Identification of High-Reliability Regions of Machine Learning Predictions Based on Materials Chemistry.

Progress in the application of machine learning (ML) methods to materials design is hindered by the lack of understanding of the reliability of ML predictions, in particular, for the application of ML to small data sets often found in materials science. Using ML prediction for transparent conductor oxide formation energy and band gap, dilute solute diffusion, and perovskite formation energy, band gap, and lattice parameter as examples, we demonstrate that (1) construction of a convex hull in feature space that encloses accurately predicted systems can be used to identify regions in feature space for which ML predictions are highly reliable; (2) analysis of the systems enclosed by the convex hull can be used to extract physical understanding; and (3) materials that satisfy all well-known chemical and physical principles that make a material physically reasonable are likely to be similar and show strong relationships between the properties of interest and the standard features used in ML. We also show that similar to the composition-structure-property relationships, inclusion in the ML training data set of materials from classes with different chemical properties will not be beneficial for the accuracy of ML prediction and that reliable results likely will be obtained by ML model for narrow classes of similar materials even in the case where the ML model will show large errors on the data set consisting of several classes of materials.

Pair correlation function based on Voronoi topology.

The pair correlation function (PCF) has proven an effective tool for analyzing many physical systems due to its simplicity and its applicability for simulated and experimental data. However, as an averaged quantity, the PCF can fail to capture subtle structural differences in particle arrangements, even when those differences can have a major impact on system properties. Here, we use Voronoi topology to introduce a discrete version of the PCF that highlights local interparticle topological configurations. The advantages of the Voronoi PCF are demonstrated in several examples including crystalline, hyperuniform, and active systems showing clustering and giant number fluctuations.

Distribution of Topological Types in Grain-Growth Microstructures.

An open question in studying normal grain growth concerns the asymptotic state to which microstructures converge. In particular, the distribution of grain topologies is unknown. We introduce a thermodynamiclike theory to explain these distributions in two- and three-dimensional systems. In particular, a bendinglike energy E_{i} is associated to each grain topology t_{i}, and the probability of observing that particular topology is proportional to [1/s(t_{i})]e^{-ßE_{i}}, where s(t_{i}) is the order of an associated symmetry group and ß is a thermodynamiclike constant. We explain the physical origins of this approach and provide numerical evidence in support.

Topological extension of the isomorph theory based on the Shannon entropy.

Isomorph theory is one of the promising theories for understanding the quasiuniversal relationship between thermodynamic, dynamic, and structural characteristics. Based on the hidden scale invariance of the inverse power law potentials, it rationalizes the excess entropy scaling law of dynamic properties. This work aims to show that this basic idea of isomorph theory can be extended by examining the microstructural features of the system. Using the topological framework in conjunction with the entropy calculation algorithm, we demonstrate that Voronoi entropy, a measure of the topological diversity of single atoms, provides a scaling law for the transport properties of soft-sphere fluids, which is comparable to the frequently used excess entropy scaling. By examining the relationship between the Voronoi entropy and the solidlike fraction of simple fluids, we suggest that the Frenkel line, a rigid-nonrigid crossover line, be a topological isomorphic line at which the scaling relation qualitatively changes.

Topological generalization of the rigid-nonrigid transition in soft-sphere and hard-sphere fluids.

A fluid particle changes its dynamics from diffusive to oscillatory as the system density increases up to the melting density. Hence the notion of the Frenkel line was introduced to demarcate the fluid region into rigid and nonrigid liquid subregions based on the collective particle dynamics. In this work, we apply a topological framework to locate the Frenkel lines of the soft-sphere and the hard-sphere models relying on the system configurations. The topological characteristics of the ideal gas and the maximally random jammed state are first analyzed, then the classification scheme designed in our earlier work is applied to soft-sphere and hard-sphere fluids. The dependence of the classification result on the bulk density is understood based on the theory of fluid polyamorphism. The percolation behavior of solid-like clusters is described based on the fraction of solid-like molecules in an integrated manner. The crossover densities are obtained by examining the percolation of solid-like clusters. The resultant crossover densities of soft-sphere fluids converge to that of hard-sphere fluid. Hence the topological method successfully highlights the generality of the Frenkel line.

