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At least 127 records · Page 7

SwitchX : Gmin-Gmax Switching for Energy-efficient and Robust Implementation of Binarized Neural Networks on ReRAM Xbars

Memristive crossbars can efficiently implement Binarized Neural Networks (BNNs) wherein the weights are stored in high-resistance states (HRS) and low-resistance states (LRS) of the synapses. We propose SwitchX mapping of BNN weights onto ReRAM crossbars such that the impact of crossbar non-idealities, that lead to degradation in computational accuracy, are minimized. Essentially, SwitchX maps the binary weights in such a manner that a crossbar instance comprises of more HRS than LRS synapses. We find BNNs mapped onto crossbars with SwitchX to exhibit better robustness against adversarial attacks than the standard crossbar mapped BNNs, the baseline. Finally, we combine SwitchX with state-aware training (that further increases the feasibility of HRS states during weight mapping) to boost the robustness of a BNN on hardware. We find that this approach yields stronger defense against adversarial attacks than adversarial training, a state-of the-art software defense. We perform experiments on a VGG16 BNN with benchmark datasets (CIFAR-10, CIFAR-100 and TinyImagenet) and use Fast Gradient Sign Method (ϵ = 0.05 to 0.3) and Projected Gradient Descent (ϵ = $\frac{2}{255}$ to $\frac{32}{255}$, α = $\frac{2}{255}$) adversarial attacks. We show that SwitchX combined with state-aware training can yield upto ~35% improvements in clean accuracy and ~6–16% in adversarial accuracies against conventional BNNs. Furthermore, an important by-product of SwitchX mapping is increased crossbar power savings, owing to an increased proportion of HRS synapses, which is furthered with state-aware training. We obtain upto ~21–22% savings in crossbar power consumption for state-aware trained BNN mapped via SwitchX on 16 × 16 and 32 × 32 crossbars using the CIFAR-10 and CIFAR-100 datasets.

97 MATHEMATICS AND COMPUTING↗

El-CID: a filter for gravitational-wave electromagnetic counterpart identification

ABSTRACT As gravitational-wave (GW) interferometers become more sensitive and probe ever more distant reaches, the number of detected binary neutron star mergers will increase. However, detecting more events farther away with GWs does not guarantee corresponding increase in the number of electromagnetic counterparts of these events. Current and upcoming wide-field surveys that participate in GW follow-up operations will have to contend with distinguishing the kilonova (KN) from the ever increasing number of transients they detect, many of which will be consistent with the GW sky-localization. We have developed a novel tool based on a temporal convolutional neural network architecture, trained on sparse early-time photometry and contextual information for Electromagnetic Counterpart Identification (El-CID). The overarching goal for El-CID is to slice through list of new transient candidates that are consistent with the GW sky localization, and determine which sources are consistent with KNe, allowing limited target-of-opportunity resources to be used judiciously. In addition to verifying the performance of our algorithm on an extensive testing sample, we validate it on AT2017gfo – the only EM counterpart of a binary neutron star merger discovered to date – and AT2019npv – a supernova that was initially suspected as a counterpart of the GW event, GW190814, but was later ruled out after further analysis.

79 ASTRONOMY AND ASTROPHYSICS↗

Physics-inspired spatiotemporal-graph AI ensemble for the detection of higher order wave mode signals of spinning binary black hole mergers

We present a new class of AI models for the detection of quasi-circular, spinning, non-precessing binary black hole mergers whose waveforms include the higher order gravitational wave modes ($\ell$, |m|) = {(2,2), (2,1), (3,3), (3,2), (4,4)}, and mode mixing effects in the $\ell$ = 3, |m| = 2 harmonics. These AI models combine hybrid dilated convolution neural networks to accurately model both short- and long-range temporal sequential information of gravitational waves; and graph neural networks to capture spatial correlations among gravitational wave observatories to consistently describe and identify the presence of a signal in a three detector network encompassing the Advanced LIGO and Virgo detectors. We first trained these spatiotemporal-graph AI models using synthetic noise, using 1.2 million modeled waveforms to densely sample this signal manifold, within 1.7 h using 256 NVIDIA A100 GPUs in the Polaris supercomputer at the Argonne Leadership Computing Facility. This distributed training approach exhibited optimal classification performance, and strong scaling up to 512 NVIDIA A100 GPUs. With these AI ensembles we processed data from a three detector network, and found that an ensemble of 4 AI models achieves state-of-the-art performance for signal detection, and reports two misclassifications for every decade of searched data. We distributed AI inference over 128 GPUs in the Polaris supercomputer and 128 nodes in the Theta supercomputer, and completed the processing of a decade of gravitational wave data from a three detector network within 3.5 h. Finally, we fine-tuned these AI ensembles to process the entire month of February 2020, which is part of the O3b LIGO/Virgo observation run, and found 6 gravitational waves, concurrently identified in Advanced LIGO and Advanced Virgo data, and zero false positives. This analysis was completed in one hour using one NVIDIA A100 GPU.

