Exploiting symmetry for the job sequencing and tool switching problem
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Here, the pressure-induced polymorphism of binary octet compounds has long been considered a settled problem although the possible atomic disordering of some phases remains a puzzling observation. Taking GaP as a case study, we conclude, through x-ray microdiffraction and first-principles calculations, that its high-pressure metallic phase (previously reported as being disordered) adopts in fact an ordered base-centered monoclinic structure previously unknown in this class of compounds. The formation of layered patterns with variable degrees of interlayer dimerization, as observed in GaP, marks a paradigm shift of our understanding of ordering in octet high-pressure phases which calls for a more extensive re-examination. A rich polymorphism with fine tuning of chemical and physical properties can be envisioned.
To accurately compute data-based prediction of Hamiltonian systems, it is essential to utilize methods that preserve the structure of the equations over time. We consider a particularly challenging case of systems with interacting parts that do not reduce to pure momentum evolution. Such systems are essential in scientific computations, such as discretization of a continuum elastic rod, which can be viewed as the group of rotations and translations $SE(3)$. The evolution involves not only the momenta but also the relative positions and orientations of the particles. The presence of Lie group-valued elements, such as relative positions and orientations, poses a problem for applying previously derived methods for data-based computing. We develop a novel method of data-based computation and complete phase space learning of such systems. We follow the original framework of SympNets (Jin et al., 2020) and LPNets (Eldred et al., 2024), building the neural network from phase space mappings that preserve the Lie–Poisson structure. We derive a novel system of mappings that are built into neural networks describing the evolution of such systems. We call such networks Coupled Lie–Poisson Neural Networks, or CLPNets. We consider increasingly complex examples for the applications of CLPNets, starting with the rotation of two rigid bodies about a common axis, progressing to the free rotation of two rigid bodies, and finally to the evolution of two connected and interacting $SE(3)$ components, describing the discretization of an elastic rod into two elements. Our method preserves all Casimir invariants to machine precision, preserves energy to high accuracy, and shows good resistance to the curse of dimensionality, requiring only a few thousand data points for all cases studied (three to eighteen dimensions). Additionally, the method is highly economical in memory requirements, requiring only about 200 parameters for the most complex case considered.
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