Meta-Heuristic Optimization Methods for Quaternion-Valued Neural Networks

In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing that hypercomplex neural networks reduce the number of total trainable parameters compared to their real-valued equivalents. Finally, this work introduces a novel training algorithm using a meta-heuristic approach that bypasses the need for analytic quaternion loss or activation functions. This algorithm allows for a broader range of activation functions over current quaternion networks and presents a proof-of-concept for future work

AFLOW-SYM: Platform for the complete, automatic and self-consistent symmetry analysis of crystals

Determination of the symmetry profile of structures is a persistent challenge in materials science. Results often vary amongst standard packages, hindering autonomous materials development by requiring continuous user attention and educated guesses. Here, we present a robust procedure for evaluating the complete suite of symmetry properties, featuring various representations for the point-, factor-, space groups, site symmetries, and Wyckoff positions. The protocol determines a system-specific mapping tolerance that yields symmetry operations entirely commensurate with fundamental crystallographic principles. The self consistent tolerance characterizes the effective spatial resolution of the reported atomic positions. The approach is compared with the most used programs and is successfully validated against the space group information provided for over 54,000 entries in the Inorganic Crystal Structure Database. Subsequently, a complete symmetry analysis is applied to all 1.7$+$ million entries of the AFLOW data repository. The AFLOW-SYM package has been implemented in, and made available for, public use through the automated, $\textit{ab-initio}$ framework AFLOW.Comment: 24 pages, 6 figure

Meta-Heuristic Optimization Methods for Quaternion-Valued Neural Networks

In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing that hypercomplex neural networks reduce the number of total trainable parameters compared to their real-valued equivalents. Finally, this work introduces a novel training algorithm using a meta-heuristic approach that bypasses the need for analytic quaternion loss or activation functions. This algorithm allows for a broader range of activation functions over current quaternion networks and presents a proof-of-concept for future work

Fitting the Light Curve of 1I/`Oumuamua with a Nonprincipal Axis Rotational Model and Outgassing Torques

In this paper, we investigate the nonprincipal axis (NPA) rotational state of 1I/`Oumuamua -- the first interstellar object discovered traversing the inner Solar System -- from its photometric light curve. Building upon Mashchenko (2019), we develop a model which incorporates NPA rotation and {Sun-induced, time-varying} outgassing torques to generate synthetic light curves of the object. The model neglects tidal forces, which are negligible compared to outgassing torques over the distances that `Oumuamua was observed. We implement an optimization scheme that incorporates the NPA rotation model to calculate the initial rotation state of the object. We find that an NPA rotation state with an average period of $\langle P \rangle\simeq7.34$ hr best reproduces the photometric data. The discrepancy between this period and previous estimates is due to continuous period modulation induced by outgassing torques in the rotational model, {as well as different periods being used}. The best fit to the October 2017 data does not reproduce the November 2017 data (although the later measurements are too sparse to fit). The light curve is consistent with no secular evolution of the angular momentum, somewhat in tension with the empirical correlations between nuclear spin-up and cometary outgassing. The complex rotation of `Oumuamua may be {the result of primordial rotation about the smallest principal axis} if (i) the object experienced hypervolatile outgassing and (ii) our idealized outgassing model is accurate.Comment: 22 pages, 8 figures, 1 animation. Accepted to the Planetary Science Journal. The animation can be found on YouTube (https://youtu.be/f5YEAMTvIeo) and in the online publication by PSJ (when available

Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds

For data given on a nonlinear space, like angles, symmetric positive matrices, the sphere, or the hyperbolic space, there is often enough structure to form a Riemannian manifold. We present the Julia package Manifolds.jl, providing a fast and easy to use library of Riemannian manifolds and Lie groups. We introduce a common interface, available in ManifoldsBase.jl, with which new manifolds, applications, and algorithms can be implemented. We demonstrate the utility of Manifolds.jl using B\'ezier splines, an optimization task on manifolds, and a principal component analysis on nonlinear data. In a benchmark, Manifolds.jl outperforms existing packages in Matlab or Python by several orders of magnitude and is about twice as fast as a comparable package implemented in C++

AFLOW-SYM: platform for the complete, automatic and self-consistent symmetry analysis of crystals

Determination of the symmetry profile of structures is a persistent challenge in materials science. Results often vary amongst standard packages, hindering autonomous materials development by requiring continuous user attention and educated guesses. This article presents a robust procedure for evaluating the complete suite of symmetry properties, featuring various representations for the point, factor and space groups, site symmetries and Wyckoff positions. The protocol determines a system-specific mapping tolerance that yields symmetry operations entirely commensurate with fundamental crystallographic principles. The self-consistent tolerance characterizes the effective spatial resolution of the reported atomic positions. The approach is compared with the most used programs and is successfully validated against the space-group information provided for over 54 000 entries in the Inorganic Crystal Structure Database (ICSD). Subsequently, a complete symmetry analysis is applied to all 1.7+ million entries of the AFLOW data repository. The AFLOW-SYM package has been implemented in, and made available for, public use through the automated ab initio framework AFLOW

Applied Mathematics and Computational Physics

As faster and more efficient numerical algorithms become available, the understanding of the physics and the mathematical foundation behind these new methods will play an increasingly important role. This Special Issue provides a platform for researchers from both academia and industry to present their novel computational methods that have engineering and physics applications