Restricted-Variance Molecular Geometry Optimization Based on Gradient-Enhanced Kriging

Abstract

Machine learning techniques, specifically Gradient-Enhanced Kriging (GEK), has been implemented for molecular geometry optimization.<br>GEK has many advantages as compared to conventional -- step-restricted second-order truncated -- molecular optimization methods.<br>In particular, the surrogate model associated with GEK can have multiple stationary points, will smoothly converge to the<br>exact model as the size of the data set increases, and contains an explicit expression for the expected average error of the model function<br>at an arbitrary point in space.<br>In this respect GEK can be of interest for methods used in molecular geometry optimizations.<br>GEK is usually, however, associated with abundance of data, contrary to the situation desired for<br>efficient geometry optimizations.<br>In the paper we will demonstrate how the GEK procedure can be utilized in a fashion such that in the presence of few data points, the<br>surrogate surface will in a robust way guide the optimization to a minimum of a molecular structure.<br>In this respect the GEK procedure will be used to mimic the behavior of a conventional second-order scheme, but retaining the<br>flexibility of the superior machine learning approach -- GEK is an exact interpolator.<br>Moreover, the expected variance will be used in the optimization to facilitate restricted-variance rational function optimizations (RV-RFO).<br>A procedure which relates the eigenvalues of the Hessian-model-function Hessian with the individual characteristic<br>lengths, used in the GEK, reduces the number of empirical parameters to two -- the value of the trend function and the<br>maximum allowed variance. These parameters are determined using the extended Baker (e-Baker) test suite, at the Hartree-Fock level of approximation,<br>and a single reaction of the Baker transition-state (Baker-TS) test suite as a training set. The so-created optimization<br>procedure -- RV-RFO-GEK -- is tested using the e-Baker, the full Baker-TS, and the S22 test suites, at the density-functional-theory level for the two Baker test suites<br>and at the second order Møller-Plesset level of approximation for the S22 test suite, respectively.<br>The tests show that the new method is generally on par with a state-of-the-art conventional method, while for difficult cases it exhibits a definite advantage.<br>

Restricted-Variance Constrained, Reaction Path, and Transition State Molecular Optimizations Using Gradient-Enhanced Kriging

“…We have extended the restricted variance optimization method, based on gradientenhanced Kriging surrogate model, 28 to work with arbitrary geometrical constraints. The resulting method combines the projected constrained optimization 43 and the RVO.…”

“…Below, a brief presentation is given of the gradient-enhanced Kriging surrogate model, the selection of the associated characteristic lengths, and restricted-variance optimization as implemented in our previous work. 28 This approach is the main engine behind the surrogate model that is used in this study for TS structure optimizations, and for performing constrained geometry optimizations. The initial presentation will be followed by brief reviews of the specific methods used for the benchmark calculations.…”

“…While the first step (computing ∆y and ∆q c ) is performed completely independent of the minimization, the latter (computing ∆x and ∆q m ) is performed in the presence of the former displacement. The global variance restriction σ RVO is defined as in the standard RVO, 28 but if ∆q c were allowed to use this whole variance, there would be no further room for minimization. Therefore, we apply a reduced threshold for the computation of ∆q c , by multiplying σ RVO by a factor that depends on the maximum gradient of any of the constraints,…”

“…Recently an alternative surrogate model -the Kriging model, 15,16 in particular the gradient-enhanced Kriging (GEK) [17][18][19] -has been proposed and explored by a number of research groups for geometry optimizations of equilibrium and transition state structures, and explorations of reactions paths. [20][21][22][23][24][25][26][27][28] These studies have demonstrated the potential of the Kriging procedure as a competitive alternative to the standard. However, these studies have also shown that in order to excel, as compared with commonly used algorithms, a GEK-based method should be based on internal coordinates, 27 but also be able to make use of the empirical and practical knowledge accumulated through decades of use and improvement of second-order methods.…”

“…However, these studies have also shown that in order to excel, as compared with commonly used algorithms, a GEK-based method should be based on internal coordinates, 27 but also be able to make use of the empirical and practical knowledge accumulated through decades of use and improvement of second-order methods. Raggi and co-workers 28 have recently demonstrated how the use of Hessian model functions 4 eliminated the required optimization through likelihood maximization of the characteristics lengths -hyperparameters of the GEK model. Moreover, the same study proposed a new type of optimization -restricted-variance optimization (RVO) -in which the explicit variance of the surrogate model is used in an efficient way as the optimization explores the multidimensional energy functional.…”