it, different data sets will yield different optimal parameters and thus different +fitted potentials. When fitting the same functional form on *Q* different data sets, we would +obtain *Q* different optimized potentials, :math:`E(\boldsymbol R,\boldsymbol Z, \boldsymbol \eta, \boldsymbol \mu_q^*), 1 \le q \le Q`. +Consequently, there exist many different sets of optimized parameters for empirical interatomic potentials. -Instead of optimizing the potential parameters, inspired by the reduced basis method -:ref:`(Grepl) ` for parametrized partial differential equations, -we view the parametrized PES as a parametric manifold of potential energies +Instead of optimizing the potential parameters, inspired by the reduced basis method +:ref:`(Grepl) ` for parametrized partial differential equations, +we view the parametrized PES as a parametric manifold of potential energies .. math:: - \mathcal{M} = \{E(\boldsymbol R, \boldsymbol Z, \boldsymbol \eta, \boldsymbol \mu) \ | \ \boldsymbol \mu \in \Omega^{\boldsymbol \mu} \} + \mathcal{M} = \{E(\boldsymbol R, \boldsymbol Z, \boldsymbol \eta, \boldsymbol \mu) \ | \ \boldsymbol \mu \in \Omega^{\boldsymbol \mu} \} -where :math:`\Omega^{\boldsymbol \mu}` is a parameter domain in which :math:`\boldsymbol \mu` resides. -The parametric manifold :math:`\mathcal{M}` contains potential energy surfaces for all values -of :math:`\boldsymbol \mu \in \Omega^{\boldsymbol \mu}`. Therefore, the parametric manifold yields a much richer -and more transferable atomic representation than any particular individual PES +where :math:`\Omega^{\boldsymbol \mu}` is a parameter domain in which :math:`\boldsymbol \mu` resides. +The parametric manifold :math:`\mathcal{M}` contains potential energy surfaces for all values +of :math:`\boldsymbol \mu \in \Omega^{\boldsymbol \mu}`. Therefore, the parametric manifold yields a much richer +and more transferable atomic representation than any particular individual PES :math:`E(\boldsymbol R, \boldsymbol Z, \boldsymbol \eta, \boldsymbol \mu^*)`. -We propose specific forms of the parametrized potentials for one-body, two-body, -and three-body interactions. We apply the Karhunen-Loeve expansion to snapshots of the parametrized potentials -to obtain sets of orthogonal basis functions. These basis functions are aggregated +We propose specific forms of the parametrized potentials for one-body, two-body, +and three-body interactions. We apply the Karhunen-Loeve expansion to snapshots of the parametrized potentials +to obtain sets of orthogonal basis functions. These basis functions are aggregated according to the chemical elements of atoms, thus leading to multi-element proper orthogonal descriptors. Proper Orthogonal Descriptors @@ -158,113 +158,113 @@ The descriptors for the one-body interaction are used to capture energy of isola .. math:: D_{ip}^{(1)} = \left\{ - \begin{array}{ll} + \begin{array}{ll} 1, & \mbox{if } Z_i = p \\ 0, & \mbox{if } Z_i \neq p - \end{array} - \right. + \end{array} + \right. -for :math:`1 \le i \le N, 1 \le p \le N_{\rm e}`. The number of one-body descriptors per atom -is equal to the number of elements. The one-body descriptors are independent of atom positions, +for :math:`1 \le i \le N, 1 \le p \le N_{\rm e}`. The number of one-body descriptors per atom +is equal to the number of elements. The one-body descriptors are independent of atom positions, but dependent on atom types. The one-body descriptors are active only when the keyword *onebody* is set to 1. -We adopt the usual assumption that the direct interaction between two atoms vanishes smoothly -when their distance is greater than the outer cutoff distance :math:`r_{\rm cut}`. Furthermore, we -assume that two atoms can not get closer than the inner cutoff distance :math:`r_{\rm in}` -due to the Pauli repulsion principle. Let :math:`r \in (r_{\rm in}, r_{\rm cut})`, we introduce the +We adopt the usual assumption that the direct interaction between two atoms vanishes smoothly +when their distance is greater than the outer cutoff distance :math:`r_{\rm cut}`. Furthermore, we +assume that two atoms can not get closer than the inner cutoff distance :math:`r_{\rm in}` +due to the Pauli repulsion principle. Let :math:`r \in (r_{\rm in}, r_{\rm cut})`, we introduce the following parametrized radial functions .. math:: - \phi(r, r_{\rm in}, r_{\rm cut}, \alpha, \beta) = \frac{\sin (\alpha \pi x) }{r - r_{\rm in}}, \qquad \varphi(r, \gamma) = \frac{1}{r^\gamma} , + \phi(r, r_{\rm in}, r_{\rm cut}, \alpha, \beta) = \frac{\sin (\alpha \pi x) }{r - r_{\rm in}}, \qquad \varphi(r, \gamma) = \frac{1}{r^\gamma} , -where the scaled distance function :math:`x` is defined below to enrich the two-body manifold +where the scaled distance function :math:`x` is defined below to enrich the two-body manifold .. math:: x(r, r_{\rm in}, r_{\rm cut}, \beta) = \frac{e^{-\beta(r - r_{\rm in})/(r_{\rm cut} - r_{\rm in})} - 1}{e^{-\beta} - 1} . -We introduce the following function as a convex combination of the two functions +We introduce the following function as a convex combination of the two functions .. math:: \psi(r, r_{\rm in}, r_{\rm cut}, \alpha, \beta, \gamma, \kappa) = \kappa \phi(r, r_{\rm in}, r_{\rm cut}, \alpha, \beta) + (1- \kappa) \varphi(r, \gamma) . -We see that :math:`\psi` is a function of distance :math:`r`, cut-off distances :math:`r_{\rm in}` -and :math:`r_{\rm cut}`, and parameters :math:`\alpha, \beta, \gamma, \kappa`. Together -these parameters allow the function :math:`\psi` to characterize a diverse spectrum of -two-body interactions within the cut-off interval :math:`(r_{\rm in}, r_{\rm cut})`. +We see that :math:`\psi` is a function of distance :math:`r`, cut-off distances :math:`r_{\rm in}` +and :math:`r_{\rm cut}`, and parameters :math:`\alpha, \beta, \gamma, \kappa`. Together +these parameters allow the function :math:`\psi` to characterize a diverse spectrum of +two-body interactions within the cut-off interval :math:`(r_{\rm in}, r_{\rm cut})`. -Next, we introduce the following parametrized potential +Next, we introduce the following parametrized potential .. math:: W^{(2)}(r_{ij}, \boldsymbol \eta, \boldsymbol \mu^{(2)}) = f_{\rm c}(r_{ij}, \boldsymbol \eta) \psi(r_{ij}, \boldsymbol \eta, \boldsymbol \mu^{(2)}) -where :math:`\eta_1 = r_{\rm in}, \eta_2 = r_{\rm cut}, \mu_1^{(2)} = \alpha, \mu_2^{(2)} = \beta, \mu_3^{(2)} = \gamma`, -and :math:`\mu_4^{(2)} = \kappa`. Here the cut-off function :math:`f_{\rm c}(r_{ij}, \boldsymbol \eta)` -proposed in [refs] is used to ensure the smooth vanishing of the potential and +where :math:`\eta_1 = r_{\rm in}, \eta_2 = r_{\rm cut}, \mu_1^{(2)} = \alpha, \mu_2^{(2)} = \beta, \mu_3^{(2)} = \gamma`, +and :math:`\mu_4^{(2)} = \kappa`. Here the cut-off function :math:`f_{\rm c}(r_{ij}, \boldsymbol \eta)` +proposed in [refs] is used to ensure the smooth vanishing of the potential and its derivative for :math:`r_{ij} \ge r_{\rm cut}`: .. math:: - f_{\rm c}(r_{ij}, r_{\rm in}, r_{\rm cut}) = \exp \left(1 -\frac{1}{\sqrt{\left(1 - \frac{(r-r_{\rm in})^3}{(r_{\rm cut} - r_{\rm in})^3} \right)^2 + 10^{-6}}} \right) + f_{\rm c}(r_{ij}, r_{\rm in}, r_{\rm cut}) = \exp \left(1 -\frac{1}{\sqrt{\left(1 - \frac{(r-r_{\rm in})^3}{(r_{\rm cut} - r_{\rm in})^3} \right)^2 + 10^{-6}}} \right) -Based on the parametrized potential, we form a set of snapshots as follows. -We assume that we are given :math:`N_{\rm s}` parameter tuples -:math:`\boldsymbol \mu^{(2)}_\ell, 1 \le \ell \le N_{\rm s}`. We introduce the +Based on the parametrized potential, we form a set of snapshots as follows. +We assume that we are given :math:`N_{\rm s}` parameter tuples +:math:`\boldsymbol \mu^{(2)}_\ell, 1 \le \ell \le N_{\rm s}`. We introduce the following set of snapshots on :math:`(r_{\rm in}, r_{\rm cut})`: .. math:: \xi_\ell(r_{ij}, \boldsymbol \eta) = W^{(2)}(r_{ij}, \boldsymbol \eta, \boldsymbol \mu^{(2)}_\ell), \quad \ell = 1, \ldots, N_{\rm s} . -To ensure adequate sampling of the PES for different parameters, we choose -:math:`N_{\rm s}` parameter points :math:`\boldsymbol \mu^{(2)}_\ell = (\alpha_\ell, \beta_\ell, \gamma_\ell, \kappa_\ell), 1 \le \ell \le N_{\rm s}` -as follows. The parameters :math:`\alpha \in [1, N_\alpha]` and :math:`\gamma \in [1, N_\gamma]` -are integers, where :math:`N_\alpha` and :math:`N_\gamma` are the highest degrees for -:math:`\alpha` and :math:`\gamma`, respectively. We next choose :math:`N_\beta` different values of -:math:`\beta` in the interval :math:`[\beta_{\min}, \beta_{\max}]`, where :math:`\beta_{\min} = 0` and -:math:`\beta_{\max} = 4`. The parameter :math:`\kappa` can be set either 0 or 1. -Hence, the total number of parameter points is :math:`N_{\rm s} = N_\alpha N_\beta + N_\gamma`. -Although :math:`N_\alpha, N_\beta, N_\gamma` can be chosen conservatively large, -we find that :math:`N_\alpha = 6, N_\beta = 3, N_\gamma = 8` are adequate for most problems. -Note that :math:`N_\alpha` and :math:`N_\gamma` correspond to *bessel_polynomial_degree* -and *inverse_polynomial_degree*, respectively. Furthermore, *bessel_scaling_parameter1*, -*bessel_scaling_parameter2*, and *bessel_scaling_parameter3* are three different +To ensure adequate sampling of the PES for different parameters, we choose +:math:`N_{\rm s}` parameter points :math:`\boldsymbol \mu^{(2)}_\ell = (\alpha_\ell, \beta_\ell, \gamma_\ell, \kappa_\ell), 1 \le \ell \le N_{\rm s}` +as follows. The parameters :math:`\alpha \in [1, N_\alpha]` and :math:`\gamma \in [1, N_\gamma]` +are integers, where :math:`N_\alpha` and :math:`N_\gamma` are the highest degrees for +:math:`\alpha` and :math:`\gamma`, respectively. We next choose :math:`N_\beta` different values of +:math:`\beta` in the interval :math:`[\beta_{\min}, \beta_{\max}]`, where :math:`\beta_{\min} = 0` and +:math:`\beta_{\max} = 4`. The parameter :math:`\kappa` can be set either 0 or 1. +Hence, the total number of parameter points is :math:`N_{\rm s} = N_\alpha N_\beta + N_\gamma`. +Although :math:`N_\alpha, N_\beta, N_\gamma` can be chosen conservatively large, +we find that :math:`N_\alpha = 6, N_\beta = 3, N_\gamma = 8` are adequate for most problems. +Note that :math:`N_\alpha` and :math:`N_\gamma` correspond to *bessel_polynomial_degree* +and *inverse_polynomial_degree*, respectively. Furthermore, *bessel_scaling_parameter1*, +*bessel_scaling_parameter2*, and *bessel_scaling_parameter3* are three different values of :math:`\beta`. -We employ the Karhunen-Loeve (KL) expansion to generate an orthogonal basis set which is known to be optimal for representation of -the snapshot family :math:`\{\xi_\ell\}_{\ell=1}^{N_{\rm s}}`. The two-body orthogonal basis +We employ the Karhunen-Loeve (KL) expansion to generate an orthogonal basis set which is known to be optimal for representation of +the snapshot family :math:`\{\xi_\ell\}_{\ell=1}^{N_{\rm s}}`. The two-body orthogonal basis functions are computed as follows .. math:: - U^{(2)}_m(r_{ij}, \boldsymbol \eta) = \sum_{\ell = 1}^{N_{\rm s}} A_{\ell m}(\boldsymbol \eta) \, \xi_\ell(r_{ij}, \boldsymbol \eta), \qquad m = 1, \ldots, N_{\rm 2b} , + U^{(2)}_m(r_{ij}, \boldsymbol \eta) = \sum_{\ell = 1}^{N_{\rm s}} A_{\ell m}(\boldsymbol \eta) \, \xi_\ell(r_{ij}, \boldsymbol \eta), \qquad m = 1, \ldots, N_{\rm 2b} , -where the matrix :math:`\boldsymbol