# Efficient global structure optimization with a machine-learned surrogate model (GOFEE)¶

Whenever global optimization of atomic structure is done with first-principles total energy methods, most of the computational effort is spent on evaluating the total energie. We propropose a scheme to actively learn a surrogate energy landscape while searhing for a global minimum. By having several length scales in the surrogate model, we arrive at a method with high accuracy for predictions in both highly sampled regions and in regions of configuration that has not yet been visited. Using Bayesian statistics the most likely next configuration to investigate with the expensive first-principles method is sampled from a distribution of random and population-derived candidate structures.

# Atomistic structure learning algorithm (ASLA)¶

Using image recognition and reinforcement learning, we let a neural network interact with a quantum mechanics solver (a density functional theory program). The network eventually trains itself into knowing the basic rules of organic and inorganic chemistry for planar compounds. After a few days of training on a few CPU cores, the network has taught itself to build a large number of sensible molecular and crystal structures.

# Proposing a new descriptor (k-CON)¶

Almost parameter free descriptor for neural network establishment of surrogate energy landscape (k-CON): Machine learning is often used to setup surrogate energy landscapes for molecules, clusters and solid surfaces. These are cheaply calculatable energy landscapes that attempt to mimic fully the true DFT energy landscapes. However, despite the name machine learning, it is often necessary for humans to design a proper descriptor for the entire structure or for each atom to compute the surrogate energy landscapes. It is unsatisfactory that humans have to prepare in depth part of the machine learning model with a lot of man made decisions on parameters. Hence we investigated if one could use the almost singular choice descriptor: all atom identities, all (reciprocal) distances for di-atoms, all sets of three (reciprocal) distances for tri-atoms, etc. as the descriptor. Used in conjunction with a neural network, it was shown that this is indeed possible and that the overall speed of an evolutionary search could be increased.

# Getting atomic energies with very small datasets (Autobag)¶

Bundling of atomic energies as a means to divide the formation energy of a structure among its constituent atoms (auto-bag): With conventional methods (kernel ridge regression and neural networks) the extraction of local energies requires large amounts of data. As data is an expensive resource in connection with DFT calculations this subproject aimed at establishing bundled local energies so that atoms with similar local environments would have to share the same local energy. This method proves very robust in the limit of few available data points. It is anticipated to form the basis for improvement of other methods being developed in the VILLUM Investigator project.

# Obtaining low energy structures by enforcing clustering in feature space¶

Artificial energy landscape (i.e. not a surrogate energy landscape) for use in cross-over and mutation operations in evolutionary algorithm (cluster-regularization, role model search): Once structures are described by other means than Cartesian coordinates, new patterns in the atomic descriptors (features) reveal themselves. In this subproject, we discovered that if, for some crystal surface structure, the global minimum energy structure is known, it shows distinct clusters of local atomic environments. Now, for non-optimal structures, the sum of distances in feature space of all atoms to their nearest such cluster center correlates with the energy of the structure. This was flipped around and used in this way: clustering was imposed in feature space to non-optimal structures, hoping this would lead to the global optimization of energy versus structure.

# On-the-fly training of a neural network as a surrogate energy landscape (LEA)¶

On-the-fly neural network establishment of surrogate energy landscape during evolutionary search (learning evolutionary algorithm = LEA): This project used an established neural network technology (Amp) to produce a surrogate energy landscape while an evolutionary structure search was carried out. Whenever the neural network and a single-point (=unrelaxed and hence cheap) DFT calculation disagree significantly on the stability of a structure, the neural network is re-trained. The project was carried out in collaboration with the author of the Amp method, Andrew Peterson, Brown University, Rhode Island, US. The speedup obtained allowed for more thorough searches and new exciting structures – never thought of by man – were discovered, including a hollow pyramidal Pt13 cluster on a magnesium oxide support.

# On-the-fly fitting of the repulsive term in DFTB (TANGO)¶

Performing global optimization with a full density functional theory (DFT) expression is computationally very costly as many relaxation steps must in general be done. Using e.g. an evolutionary approach, new structures originating from cross-over and mutation operations may require 50-100 such steps each implying many energy and force evaluations. In this work, we use the faster but more approximate density functional tight-binding method (DFTB) for the relaxation steps. In order to obtain near-DFT level of accuracy, we optimize some of the DFTB parameters on-the-fly by comparing to selected single-point DFT calculations.

# Balancing exploitation versus exploration¶

Parent selection in evolutionary algorithm: In this subproject, Bayesian optimization techniques are used to estimate the uncertainty by which the stability of a structure can be inferred from all other known structure/stability pairs. Parents are subsequently selected with a view at both stability (exploitation) and uncertainty (exploration). A paper was published with a volcano-plot showing a maximum in speed for an evolutionary search as a function of a single parameter.

# On-the-fly training of GPR model to predict potential energy of atoms¶

On-the-fly kernel ridge regression estimation of local potential energy during evolutionary search: When quantum mechanical calculations are performed (e.g. DFT calculations), a total energy is obtained for the entire structure and no information is available (or even well-defined) on how the individual atoms contribute to this energy. In this subproject, we investigated if a type of sensitivity analysis could reveal a clue about the potential of each atom in the final structure. Thus a kernel ridge regression model was used to estimate the energy cost of removing just one atom from a structure. The method works when applied to a small set of structures, where sometimes some atoms are detached from the others. This is the situation for evolutionary searches where the method proved successful. Was published in Phys. Rev. Lett. and is attracting quite a few citations.

# Clustering to optimize parent selection in evolutionary algorithm¶

We introduce the unsupervised machine learning technique, clustering, to an evolutionary algorithm and thereby speed it up. The clustering technique allows us to label potential parent candidates in the population. It turns out that by more often using outlier candidates, i.e. the odd type parents, the evolutionary search for an optimal structure speeds up significantly.