Conrad W. Rosenbrock

Discovering the building blocks of atomic systems using machine learning: application to grain boundaries

Machine learning has proven to be a valuable tool to approximate functions in high-dimensional spaces. Unfortunately, analysis of these models to extract the relevant physics is never as easy as applying machine learning to a large data set in the first place. Here we present a description of atomic systems that generates machine learning representations with a direct path to physical interpretation. As an example, we demonstrate its usefulness as a universal descriptor of grain boundary systems. Grain boundaries in crystalline materials are a quintessential example of a complex, high-dimensional system with broad impact on many physical properties including strength, ductility, corrosion resistance, crack resistance, and conductivity. In addition to modeling such properties, the method also provides insight into the physical “building blocks” that influence them. This opens the way to discover the underlying physics behind behaviors by understanding which building blocks map to particular properties. Once the structures are understood, they can then be optimized for desirable behaviors.Machine learning: Modelling atomic systems to make property predictionsA method for representing atomic systems for machine learning is shown that can provide access to the physical properties of these systems. Machine learning is a powerful tool for finding correlations but when used to look at real-word systems, the complexity of the models often limits the amount of information that can be extracted about the underlying physics. An international team of researchers led by Conrad Rosenbrock from Brigham Young University now present a machine learning-based approach for modelling atomic systems that can provide insight into the physical building blocks that influence them. They demonstrate the power of their approach by examining the predictive performance of several machine learning models, providing connections between the structure and behaviour of grain boundaries in crystalline materials, which could be extended to other systems that involve local structural changes.

Numerical Algorithm for Pólya Enumeration Theorem

Although the Pólya enumeration theorem has been used extensively for decades, an optimized, purely numerical algorithm for calculating its coefficients is not readily available. We present such an algorithm for finding the number of unique colorings of a finite set under the action of a finite group.

Robustness of the cluster expansion: Assessing the roles of relaxation and numerical error

Cluster expansion (CE) is effective in modeling the stability of metallic alloys, but sometimes cluster expansions fail. Failures are often attributed to atomic relaxation in the DFT-calculated data, but there is no metric for quantifying the degree of relaxation. Additionally, numerical errors can also be responsible for slow CE convergence. We studied over one hundred different Hamiltonians and identified a heuristic, based on a normalized mean-squared displacement of atomic positions in a crystal, to determine if the effects of relaxation in CE data are too severe to build a reliable CE model. Using this heuristic, CE practitioners can determine a priori whether or not an alloy system can be reliably expanded in the cluster basis. We also examined the error distributions of the fitting data. We find no clear relationship between the type of error distribution and CE prediction ability, but there are clear correlations between CE formalism reliability, model complexity, and the number of significant terms in the model. Our results show that the \emph{size} of the errors is much more important than their distribution.