A \in \mathbb{R}^{N_{\rm s} \times N_{\rm s}}` consists of -eigenvectors of the eigenvalue problem +where the matrix :math:`\boldsymbol A \in \mathbb{R}^{N_{\rm s} \times N_{\rm s}}` consists of +eigenvectors of the eigenvalue problem .. math:: - \boldsymbol C \boldsymbol a = \lambda \boldsymbol a + \boldsymbol C \boldsymbol a = \lambda \boldsymbol a -with the entries of :math:`\boldsymbol C \in \mathbb{R}^{N_{\rm s} \times N_{\rm s}}` being given by +with the entries of :math:`\boldsymbol C \in \mathbb{R}^{N_{\rm s} \times N_{\rm s}}` being given by .. math:: - C_{ij} = \frac{1}{N_{\rm s}} \int_{r_{\rm in}}^{r_{\rm cut}} \xi_i(x, \boldsymbol \eta) \xi_j(x, \boldsymbol \eta) dx, \quad 1 \le i, j \le N_{\rm s} + C_{ij} = \frac{1}{N_{\rm s}} \int_{r_{\rm in}}^{r_{\rm cut}} \xi_i(x, \boldsymbol \eta) \xi_j(x, \boldsymbol \eta) dx, \quad 1 \le i, j \le N_{\rm s} -Note that the eigenvalues :math:`\lambda_\ell, 1 \le \ell \le N_{\rm s}`, are ordered such -that :math:`\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_{N_{\rm s}}`, and that the -matrix :math:`\boldsymbol A` is pe-computed and stored for any given :math:`\boldsymbol \eta`. -Owing to the rapid convergence of the KL expansion, only a small number of orthogonal -basis functions is needed to obtain accurate approximation. The value of :math:`N_{\rm 2b}` -corresponds to *twobody_number_radial_basis_functions*. +Note that the eigenvalues :math:`\lambda_\ell, 1 \le \ell \le N_{\rm s}`, are ordered such +that :math:`\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_{N_{\rm s}}`, and that the +matrix :math:`\boldsymbol A` is pe-computed and stored for any given :math:`\boldsymbol \eta`. +Owing to the rapid convergence of the KL expansion, only a small number of orthogonal +basis functions is needed to obtain accurate approximation. The value of :math:`N_{\rm 2b}` +corresponds to *twobody_number_radial_basis_functions*. -The two-body proper orthogonal descriptors at each atom *i* are computed by -summing the orthogonal basis functions over the neighbors of atom *i* and numerating on +The two-body proper orthogonal descriptors at each atom *i* are computed by +summing the orthogonal basis functions over the neighbors of atom *i* and numerating on the atom types as follows .. math:: @@ -273,84 +273,84 @@ the atom types as follows \begin{array}{ll} \displaystyle \sum_{\{j | Z_j = q\}} U^{(2)}_m(r_{ij}, \boldsymbol \eta), & \mbox{if } Z_i = p \\ 0, & \mbox{if } Z_i \neq p - \end{array} - \right. + \end{array} + \right. -for :math:`1 \le i \le N, 1 \le m \le N_{\rm 2b}, 1 \le q, p \le N_{\rm e}`. Here :math:`l(p,q)` is a -symmetric index mapping such that +for :math:`1 \le i \le N, 1 \le m \le N_{\rm 2b}, 1 \le q, p \le N_{\rm e}`. Here :math:`l(p,q)` is a +symmetric index mapping such that .. math:: l(p,q) = \left\{ \begin{array}{ll} q + (p-1) N_{\rm e} - p(p-1)/2, & \mbox{if } q \ge p \\ - p + (q-1) N_{\rm e} - q(q-1)/2, & \mbox{if } q < p . - \end{array} - \right. + p + (q-1) N_{\rm e} - q(q-1)/2, & \mbox{if } q < p . + \end{array} + \right. The number of two-body descriptors per atom is thus :math:`N_{\rm 2b} N_{\rm e}(N_{\rm e}+1)/2`. - -It is important to note that the orthogonal basis functions -do not depend on the atomic numbers :math:`Z_i` and :math:`Z_j`. Therefore, the cost of evaluating -the basis functions and their derivatives with respect to :math:`r_{ij}` is independent of the -number of elements :math:`N_{\rm e}`. Consequently, even though the two-body proper orthogonal -descriptors depend on :math:`\boldsymbol Z`, their computational complexity -is independent of :math:`N_{\rm e}`. -In order to provide proper orthogonal descriptors for three-body interactions, -we need to introduce a three-body parametrized potential. In particular, the +It is important to note that the orthogonal basis functions +do not depend on the atomic numbers :math:`Z_i` and :math:`Z_j`. Therefore, the cost of evaluating +the basis functions and their derivatives with respect to :math:`r_{ij}` is independent of the +number of elements :math:`N_{\rm e}`. Consequently, even though the two-body proper orthogonal +descriptors depend on :math:`\boldsymbol Z`, their computational complexity +is independent of :math:`N_{\rm e}`. + +In order to provide proper orthogonal descriptors for three-body interactions, +we need to introduce a three-body parametrized potential. In particular, the three-body potential is defined as a product of radial and angular functions as follows .. math:: W^{(3)}(r_{ij}, r_{ik}, \theta_{ijk}, \boldsymbol \eta, \boldsymbol \mu^{(3)}) = \psi(r_{ij}, r_{\rm min}, r_{\rm max}, \alpha, \beta, \gamma, \kappa) f_{\rm c}(r_{ij}, r_{\rm min}, r_{\rm max}) \\ \psi(r_{ik}, r_{\rm min}, r_{\rm max}, \alpha, \beta, \gamma, \kappa) f_{\rm c}(r_{ik}, r_{\rm min}, r_{\rm max}) \\ - \cos (\sigma \theta_{ijk} + \zeta) + \cos (\sigma \theta_{ijk} + \zeta) -where :math:`\sigma` is the periodic multiplicity, :math:`\zeta` is the equilibrium angle, -:math:`\boldsymbol \mu^{(3)} = (\alpha, \beta, \gamma, \kappa, \sigma, \zeta)`. The three-body -potential provides an angular fingerprint of the atomic environment through the -bond angles :math:`\theta_{ijk}` formed with each pair of neighbors :math:`j` and :math:`k`. -Compared to the two-body potential, the three-body potential -has two extra parameters :math:`(\sigma, \zeta)` associated with the angular component. +where :math:`\sigma` is the periodic multiplicity, :math:`\zeta` is the equilibrium angle, +:math:`\boldsymbol \mu^{(3)} = (\alpha, \beta, \gamma, \kappa, \sigma, \zeta)`. The three-body +potential provides an angular fingerprint of the atomic