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Fast solution of wave propagation in periodic structures usually relies on simplified approaches, such as analytical methods, transmission line models, scattering matrix approaches, plane wave methods, etc. For complex multi-dimensional problems, computationally intensive direct numerical simulation (DNS) is always needed. This study demonstrates a fast and accurate simulation methodology enabled by a physics-based learning methodology, derived from proper orthogonal decomposition (POD) and Galerkin projection, for periodic quantum nanostructure and photonic crystals. POD is a projection-based method that generates optimal basis functions (or POD modes) via solution data collected from DNSs. This process trains the POD modes to adapt parametric variations of the system and offers the best least squares (LS) fit to the solution using the smallest number of modes. This is very different from other projection approaches, e.g., Fourier, Legendre, Bessel, Airy functions, etc., that adopt assumed basis functions selected for the problem based on the solution form. After generating the optimal POD modes, Galerkin projection of the wave equation onto each of the POD modes is performed to close the model and incorporate physical principles guided by the wave equation. Such a rigorous approach offers efficient simulations with high accuracy and exhibits the extrapolation ability in cases reasonably beyond the training bounds. The POD-Galerkin methodology is applied in this study to predict band structures and wave solutions for 2D periodic quantum-dot and photonic-lattice structures. The plane-wave approach is also included in a periodic quantum-dot structure to illustrate the superior performance of the POD-Galerkin methodology. The POD-Galerkin approach offers a 2-order computing speedup for both nanostructure and optical superlattices, compared to DNS, when solving both the wave solution and band structure. If the band structure is the only concern, a 4-order improvement in computational efficiency can be achieved. Fig. 1(a) shows the optical superlattice in a demonstration, where a unit cell includes 22 discs with diagonally symmetrical refractive indices and the background index n = 1. The POD modes for this case are trained by TE mode electric field data collected from DNSs with variation of diagonally symmetrical refractive indices. The LS error of the predicted electric field wave solution from the POD-Galerkin approach, shown in Fig. 1(b) compared to DNS, is below 1% with just 8 POD modes that offer a more than 4-order reduction in the degrees of freedom, compared to DNS. In addition, an extremely accurate prediction of band structure is illustrated in Fig. 1(c) with a maximum error below 0.1% in the entire Brillouin zone. 

A rigorous physics-informed learning methodology is proposed for predictions of wave solutions and band structures in electronic and optical superlattice structures. The methodology is enabled by proper orthogonal decomposition (POD) and Galerkin projection of the wave equation. The approach solves the wave eigenvalue problem in POD space constituted by a finite set of basis functions (or POD modes). The POD ensures that the generated modes are optimized and tailored to the parametric variations of the system. Galerkin projection however enforces physical principles in the methodology to further enhance the accuracy and efficiency of the developed model. It has been demonstrated that the POD-Galerkin methodology offers an approach with a reduction in degrees of freedom by 4 orders of magnitude, compared to direct numerical simulation (DNS). A computing speedup near 15,000 times over DNS can be achieved with high accuracy for either of the superlattice structures if only the band structure is calculated without the wave solution. If both wave function solution and band structure are needed, a 2-order reduction in computational time can be achieved with a relative least square error (LSE) near 1%. When the training is incomplete or the desired eigenstates are slightly beyond the training bounds, an accurate prediction with an LSE near 1%-2% still can be reached if more POD modes are included. This reveals its remarkable learning ability to reach correct solutions with the guidance of physical principles provided by Galerkin projection. 

An ensemble data-learning approach based on proper orthogonal decomposition (POD) and Galerkin projection (EnPOD-GP) is proposed for thermal simulations of multi-core CPUs to improve training efficiency and the model accuracy for a previously developed global POD-GP method (GPOD-GP). GPOD-GP generates one set of basis functions (or POD modes) to account for thermal behavior in response to variations in dynamic power maps (PMs) in the entire chip, which is computationally intensive to cover possible variations of all power sources. EnPOD-GP however acquires multiple sets of POD modes to significantly improve training efficiency and effectiveness, and its simulation accuracy is independent of any dynamic PM. Compared to finite element simulation, both GPOD-GP and EnPOD-GP offer a computational speedup over 3 orders of magnitude. For a processor with a small number of cores, GPOD-GP provides a more efficient approach. When high accuracy is desired and/or a processor with more cores is involved, EnPOD-GP is more preferable in terms of training effort and simulation accuracy and efficiency. Additionally, the error resulting from EnPOD-GP can be precisely predicted for any random spatiotemporal power excitation. 