Topological Characterization of Rigid-Nonrigid Transition across the Frenkel Line.

The dynamics of supercritical fluids, a state of matter beyond the gas-liquid critical point, changes from diffusive to oscillatory motions at high pressure. This transition is believed to occur across a locus of thermodynamic states called the Frenkel line. The Frenkel line has been extensively investigated from the viewpoint of the dynamics, but its structural meaning is still not well-understood. This Letter interprets the mesoscopic picture of the Frenkel line entirely based on a topological and geometrical framework. This discovery makes it possible to understand the mechanism of rigid-nonrigid transition based not on the dynamics of individual atoms but on their instantaneous configurations. The topological classification method reveals that the percolation of solid-like structures occurs above the rigid-nonrigid crossover densities.

Topological framework for local structure analysis in condensed matter.

Physical systems are frequently modeled as sets of points in space, each representing the position of an atom, molecule, or mesoscale particle. As many properties of such systems depend on the underlying ordering of their constituent particles, understanding that structure is a primary objective of condensed matter research. Although perfect crystals are fully described by a set of translation and basis vectors, real-world materials are never perfect, as thermal vibrations and defects introduce significant deviation from ideal order. Meanwhile, liquids and glasses present yet more complexity. A complete understanding of structure thus remains a central, open problem. Here we propose a unified mathematical framework, based on the topology of the Voronoi cell of a particle, for classifying local structure in ordered and disordered systems that is powerful and practical. We explain the underlying reason why this topological description of local structure is better suited for structural analysis than continuous descriptions. We demonstrate the connection of this approach to the behavior of physical systems and explore how crystalline structure is compromised at elevated temperatures. We also illustrate potential applications to identifying defects in plastically deformed polycrystals at high temperatures, automating analysis of complex structures, and characterizing general disordered systems.

Geometric and topological properties of the canonical grain-growth microstructure.

Many physical systems can be modeled as large sets of domains "glued" together along boundaries-biological cells meet along cell membranes, soap bubbles meet along thin films, countries meet along geopolitical boundaries, and metallic crystals meet along grain interfaces. Each class of microstructures results from a complex interplay of initial conditions and particular evolutionary dynamics. The statistical steady-state microstructure resulting from isotropic grain growth of a polycrystalline material is canonical in that it is the simplest example of a cellular microstructure resulting from a gradient flow of an energy that is directly proportional to the total length or area of all cell boundaries. As many properties of polycrystalline materials depend on their underlying microstructure, a more complete understanding of the grain growth steady state can provide insight into the physics of a broad range of everyday materials. In this paper we report geometric and topological features of these canonical two- and three-dimensional steady-state microstructures obtained through extensive simulations of isotropic grain growth.

Statistical topology of three-dimensional Poisson-Voronoi cells and cell boundary networks.

Voronoi tessellations of Poisson point processes are widely used for modeling many types of physical and biological systems. In this paper, we analyze simulated Poisson-Voronoi structures containing a total of 250000000 cells to provide topological and geometrical statistics of this important class of networks. We also report correlations between some of these topological and geometrical measures. Using these results, we are able to corroborate several conjectures regarding the properties of three-dimensional Poisson-Voronoi networks and refute others. In many cases, we provide accurate fits to these data to aid further analysis. We also demonstrate that topological measures represent powerful tools for describing cellular networks and for distinguishing among different types of networks.

Complete topology of cells, grains, and bubbles in three-dimensional microstructures.

We introduce a general, efficient method to completely describe the topology of individual grains, bubbles, and cells in three-dimensional polycrystals, foams, and other multicellular microstructures. This approach is applied to a pair of three-dimensional microstructures that are often regarded as close analogues in the literature: one resulting from normal grain growth (mean curvature flow) and another resulting from a random Poisson-Voronoi tessellation of space. Grain growth strongly favors particular grain topologies, compared with the Poisson-Voronoi model. Moreover, the frequencies of highly symmetric grains are orders of magnitude higher in the grain growth microstructure than they are in the Poisson-Voronoi one. Grain topology statistics provide a strong, robust differentiator of different cellular microstructures and provide hints to the processes that drive different classes of microstructure evolution.