environment through the +bond angles :math:`\theta_{ijk}` formed with each pair of neighbors :math:`j` and :math:`k`. +Compared to the two-body potential, the three-body potential +has two extra parameters :math:`(\sigma, \zeta)` associated with the angular component. -Let :math:`\boldsymbol \varrho = (\alpha, \beta, \gamma, \kappa)`. We assume that -we are given :math:`L_{\rm r}` parameter tuples :math:`\boldsymbol \varrho_\ell, 1 \le \ell \le L_{\rm r}`. +Let :math:`\boldsymbol \varrho = (\alpha, \beta, \gamma, \kappa)`. We assume that +we are given :math:`L_{\rm r}` parameter tuples :math:`\boldsymbol \varrho_\ell, 1 \le \ell \le L_{\rm r}`. We introduce the following set of snapshots on :math:`(r_{\min}, r_{\max})`: .. math:: \zeta_\ell(r_{ij}, r_{\rm min}, r_{\rm max} ) = \psi(r_{ij}, r_{\rm min}, r_{\rm max}, \boldsymbol \varrho_\ell) f_{\rm c}(r_{ij}, r_{\rm min}, r_{\rm max}), \quad 1 \le \ell \le L_{\rm r} . -We apply the Karhunen-Lo\`eve (KL) expansion to this set of snapshots to +We apply the Karhunen-Lo\`eve (KL) expansion to this set of snapshots to obtain orthogonal basis functions as follows .. math:: - U^{r}_m(r_{ij}, r_{\rm min}, r_{\rm max} ) = \sum_{\ell = 1}^{L_{\rm r}} A_{\ell m} \, \zeta_\ell(r_{ij}, r_{\rm min}, r_{\rm max} ), \qquad m = 1, \ldots, N_{\rm r} , + U^{r}_m(r_{ij}, r_{\rm min}, r_{\rm max} ) = \sum_{\ell = 1}^{L_{\rm r}} A_{\ell m} \, \zeta_\ell(r_{ij}, r_{\rm min}, r_{\rm max} ), \qquad m = 1, \ldots, N_{\rm r} , -where the matrix :math:`\boldsymbol A \in \mathbb{R}^{L_{\rm r} \times L_{\rm r}}` consists -of eigenvectors of the eigenvalue problem. For the parametrized angular function, -we consider angular basis functions +where the matrix :math:`\boldsymbol A \in \mathbb{R}^{L_{\rm r} \times L_{\rm r}}` consists +of eigenvectors of the eigenvalue problem. For the parametrized angular function, +we consider angular basis functions .. math:: - U^{a}_n(\theta_{ijk}) = \cos ((n-1) \theta_{ijk}), \qquad n = 1,\ldots, N_{\rm a}, + U^{a}_n(\theta_{ijk}) = \cos ((n-1) \theta_{ijk}), \qquad n = 1,\ldots, N_{\rm a}, -where :math:`N_{\rm a}` is the number of angular basis functions. The orthogonal +where :math:`N_{\rm a}` is the number of angular basis functions. The orthogonal basis functions for the parametrized potential are computed as follows .. math:: U^{(3)}_{mn}(r_{ij}, r_{ik}, \theta_{ijk}, \boldsymbol \eta) = U^{r}_m(r_{ij}, \boldsymbol \eta) U^{r}_m(r_{ik}, \boldsymbol \eta) U^{a}_n(\theta_{ijk}), -for :math:`1 \le m \le N_{\rm r}, 1 \le n \le N_{\rm a}`. The number of three-body -orthogonal basis functions is equal to :math:`N_{\rm 3b} = N_{\rm r} N_{\rm a}` and +for :math:`1 \le m \le N_{\rm r}, 1 \le n \le N_{\rm a}`. The number of three-body +orthogonal basis functions is equal to :math:`N_{\rm 3b} = N_{\rm r} N_{\rm a}` and independent of the number of elements. The value of :math:`N_{\rm r}` corresponds to -*threebody_number_radial_basis_functions*, while that of :math:`N_{\rm a}` to -*threebody_number_angular_basis_functions*. +*threebody_number_radial_basis_functions*, while that of :math:`N_{\rm a}` to +*threebody_number_angular_basis_functions*. -The three-body proper orthogonal descriptors at each atom *i* +The three-body proper orthogonal descriptors at each atom *i* are obtained by summing over the neighbors *j* and *k* of atom *i* as .. math:: @@ -359,10 +359,10 @@ are obtained by summing over the neighbors *j* and *k* of atom *i* as \begin{array}{ll} \displaystyle \sum_{\{j | Z_j = q\}} \sum_{\{k | Z_k = s\}} U^{(3)}_{mn}(r_{ij}, r_{ik}, \theta_{ijk}, \boldsymbol \eta), & \mbox{if } Z_i = p \\ 0, & \mbox{if } Z_i \neq p - \end{array} - \right. + \end{array} + \right. -for :math:`1 \le i \le N, 1 \le m \le N_{\rm r}, 1 \le n \le N_{\rm a}, 1 \le q, p, s \le N_{\rm e}`, +for :math:`1 \le i \le N, 1 \le m \le N_{\rm r}, 1 \le n \le N_{\rm a}, 1 \le q, p, s \le N_{\rm e}`, where .. math:: @@ -370,20 +370,20 @@ where \ell(p,q,s) = \left\{ \begin{array}{ll} s + (q-1) N_{\rm e} - q(q-1)/2 + (p-1)N_{\rm e}(1+N_{\rm e})/2 , & \mbox{if } s \ge q \\ - q + (s-1) N_{\rm e} - s(s-1)/2 + (p-1)N_{\rm e}(1+N_{\rm e})/2, & \mbox{if } s < q . - \end{array} - \right. + q + (s-1) N_{\rm e} - s(s-1)/2 + (p-1)N_{\rm e}(1+N_{\rm e})/2, & \mbox{if } s < q . + \end{array} + \right. -The number of three-body descriptors per atom is thus :math:`N_{\rm 3b} N_{\rm e}^2(N_{\rm e}+1)/2`. -While the number of three-body PODs increases cubically as a function of the number of elements, -the computational complexity of the three-body PODs is independent of the number of elements. +The number of three-body descriptors per atom is thus :math:`N_{\rm 3b} N_{\rm e}^2(N_{\rm e}+1)/2`. +While the number of three-body PODs increases cubically as a function of the number of elements, +the computational complexity of the three-body PODs is independent of the number of elements. -Four-Body SNAP Descriptors +Four-Body SNAP Descriptors """""""""""""""""""""""""" In addition to the proper orthogonal descriptors described above, we also employ the spectral neighbor analysis potential (SNAP) descriptors. SNAP uses bispectrum components -to characterize the local neighborhood of each atom in a very general way. The mathematical definition +to characterize the local neighborhood of each atom in a very general way. The mathematical definition of the bispectrum calculation and its derivatives w.r.t. atom positions is described in :doc:`compute snap `. In SNAP, the total energy is decomposed into a sum over atom energies. The energy of @@ -402,54 +402,54 @@ where the SNAP descriptors are related to the bispectrum components by \begin{array}{ll} \displaystyle