Algorithms for Finding Saddle Points and Minimum Energy Paths Using Gaussian Process Regression

Olli-Pekka Koistinen

Research output: ThesisDoctoral ThesisCollection of Articles


Chemical reactions and other transitions involving rearrangements of atoms can be studied theoretically by analyzing a potential energy surface defined in a high-dimensional space of atom coordinates. Local minimum points of the energy surface correspond to stable states of the system, and minimum energy paths connecting these states characterize mechanisms of possible transitions. Of particular interest is often the maximum point of the minimum energy path, which is located at a first-order saddle point of the energy surface and can be used to estimate the activation energy and rate of the particular transition. Minimum energy paths and saddle points between two known states have been traditionally searched with iterative methods where a chain of discrete points of the coordinate space is moved and stretched towards a minimum energy path according to imaginary forces based on gradient vectors of the potential energy surface. The actual saddle point can be found by reversing the component of the gradient vector parallel to the path at one of the points of the chain and letting this point climb along the path towards the saddle point. If the end state of the transition is unknown, the saddle point can be searched correspondingly by rotating a pair of closely spaced points towards the orientation of the lowest curvature, reversing the gradient component corresponding to this direction, and moving the pair towards the saddle point. These methods may, however, require hundreds of iterations, and since accurate evaluation of the gradient vector is often computationally expensive, the information obtained from previous iterations should be utilized as efficiently as possible to decrease the number of iterations. Using statistical models, an approximation to the energy surface can be constructed, and a minimum energy path or a saddle point can be searched on the approximate surface. The accuracy of the solution can be checked with further evaluations, which can be then used to update the model for following iterations. In this dissertation, machine learning algorithms based on Gaussian process regression are developed to enhance searches of minimum energy paths and saddle points. Gaussian process models serve here as flexible prior probability models for potential energy surfaces. Observed values of both energy and its derivatives can be used to update the model, and the posterior predictive distribution obtained as a result of Bayesian inference provides also an uncertainty estimate, which can be utilized when selecting new observation points. Separate methods are presented both for finding a minimum energy path between two known states and a saddle point located in the vicinity of a given start point. Based on simple test examples, the methods utilizing Gaussian processes may reduce the number of evaluations to a fraction of what is required by conventional methods.
Translated title of the contributionGaussisia prosesseja hyödyntäviä menetelmiä satulapisteiden ja minimienergiapolkujen etsintään
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
  • Vehtari, Aki, Supervising Professor
  • Jonsson, Hannes, Supervising Professor
Print ISBNs978-952-60-8850-1
Electronic ISBNs978-952-60-8851-8
Publication statusPublished - 2019
MoE publication typeG5 Doctoral dissertation (article)


  • saddle point
  • minimum energy path
  • Gaussian process
  • machine learning

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