**Olli-Pekka Koistinen gives a lecture about his doctoral thesis titled: Algorithms for finding saddle points and minimum energy paths using Gaussian process regression.**

**The lecture takes place on 31th of January 2020 in VR-II, room 157 at 12:30**

The defense was delivered at Aalto University on January 9th 2020. This is a joint degree with the Aalto University, Finland.

**Opponents were:**

Dr. Thomas Bligaard, Professor of Physics, Technical University of Denmark, DTU.

**Advisor: **Dr. Hannes Jónsson, Professor at the Faculty of Physical Sciences, University of Iceland.

**Doctoral committee:**

Dr. Aki Vehtari, Associate professor at the Department of Computer Science, Aalto University, Finland.

Dr. Egill Skúlason, Professor at the Faculty of Industrial Engineering, Mechanical Engineering and Computer Science, University of Iceland.

**Chair of Ceremony:** Dr. Oddur Ingólfsson, Professor and the Head of the Faculty of Physical Sciences, University of Iceland.

**Abstract:**

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 accurate saddle point is 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 two closely spaced points towards the orientation of 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 the 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. With 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 correct the model for following iterations.

In this dissertation, machine learning algorithms based on Gaussian process regression are developed to enhance search 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 the gradient components 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.

**About the doctoral candidate:**

The candidate, Olli-Pekka Koistinen, has carried out all his university studies at Aalto University in Espoo, Finland. He completed a B.S. degree in Bioinformation Technology in 2011, and a M.S. degree in the same discipline in 2012. After that he enrolled in the Ph.D. program in Computational Science and is now completing this study as a joint study in Chemistry with the University of Iceland.