B_{ik}, & \mbox{if } Z_i = p \\ 0, & \mbox{if } Z_i \neq p - \end{array} - \right. + \end{array} + \right. Here :math:`B_{ik}` is the *k*\ -th bispectrum component of atom *i*. The number of -bispectrum components :math:`N_{\rm 4b}` depends on the value of *fourbody_snap_twojmax* :math:`= 2 J_{\rm max}` -and *fourbody_snap_chemflag*. If *fourbody_snap_chemflag* = 0 -then :math:`N_{\rm 4b} = (J_{\rm max}+1)(J_{\rm max}+2)(J_{\rm max}+1.5)/3`. -If *fourbody_snap_chemflag* = 1 then :math:`N_{\rm 4b} = N_{\rm e}^3 (J_{\rm max}+1)(J_{\rm max}+2)(J_{\rm max}+1.5)/3`. -The bispectrum calculation is described in more detail in :doc:`compute sna/atom `. +bispectrum components :math:`N_{\rm 4b}` depends on the value of *fourbody_snap_twojmax* :math:`= 2 J_{\rm max}` +and *fourbody_snap_chemflag*. If *fourbody_snap_chemflag* = 0 +then :math:`N_{\rm 4b} = (J_{\rm max}+1)(J_{\rm max}+2)(J_{\rm max}+1.5)/3`. +If *fourbody_snap_chemflag* = 1 then :math:`N_{\rm 4b} = N_{\rm e}^3 (J_{\rm max}+1)(J_{\rm max}+2)(J_{\rm max}+1.5)/3`. +The bispectrum calculation is described in more detail in :doc:`compute sna/atom `. Linear Proper Orthogonal Descriptor Potentials """""""""""""""""""""""""""""""""""""""""""""" -The proper orthogonal descriptors and SNAP descriptors are used to define the atomic energies -in the following expansion +The proper orthogonal descriptors and SNAP descriptors are used to define the atomic energies +in the following expansion .. math:: - E_{i}(\boldsymbol \eta) = \sum_{p=1}^{N_{\rm e}} c^{(1)}_p D^{(1)}_{ip} + \sum_{m=1}^{N_{\rm 2b}} \sum_{l=1}^{N_{\rm e}(N_{\rm e}+1)/2} c^{(2)}_{ml} D^{(2)}_{iml}(\boldsymbol \eta) + \sum_{m=1}^{N_{\rm r}} \sum_{n=1}^{N_{\rm a}} \sum_{\ell=1}^{N_{\rm e}^2(N_{\rm e}+1)/2} c^{(3)}_{mn\ell} D^{(3)}_{imn\ell}(\boldsymbol \eta) + \sum_{k=1}^{N_{\rm 4b}} \sum_{p=1}^{N_{\rm e}} c_{kp}^{(4)} D_{ikp}^{(4)}(\boldsymbol \eta), + E_{i}(\boldsymbol \eta) = \sum_{p=1}^{N_{\rm e}} c^{(1)}_p D^{(1)}_{ip} + \sum_{m=1}^{N_{\rm 2b}} \sum_{l=1}^{N_{\rm e}(N_{\rm e}+1)/2} c^{(2)}_{ml} D^{(2)}_{iml}(\boldsymbol \eta) + \sum_{m=1}^{N_{\rm r}} \sum_{n=1}^{N_{\rm a}} \sum_{\ell=1}^{N_{\rm e}^2(N_{\rm e}+1)/2} c^{(3)}_{mn\ell} D^{(3)}_{imn\ell}(\boldsymbol \eta) + \sum_{k=1}^{N_{\rm 4b}} \sum_{p=1}^{N_{\rm e}} c_{kp}^{(4)} D_{ikp}^{(4)}(\boldsymbol \eta), -where :math:`D^{(1)}_{ip}, D^{(2)}_{iml}, D^{(3)}_{imn\ell}, D^{(4)}_{ikp}` are the one-body, two-body, three-body, four-body descriptors, -respectively, and :math:`c^{(1)}_p, c^{(2)}_{ml}, c^{(3)}_{mn\ell}, c^{(4)}_{kp}` are their respective expansion -coefficients. In a more compact notation that implies summation over descriptor indices +where :math:`D^{(1)}_{ip}, D^{(2)}_{iml}, D^{(3)}_{imn\ell}, D^{(4)}_{ikp}` are the one-body, two-body, three-body, four-body descriptors, +respectively, and :math:`c^{(1)}_p, c^{(2)}_{ml}, c^{(3)}_{mn\ell}, c^{(4)}_{kp}` are their respective expansion +coefficients. In a more compact notation that implies summation over descriptor indices the atomic energies can be written as .. math:: - E_i(\boldsymbol \eta) = \sum_{m=1}^{N_{\rm e}} c^{(1)}_m D^{(1)}_{im} + \sum_{m=1}^{N_{\rm d}^{(2)}} c^{(2)}_k D^{(2)}_{im} + \sum_{m=1}^{N_{\rm d}^{(3)}} c^{(3)}_m D^{(3)}_{im} + \sum_{m=1}^{N_{\rm d}^{(4)}} c^{(4)}_m D^{(4)}_{im} + E_i(\boldsymbol \eta) = \sum_{m=1}^{N_{\rm e}} c^{(1)}_m D^{(1)}_{im} + \sum_{m=1}^{N_{\rm d}^{(2)}} c^{(2)}_k D^{(2)}_{im} + \sum_{m=1}^{N_{\rm d}^{(3)}} c^{(3)}_m D^{(3)}_{im} + \sum_{m=1}^{N_{\rm d}^{(4)}} c^{(4)}_m D^{(4)}_{im} -where :math:`N_{\rm d}^{(2)} = N_{\rm 2b} N_{\rm e} (N_{\rm e}+1)/2`, +where :math:`N_{\rm d}^{(2)} = N_{\rm 2b} N_{\rm e} (N_{\rm e}+1)/2`, :math:`N_{\rm d}^{(3)} = N_{\rm 3b} N_{\rm e}^2 (N_{\rm e}+1)/2`, and -:math:`N_{\rm d}^{(4)} = N_{\rm 4b} N_{\rm e}` are +:math:`N_{\rm d}^{(4)} = N_{\rm 4b} N_{\rm e}` are the number of two-body, three-body, and four-body descriptors, respectively. -The potential energy is then obtained by summing local atomic energies :math:`E_i` +The potential energy is then obtained by summing local atomic energies :math:`E_i` for all atoms :math:`i` in the system .. math:: - E(\boldsymbol \eta) = \sum_{i}^N E_{i}(\boldsymbol \eta) + E(\boldsymbol \eta) = \sum_{i}^N E_{i}(\boldsymbol \eta) -Because the descriptors are one-body, two-body, and three-body terms, -the resulting POD potential is a three-body PES. We can express the potential +Because the descriptors are one-body, two-body, and three-body terms, +the resulting POD potential is a three-body PES. We can express the potential energy as a linear combination of the global descriptors as follows .. math:: - E(\boldsymbol \eta) = \sum_{m=1}^{N_{\rm e}} c^{(1)}_m d^{(1)}_{m} + \sum_{m=1}^{N_{\rm d}^{(2)}} c^{(2)}_m d^{(2)}_{m} + \sum_{m=1}^{N_{\rm d}^{(3)}} c^{(3)}_m d^{(3)}_{m} + \sum_{m=1}^{N_{\rm d}^{(4)}} c^{(4)}_m d^{(4)}_{m} + E(\boldsymbol \eta) = \sum_{m=1}^{N_{\rm e}} c^{(1)}_m d^{(1)}_{m} + \sum_{m=1}^{N_{\rm d}^{(2)}} c^{(2)}_m d^{(2)}_{m} + \sum_{m=1}^{N_{\rm d}^{(3)}} c^{(3)}_m d^{(3)}_{m} + \sum_{m=1}^{N_{\rm d}^{(4)}} c^{(4)}_m d^{(4)}_{m} where the global descriptors are given by @@ -461,54 +461,54 @@ Hence, we obtain the atomic forces as .. math:: - \boldsymbol F = -\nabla E(\boldsymbol \eta) = - \sum_{m=1}^{N_{\rm d}^{(2)}} c^{(2)}_m \nabla d_m^{(2)} - \sum_{m=1}^{N_{\rm d}^{(3)}} c^{(3)}_m \nabla d_m^{(3)} - \sum_{m=1}^{N_{\rm d}^{(4)}} c^{(4)}_m \nabla d_m^{(4)} + \boldsymbol F = -\nabla E(\boldsymbol \eta) = - \sum_{m=1}^{N_{\rm d}^{(2)}} c^{(2)}_m \nabla d_m^{(2)} - \sum_{m=1}^{N_{\rm d}^{(3)}} c^{(3)}_m \nabla d_m^{(3)} - \sum_{m=1}^{N_{\rm d}^{(4)}} c^{(4)}_m \nabla d_m^{(4)} -where :math:`\nabla