Abstract In this paper, we consider an application of lot-streaming for processing a lot of multiple items in a hybrid flow shop (HFS) for the objective of minimizing makespan. The HFS that we consider consists of two stages with a single machine available for processing in Stage 1 andmidentical parallel machines in Stage 2. We call this problem a 1 + mTSHFS-LSP (two-stage hybrid flow shop, lot streaming problem), and show it to be NP-hard in general, except for the case when the sublot sizes are treated to be continuous. The novelty of our work is in obtaining closed-form expressions for optimal continuous sublot sizes that can be solved in polynomial time, for a given number of sublots. A fast linear search algorithm is also developed for determining the optimal number of sublots for the case of continuous sublot sizes. For the case when the sublot sizes are discrete, we propose a branch-and-bound-based heuristic to determine both the number of sublots and sublot sizes and demonstrate its efficacy by comparing its performance against that of a direct solution of a mixed-integer formulation of the problem by CPLEX^®. 

The classical proper orthogonal decomposition (POD) with the Galerkin projection (GP) has been revised for chip-level thermal simulation of microprocessors with a large number of cores. An ensemble POD-GP methodology (EnPODGP) is introduced to significantly improve the training effectiveness and prediction accuracy by dividing a large number of heat sources into heat source blocks (HSBs) each of which may contains one or a very small number of heat sources. Although very accurate, efficient and robust to any power map, EnPOD-GP suffers from intensive training for microprocessors with an enormous number of cores. A local-domain EnPOD-GP model (LEnPOD-GP) is thus proposed to further minimize the training burden. LEnPOD-GP utilizes the concepts of local domain truncation and generic building blocks to reduce the massive training data. LEnPOD-GP has been demonstrated on thermal simulation of NVIDIA Tesla Volta™ GV100, a GPU with more than 13,000 cores including FP32, FP64, INT32, and Tensor Cores. Due to the domain truncation for LEnPOD-GP, the least square error (LSE) is degraded but is still as small as 1.6% over the entire space and below 1.4% in the device layer when using 4 modes per HSB. When only the maximum temperature of the entire GPU is of interest, LEnPOD-GP offers a computing speed 1.1 million times faster than the FEM with a maximum error near 1.2oC over the entire simulation time. 

Diffusion of native defects such as vacancies and their interactions with impurities are fundamental to semiconductor crystal growth, device processing, and design. However, the transient equilibration of native defects is difficult to directly measure. We used (AlxGa1−x)2O3/Ga2O3 superlattices (SLs) to detect and analyze transient diffusion of cation vacancies during annealing in O2 at 1000–1100 °C. Using a novel finite difference scheme for diffusion with time- and space-varying diffusion constants, we determined diffusion constants for Al, Fe, and cation vacancies, including the vacancy concentration dependence for Al. In the case of SLs grown on Sn-doped β-Ga2O3 (010) substrates, gradients observed in the extent of Al diffusion indicate a supersaturation of vacancies in the substrates that transiently diffuse through the SLs coupled strongly to Sn and thus slowed compared to undoped cases. In the case of SLs grown on (010) Fe-doped substrates, the Al diffusion is uniform through the SLs, indicating a depth-uniform concentration of vacancies. We find no evidence for the introduction of VGa from the free surface at rates sufficient to affect Al diffusion at at. % concentrations, establishing an upper bound on surface injection. In addition, we show that unintentional impurities in Sn-doped Ga2O3 such as Fe, Ni, Mn, Cu, and Li also diffuse toward the surface and accumulate. Many of these likely have fast interstitial diffusion modes capable of destabilizing devices, thus suggesting that impurities may require further reduction. This work provides a method to measure transients in diffusion-mediating native defects otherwise hidden in common processes such as ion implantation, etching, and film growth. 

The need for full-chip dynamic thermal simulation for effective runtime thermal management of multicore processors has been growing in recent years due to the rising demand for high-performance computing. In addition to simulation efficiency and accuracy, a high resolution is desirable in order to accurately predict crucial hot spots in the chip. This work investigates a simulation technique derived from proper orthogonal decomposition (POD) for full-chip dynamic thermal simulation of a multicore processor. The POD projects a heat transfer problem onto a mathematical space constituted by a finite set of basis functions (or POD modes) that are generated (or trained) by thermal solution data collected from direct numerical simulation (DNS). Accuracy and efficiency of the POD simulation technique influenced by the quality of thermal data are examined thoroughly, especially in the areas with high thermal gradients. The results show that if the POD modes are trained by good-quality data, the POD simulation offers an accurate prediction of the dynamic thermal distribution in the multicore processor with an extremely small degree of freedom (DoF). A reduction in computational time over four orders of magnitude, compared to the DNS, can be achieved for full-chip dynamic thermal simulation with a resolution as fine as the DNS. The study has also demonstrated that the POD approach can be used to rigorously verify the accuracy of solutions offered by DNS tools. A practical approach is proposed to further enhance the accuracy and efficiency of the proposed full-chip thermal simulation technique.