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

Machine learning techniques, specifically gradient-enhanced Kriging (GEK), have been implemented for molecular geometry optimization. GEK-based optimization has many advantages compared to conventional - step-restricted second-order truncated expansion - molecular optimization methods. In particular, the surrogate model given by GEK can have multiple stationary points, will smoothly converge to the exact model as the number of sample points increases, and contains an explicit expression for the expected error of the model function at an arbitrary point. Machine learning is, however, associated with abundance of data, contrary to the situation desired for efficient geometry optimizations. In the paper we demonstrate how the GEK procedure can be utilized in a fashion such that in the presence of few data points, the surrogate surface will in a robust way guide the optimization to a minimum of a potential energy surface. In this respect the GEK procedure will be used to mimic the behavior of a conventional second-order scheme, but retaining the flexibility of the superior machine learning approach. Moreover, the expected error will be used in the optimization to facilitate restricted-variance optimizations (RVO). A procedure which relates the eigenvalues of the approximate guessed Hessian with the individual characteristic lengths, used in the GEK model, reduces the number of empirical parameters to optimize to two - the value of the trend function and the maximum allowed variance. These parameters are determined using the extended Baker (e-Baker) and part of the Baker transition-state (Baker-TS) test suites as a training set. The so-created optimization procedure is tested using the e-Baker, the full Baker-TS, and the S22 test suites, at the density-functional-theory and second order Møller-Plesset levels of approximation. The results show that the new method is generally of similar or better performance than a state-of-the-art conventional method, even for cases where no significant improvement was expected.

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

Machine learning techniques, specifically gradient-enhanced Kriging (GEK), have been implemented for molecular geometry optimization. GEK-based optimization has many advantages compared to conventional - step-restricted second-order truncated expansion - molecular optimization methods. In particular, the surrogate model given by GEK can have multiple stationary points, will smoothly converge to the exact model as the number of sample points increases, and contains an explicit expression for the expected error of the model function at an arbitrary point. Machine learning is, however, associated with abundance of data, contrary to the situation desired for efficient geometry optimizations. In the paper we demonstrate how the GEK procedure can be utilized in a fashion such that in the presence of few data points, the surrogate surface will in a robust way guide the optimization to a minimum of a potential energy surface. In this respect the GEK procedure will be used to mimic the behavior of a conventional second-order scheme, but retaining the flexibility of the superior machine learning approach. Moreover, the expected error will be used in the optimization to facilitate restricted-variance optimizations (RVO). A procedure which relates the eigenvalues of the approximate guessed Hessian with the individual characteristic lengths, used in the GEK model, reduces the number of empirical parameters to optimize to two - the value of the trend function and the maximum allowed variance. These parameters are determined using the extended Baker (e-Baker) and part of the Baker transition-state (Baker-TS) test suites as a training set. The so-created optimization procedure is tested using the e-Baker, the full Baker-TS, and the S22 test suites, at the density-functional-theory and second order Møller-Plesset levels of approximation. The results show that the new method is generally of similar or better performance than a state-of-the-art conventional method, even for cases where no significant improvement was expected.

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

Machine learning techniques, specifically gradient-enhanced Kriging (GEK), have been implemented for molecular geometry optimization. GEK-based optimization has many advantages compared to conventional - step-restricted second-order truncated expansion - molecular optimization methods. In particular, the surrogate model given by GEK can have multiple stationary points, will smoothly converge to the exact model as the number of sample points increases, and contains an explicit expression for the expected error of the model function at an arbitrary point. Machine learning is, however, associated with abundance of data, contrary to the situation desired for efficient geometry optimizations. In the paper we demonstrate how the GEK procedure can be utilized in a fashion such that in the presence of few data points, the surrogate surface will in a robust way guide the optimization to a minimum of a potential energy surface. In this respect the GEK procedure will be used to mimic the behavior of a conventional second-order scheme, but retaining the flexibility of the superior machine learning approach. Moreover, the expected error will be used in the optimization to facilitate restricted-variance optimizations (RVO). A procedure which relates the eigenvalues of the approximate guessed Hessian with the individual characteristic lengths, used in the GEK model, reduces the number of empirical parameters to optimize to two - the value of the trend function and the maximum allowed variance. These parameters are determined using the extended Baker (e-Baker) and part of the Baker transition-state (Baker-TS) test suites as a training set. The so-created optimization procedure is tested using the e-Baker, the full Baker-TS, and the S22 test suites, at the density-functional-theory and second order Møller-Plesset levels of approximation. The results show that the new method is generally of similar or better performance than a state-of-the-art conventional method, even for cases where no significant improvement was expected.

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

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>