d_m^{(2)}`, :math:`\nabla d_m^{(3)}` and :math:`\nabla d_m^{(4)}` are derivatives of the two-body -three-body, and four-body global descriptors with respect to atom positions, respectively. +where :math:`\nabla d_m^{(2)}`, :math:`\nabla d_m^{(3)}` and :math:`\nabla d_m^{(4)}` are derivatives of the two-body +three-body, and four-body global descriptors with respect to atom positions, respectively. Note that since the first-body global descriptors are constant, their derivatives are zero. Quadratic Proper Orthogonal Descriptor Potentials """"""""""""""""""""""""""""""""""""""""""""""""" -We recall two-body PODs :math:`D^{(2)}_{ik}, 1 \le k \le N_{\rm d}^{(2)}`, -and three-body PODs :math:`D^{(3)}_{im}, 1 \le m \le N_{\rm d}^{(3)}`, -with :math:`N_{\rm d}^{(2)} = N_{\rm 2b} N_{\rm e} (N_{\rm e}+1)/2` and -:math:`N_{\rm d}^{(3)} = N_{\rm 3b} N_{\rm e}^2 (N_{\rm e}+1)/2` being -the number of descriptors per atom for the two-body PODs and three-body PODs, -respectively. We employ them to define a new set of atomic descriptors as follows +We recall two-body PODs :math:`D^{(2)}_{ik}, 1 \le k \le N_{\rm d}^{(2)}`, +and three-body PODs :math:`D^{(3)}_{im}, 1 \le m \le N_{\rm d}^{(3)}`, +with :math:`N_{\rm d}^{(2)} = N_{\rm 2b} N_{\rm e} (N_{\rm e}+1)/2` and +:math:`N_{\rm d}^{(3)} = N_{\rm 3b} N_{\rm e}^2 (N_{\rm e}+1)/2` being +the number of descriptors per atom for the two-body PODs and three-body PODs, +respectively. We employ them to define a new set of atomic descriptors as follows .. math:: D^{(2*3)}_{ikm} = \frac{1}{2}\left( D^{(2)}_{ik} \sum_{j=1}^N D^{(3)}_{jm} + D^{(3)}_{im} \sum_{j=1}^N D^{(2)}_{jk} \right) -for :math:`1 \le i \le N, 1 \le k \le N_{\rm d}^{(2)}, 1 \le m \le N_{\rm d}^{(3)}`. -The new descriptors are four-body because they involve central atom :math:`i` together -with three neighbors :math:`j, k` and :math:`l`. The total number of new descriptors per atom is equal to +for :math:`1 \le i \le N, 1 \le k \le N_{\rm d}^{(2)}, 1 \le m \le N_{\rm d}^{(3)}`. +The new descriptors are four-body because they involve central atom :math:`i` together +with three neighbors :math:`j, k` and :math:`l`. The total number of new descriptors per atom is equal to .. math:: N_{\rm d}^{(2*3)} = N_{\rm d}^{(2)} * N_{\rm d}^{(3)} = N_{\rm 2b} N_{\rm 3b} N_{\rm e}^3 (N_{\rm e}+1)^2/4 . -The new global descriptors are calculated as +The new global descriptors are calculated as .. math:: - d^{(2*3)}_{km} = \sum_{i=1}^N D^{(2*3)}_{ikm} = \left( \sum_{i=1}^N D^{(2)}_{ik} \right) \left( \sum_{i=1}^N D^{(3)}_{im} \right) = d^{(2)}_{k} d^{(3)}_m, + d^{(2*3)}_{km} = \sum_{i=1}^N D^{(2*3)}_{ikm} = \left( \sum_{i=1}^N D^{(2)}_{ik} \right) \left( \sum_{i=1}^N D^{(3)}_{im} \right) = d^{(2)}_{k} d^{(3)}_m, -for :math:`1 \le k \le N_{\rm d}^{(2)}, 1 \le m \le N_{\rm d}^{(3)}`. Hence, the gradient +for :math:`1 \le k \le N_{\rm d}^{(2)}, 1 \le m \le N_{\rm d}^{(3)}`. Hence, the gradient of the new global descriptors with respect to atom positions is calculated as .. math:: \nabla d^{(2*3)}_{km} = d^{(3)}_m \nabla d^{(2)}_{k} + d^{(2)}_{k} \nabla d^{(3)}_m, \quad 1 \le k \le N_{\rm d}^{(2)}, 1 \le m \le N_{\rm d}^{(3)} . -Instead of using all the new descriptors, we allow the user to choose a subset as :math:`{N}_{\rm 2d}^{(2*3)} = N_{\rm 2b}^{2*3} N_{\rm e} (N_{\rm e}+1)/2` and -:math:`{N}_{\rm 3d}^{(2*3)} = N^{2*3}_{\rm 3b} N_{\rm e}^2 (N_{\rm e}+1)/2`. Here -:math:`N_{\rm 2b}^{2*3}` and :math:`N_{\rm 3b}^{2*3}` correspond to *quadratic23_number_twobody_basis_functions* and +Instead of using all the new descriptors, we allow the user to choose a subset as :math:`{N}_{\rm 2d}^{(2*3)} = N_{\rm 2b}^{2*3} N_{\rm e} (N_{\rm e}+1)/2` and +:math:`{N}_{\rm 3d}^{(2*3)} = N^{2*3}_{\rm 3b} N_{\rm e}^2 (N_{\rm e}+1)/2`. Here +:math:`N_{\rm 2b}^{2*3}` and :math:`N_{\rm 3b}^{2*3}` correspond to *quadratic23_number_twobody_basis_functions* and *quadratic23_number_threebody_basis_functions*, respectively. -The (2*3) quadratic potential is defined as a linear combination of the +The (2*3) quadratic potential is defined as a linear combination of the original and new global descriptors as follows .. math:: @@ -525,7 +525,7 @@ which is simplified to .. math:: - E^{(2*3)} = 0.5 \sum_{k=1}^{N_{\rm 2d}^{(2*3)}} b_k^{(2)} d_k^{(2)} + 0.5 \sum_{m=1}^{N_{\rm 3d}^{(2*3)}} b_m^{(3)} d_m^{(3)} + E^{(2*3)} = 0.5 \sum_{k=1}^{N_{\rm 2d}^{(2*3)}} b_k^{(2)} d_k^{(2)} + 0.5 \sum_{m=1}^{N_{\rm 3d}^{(2*3)}} b_m^{(3)} d_m^{(3)} where @@ -534,7 +534,7 @@ where b_k^{(2)} & = \sum_{m=1}^{N_{\rm 3d}^{(2*3)}} c^{(2*3)}_{km} d_m^{(3)}, \quad k = 1,\ldots, N_{\rm 2d}^{(2*3)}, \\ b_m^{(3)} & = \sum_{k=1}^{N_{\rm 2d}^{(2*3)}} c^{(2*3)}_{km} d_k^{(2)}, \quad m = 1,\ldots, N_{\rm 3d}^{(2*3)} . -The (2*3) quadratic potential results in the following atomic forces +The (2*3) quadratic potential results in the following atomic forces .. math:: @@ -546,16 +546,16 @@ It can be shown that \boldsymbol F^{(2*3)} = - \sum_{k=1}^{N_{\rm 2d}^{(2*3)}} b^{(2)}_k \nabla d_k^{(2)} - \sum_{m=1}^{N_{\rm 3d}^{(2*3)}} b^{(3)}_m \nabla d_m^{(3)} . -The calculation of the atomic forces for the (2*3) quadratic potential -only requires the extra calculation of :math:`b_k^{(2)}` and :math:`b_m^{(3)}` which can be negligible. +The calculation of the atomic forces for the (2*3) quadratic potential +only requires the extra calculation of :math:`b_k^{(2)}` and :math:`b_m^{(3)}` which can be negligible. As a result, the (2*3) quadratic potential does not increase the computational complexity. -A similar procedure can be used to form other quadratic potentials. -For instance, we may combine the three-body descriptors with the four-body -descriptors to generate the (3*4) quadratic potential. We can also combine -the three-body descriptors with themselves to generate the (3*3) quadratic potential. -It is important to know that because quadratic potentials have a large number of coefficients -they require large training data set in order to avoid overfitting. +A similar procedure can be used to form other quadratic potentials. +For instance, we may combine the three-body descriptors with the four-body +descriptors to generate the (3*4) quadratic potential. We can also combine +the three-body descriptors with themselves to generate the (3*3) quadratic potential. +It is important to know that because quadratic potentials have a large number of coefficients +they require large training data set in order to avoid overfitting. Cubic Proper Orthogonal Descriptor Potentials """"""""""""""""""""""""""""""""""""""""""""" @@ -570,7 +570,7 @@ It thus follows that .. math:: - E^{(2*3*4)} = \frac13 \sum_{k=1}^{N_{\rm 2d}^{(2*3*4)}} b_k^{(2)} d_k^{(2)} + \frac13 \sum_{m=1}^{N_{\rm 3d}^{(2*3*4)}} b_m^{(3)} d_m^{(3)} + \frac13 \sum_{n=1}^{N_{\rm 4d}^{(2*3*4)}} b_n^{(4)} d_n^{(4)} + E^{(2*3*4)} = \frac13 \sum_{k=1}^{N_{\rm 2d}^{(2*3*4)}} b_k^{(2)} d_k^{(2)} + \frac13 \sum_{m=1}^{N_{\rm 3d}^{(2*3*4)}} b_m^{(3)} d_m^{(3)} + \frac13 \sum_{n=1}^{N_{\rm 4d}^{(2*3*4)}} b_n^{(4)} d_n^{(4)} where @@ -578,65 +578,65 @@ where b_k^{(2)} & = \sum_{m=1}^{N_{\rm 3d}^{(2*3*4)}} \sum_{n=1}^{N_{\rm 4d}^{(2*3*4)}} c^{(2*3*4)}_{kmn} d_m^{(3)} d_n^{(4)}, \quad k = 1,\ldots, N_{\rm 2d}^{(2*3*4)} \\ b_m^{(3)} & = \sum_{k=1}^{N_{\rm 2d}^{(2*3*4)}} \sum_{n=1}^{N_{\rm 4d}^{(2*3*4)}} c^{(2*3*4)}_{kmn} d_k^{(2)} d_n^{(4)}, \quad m = 1,\ldots, N_{\rm 3d}^{(2*3*4)} \\ - b_n^{(4)} & = \sum_{k=1}^{N_{\rm 2d}^{(2*3*4)}} \sum_{m=1}^{N_{\rm 3d}^{(2*3*4)}} c^{(2*3*4)}_{kmn} d_k^{(2)} d_m^{(3)}, \quad n = 1,\ldots, N_{\rm 4d}^{(2*3*4)} + b_n^{(4)} & = \sum_{k=1}^{N_{\rm 2d}^{(2*3*4)}} \sum_{m=1}^{N_{\rm 3d}^{(2*3*4)}} c^{(2*3*4)}_{kmn} d_k^{(2)} d_m^{(3)}, \quad n = 1,\ldots, N_{\rm 4d}^{(2*3*4)} -The (2*3*4) cubic potential results in the following atomic forces +The (2*3*4) cubic potential results in the following atomic forces .. math:: \boldsymbol F^{(2*3*4)} = - \sum_{k=1}^{N_{\rm 2d}^{(2*3*4)}} b^{(2)}_k \nabla d_k^{(2)} - \sum_{m=1}^{N_{\rm 3d}^{(2*3*4)}} b^{(3)}_m \nabla d_m^{(3)} - \sum_{n=1}^{N_{\rm 4d}^{(2*3*4)}} b^{(4)}_n \nabla d_n^{(4)} . -Note that :math:`{N}_{\rm 2d}^{(2*3*4)} = N_{\rm 2b}^{2*3*4} N_{\rm e} (N_{\rm e}+1)/2`, -:math:`{N}_{\rm 3d}^{(2*3*4)} = N^{2*3*4}_{\rm 3b} N_{\rm e}^2 (N_{\rm e}+1)/2`, and -:math:`{N}_{\rm 4d}^{(2*3*4)} = N^{2*3*4}_{\rm 4b} N_{\rm e}`. Here -:math:`N_{\rm 2b}^{2*3*4}`, :math:`N_{\rm 3b}^{2*3*4}`, and :math:`N_{\rm 4b}^{2*3*4}` correspond to -*cubic234_number_twobody_basis_functions*, +Note that :math:`{N}_{\rm 2d}^{(2*3*4)} = N_{\rm 2b}^{2*3*4} N_{\rm e} (N_{\rm e}+1)/2`, +:math:`{N}_{\rm 3d}^{(2*3*4)} = N^{2*3*4}_{\rm 3b} N_{\rm e}^2 (N_{\rm e}+1)/2`, and +:math:`{N}_{\rm 4d}^{(2*3*4)} = N^{2*3*4}_{\rm 4b} N_{\rm e}`. Here +:math:`N_{\rm 2b}^{2*3*4}`, :math:`N_{\rm 3b}^{2*3*4}`, and :math:`N_{\rm 4b}^{2*3*4}` correspond to +*cubic234_number_twobody_basis_functions*, *cubic234_number_threebody_basis_functions*, and *cubic234_number_fourbody_basis_functions*, respectively. -The calculation of the atomic forces for the (2*3*4) cubic potential -only requires the extra calculation of :math:`b_k^{(2)}`, :math:`b_m^{(3)}`, and :math:`b_n^{(4)}` which can be negligible. +The calculation of the atomic forces for the (2*3*4) cubic potential +only requires the extra calculation of :math:`b_k^{(2)}`, :math:`b_m^{(3)}`, and :math:`b_n^{(4)}` which can be negligible. As a result, the (2*3*4) cubic potential does not increase the computational complexity. -Similarly, other cubic potentials can be formed by combining three sets of descriptors. +Similarly, other cubic potentials can be formed by combining three sets of descriptors. Training """""""""""" -POD potentials are trained using the least-squares regression against density functional theory (DFT) data. -Let :math:`J` be the number of training configurations, with :math:`N_j` being the number of -atoms in the jth configuration. Let :math:`\{E^{\star}_j\}_{j=1}^{J}` -and :math:`\{\boldsymbol F^{\star}_j\}_{j=1}^{J}` be the DFT energies and forces -for :math:`J` configurations. Next, we calculate the global descriptors -and their derivatives for all training configurations. Let :math:`d_{jm}, 1 \le m \le M`, be the -global descriptors associated with the jth configuration, where :math:`M` is the number of global +POD potentials are trained using the least-squares regression against density functional theory (DFT) data. +Let :math:`J` be the number of training configurations, with :math:`N_j` being the number of +atoms in the jth configuration. Let :math:`\{E^{\star}_j\}_{j=1}^{J}` +and :math:`\{\boldsymbol F^{\star}_j\}_{j=1}^{J}` be the DFT energies and forces +for :math:`J` configurations. Next, we calculate the global descriptors +and their derivatives for all training configurations. Let :math:`d_{jm}, 1 \le m \le M`, be the +global descriptors associated with the jth configuration, where :math:`M` is the number of global descriptors. We then form a matrix :math:`\boldsymbol A \in \mathbb{R}^{J \times M}` -with entries :math:`A_{jm} = d_{jm}/ N_j` for :math:`j=1,\ldots,J` and :math:`m=1,\ldots,M`. -Moreover, we form a matrix :math:`\boldsymbol B \in \mathbb{R}^{\mathcal{N} \times M}` by stacking -the derivatives of the global descriptors for all training configurations from top -to bottom, where :math:`\mathcal{N} = 3\sum_{j=1}^{J} N_j`. +with entries :math:`A_{jm} = d_{jm}/ N_j` for :math:`j=1,\ldots,J` and :math:`m=1,\ldots,M`. +Moreover, we form a matrix :math:`\boldsymbol B \in \mathbb{R}^{\mathcal{N} \times M}` by stacking +the derivatives of the global descriptors for all training configurations from top +to bottom, where :math:`\mathcal{N} = 3\sum_{j=1}^{J} N_j`. -The coefficient vector :math:`\boldsymbol c` of the POD potential is found by solving -the following least-squares problem +The coefficient vector :math:`\boldsymbol c` of the POD potential is found by solving +the following least-squares problem .. math:: - {\min}_{\boldsymbol c \in \mathbb{R}^{M}} \ w_E \|\boldsymbol A(\boldsymbol \eta) \boldsymbol c - \bar{\boldsymbol E}^{\star} \|^2 + w_F \|\boldsymbol B(\boldsymbol \eta) \boldsymbol c + \boldsymbol F^{\star} \|^2, + {\min}_{\boldsymbol c \in \mathbb{R}^{M}} \ w_E \|\boldsymbol A(\boldsymbol \eta) \boldsymbol c - \bar{\boldsymbol E}^{\star} \|^2 + w_F \|\boldsymbol B(\boldsymbol \eta) \boldsymbol c + \boldsymbol F^{\star} \|^2, -where :math:`w_E` and :math:`w_F` are weights for the energy (*fitting_weight_energy*) and -force (*fitting_weight_force*), respectively. -Here :math:`\bar{\boldsymbol E}^{\star} \in \mathbb{R}^{J}` is a vector of with entries -:math:`\bar{E}^{\star}_j = E^{\star}_j/N_j` and :math:`\boldsymbol F^{\star}` is a vector of :math:`\mathcal{N}` -entries obtained by stacking :math:`\{\boldsymbol F^{\star}_j\}_{j=1}^{J}` from top to bottom. +where :math:`w_E` and :math:`w_F` are weights for the energy (*fitting_weight_energy*) and +force (*fitting_weight_force*), respectively. +Here :math:`\bar{\boldsymbol E}^{\star} \in \mathbb{R}^{J}` is a vector of with entries +:math:`\bar{E}^{\star}_j = E^{\star}_j/N_j` and :math:`\boldsymbol F^{\star}` is a vector of :math:`\mathcal{N}` +entries obtained by stacking :math:`\{\boldsymbol F^{\star}_j\}_{j=1}^{J}` from top to bottom. -The training procedure is the same for both the linear and quadratic POD potentials. -However, since the quadratic POD potential has a significantly larger number of the global -descriptors, it is more expensive to train the linear POD potential. This is -because the training of the quadratic POD potential -still requires us to calculate and store the quadratic global descriptors and -their gradient. Furthermore, the quadratic POD potential may require more training -data in order to prevent overfitting. In order to reduce the computational cost of fitting -the quadratic POD potential and avoid overfitting, we can use subsets of two-body and three-body -PODs for constructing the new descriptors. +The training procedure is the same for both the linear and quadratic POD potentials. +However, since the quadratic POD potential has a significantly larger number of the global +descriptors, it is more expensive to train the linear POD potential. This is +because the training of the quadratic POD potential +still requires us to calculate and store the quadratic global descriptors and +their gradient. Furthermore, the quadratic POD potential may require more training +data in order to prevent overfitting. In order to reduce the computational cost of fitting +the quadratic POD potential and avoid overfitting, we can use subsets of two-body and three-body +PODs for constructing the new descriptors. Restrictions @@ -644,7 +644,7 @@ Restrictions This compute is part of the ML-POD package. It is only enabled if LAMMPS was built with that package by setting -D PKG_ML-POD=on. See the :doc:`Build package -` page for more info. +` page for more info. Related commands """""""""""""""" @@ -665,6 +665,3 @@ The keyword defaults are also given in the description of the input files. .. _Nguyen20222: **(Nguyen)** Nguyen and Rohskopf, arXiv preprint arXiv:2209.02362 (2022). - - - diff --git a/doc/src/pair_pod.rst b/doc/src/pair_pod.rst index f915d1a858..0930f1e2ee 100644 --- a/doc/src/pair_pod.rst +++ b/doc/src/pair_pod.rst @@ -21,27 +21,27 @@ Examples Description """"""""""" -Pair style *pod* defines the proper orthogonal descriptor (POD) potential +Pair style *pod* defines the proper orthogonal descriptor (POD) potential :ref:`(Nguyen) `. The mathematical definition of the POD potential is described from :doc:`compute podfit `, which is used to fit the POD -potential to *ab initio* energy and force data. +potential to *ab initio* energy and force data. Only a single pair_coeff command is used with the *pod* style which specifies a POD parameter file followed by a coefficient file. -The coefficient file (coefficient.txt) contains coefficients for the POD potential. The top of the coefficient -file can contain any number of blank and comment lines (start with #), but follows a +The coefficient file (coefficient.txt) contains coefficients for the POD potential. The top of the coefficient +file can contain any number of blank and comment lines (start with #), but follows a strict format after that. The first non-blank non-comment line must contain: * POD_coefficients: *ncoeff* This is followed by *ncoeff* coefficients, one per line. The coefficient file -is generated after training the POD potential using :doc:`compute podfit `. +is generated after training the POD potential using :doc:`compute podfit `. The POD parameter file (pod.txt) can contain blank and comment lines (start with #) anywhere. Each non-blank non-comment line must contain one -keyword/value pair. See :doc:`compute podfit ` for the description -of all the keywords that can be assigned in the parameter file. +keyword/value pair. See :doc:`compute podfit ` for the description +of all the keywords that can be assigned in the parameter file. ----------