79 ASTRONOMY AND ASTROPHYSICS↗

Fast decay of classification error in variational quantum circuits

Variational quantum circuits (VQCs) have shown great potential in near-term applications. However, the discriminative power of a VQC, in connection to its circuit architecture and depth, is not understood. To unleash the genuine discriminative power of a VQC, we propose a VQC system with the optimal classical post-processing—maximum-likelihood estimation on measuring all VQC output qubits. Via extensive numerical simulations, we find that the error of VQC quantum data classification typically decays exponentially with the circuit depth, when the VQC architecture is extensive—the number of gates does not shrink with the circuit depth. This fast error suppression ends at the saturation towards the ultimate Helstrom limit of quantum state discrimination. On the other hand, non-extensive VQCs such as quantum convolutional neural networks are sub-optimal and fail to achieve the Helstrom limit, demonstrating a trade-off between ansatz complexity and classification performance in general. To achieve the best performance for a given VQC, the optimal classical post-processing is crucial even for a binary classification problem. To simplify VQCs for near-term implementations, we find that utilizing the symmetry of the input properly can improve the performance, while oversimplification can lead to degradation.

Zhang, Bingzhi↗

FY24 Progress Report: SRNL Analysis of ICCWR LCM and WAMS data for Corrosion and Cracking

Algorithms for Machine Learning (ML) and data analysis for the 3013 Surveillance Program have been developed in an ongoing collaborative effort by the Savannah River National Laboratory (SRNL) and the University of South Carolina (USC). The objective of the algorithms is to automate the identification of corrosion and crack formation in the Inner Container Closure Weld Region (ICCWR) of the canister system used to store Pu-bearing material. Data for corrosion and cracking is collected from large binary files generated by a Laser Confocal Microscope (LCM), the Wide Area 3D Measurement System (WAMS), or,in a recent proposal, by a Scanning Electron Microscope (SEM). The ML software uses the physical attributes in the data files (e.g., one or more of: height, color, and 16-bit grayscale values as functions of position in a plane projection) to detect signs of surface corrosion and cracking after being trained on similar data, with the features to be detected. Although the initial scope included screening for broader indicators of corrosion, e.g., pitting, identification of potential cracks was prioritized for the past several years at the request of program leadership. Labeled training data is essential to developing the ML algorithm, and enhancements to data labeling capability have been developed to address this essential precursor to application of ML routines. Efficient labeling is particularly important in view of the large volume of data required to train ML algorithms and the relative rarity of cracks in the ICCWR data set. The updated program will read binary data from either LCM, WAMS or SEM files, interrogate data attributes, facilitate user labeling of data for training ML algorithms, execute ML algorithms, output parameters from trained ML algorithms, report ML model accuracy with respect to labeled data, and generate graphical representations for various analyses. In FY24, hourglass neural networks (HNNs) that were initiated in FY22 were further developed and tested using available LCM data, and their performance was tested against that of the alternative U-Net Neural Network algorithm structure. HNNs along with previously developed Convolutional Neural Networks (CNNs) and Deep Neural Networks (DNNs) comprise a suite of ML tools for identification of cracks in the ICCWR

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Feedforward, high density, programmable read only neural network based memory system

Neural network-inspired, nonvolatile, programmable associative memory using thin-film technology is demonstrated. The details of the architecture, which uses programmable resistive connection matrices in synaptic arrays and current summing and thresholding amplifiers as neurons, are described. Several synapse configurations for a high-density array of a binary connection matrix are also described. Test circuits are evaluated for operational feasibility and to demonstrate the speed of the read operation. The results are discussed to highlight the potential for a read data rate exceeding 10 megabits/sec.

Daud, Taher↗

Machine learning approaches to streamline and enhance the analysis of multiscale imaging data for bioaerosol and soil particles

Bioaerosol and soil particles are ubiquitous in the environment. They are multicomponent and complex in nature displaying mixed inorganic and organic components. The way components are mixed in a bioaerosol sample is referred to as its mixing state. Soil particles are also a mixture of inorganic (mineral) and organic (soil organic matter) components. Bioaerosol particles contribute to a major fraction of coarse mode atmospheric particles, especially in the tropical areas, contributing up to 80 % of the particle mass concentration. The mixing state of particles is crucial to evaluate because it impacts several important environmental processes such as warm and cold cloud formation and radiation budget. Mixing states in aerosols are accompanied by chemical reactions across solid-liquid-gas interfaces. In this study, we utilized elemental compositions and microcopy images of thousands of atmospheric particles acquired by computer-controlled scanning electron microscope equipped with an energy-dispersive x-ray spectrometer to compute the mixing state of atmospheric particles. A 2D convolutional neural network (CNN), also known as convnet, was used to model the relationship between low resolution imaging data and higher resolution spectroscopy data, with the former as training input and the latter as target output. Two types of CNNs were implemented and tested; a basic CNN and an Inception-v3 network. For binary classification, the basic CNN achieved an accuracy of 84.29 % across all atom types, and the Inception-v3-like network achieved an accuracy 85.51 %. This study demonstrates the applicability of deep learning to handle large amounts of imaging/chemical spectroscopy data efficiently and evaluate particle mixing state from a range of environmental samples.

54 ENVIRONMENTAL SCIENCES↗

Unraveling Adsorbate-Induced Structural Evolution of Iron Carbide Nanoparticles

Iron carbide (Fe x C y ) nanoparticles (NPs) are promising candidates for replacing platinum group metals in industrial applications, such as high-temperature Fischer–Tropsch synthesis. However, due to their amorphous nature, characterization of the active sites has been challenging experimentally and computationally. Here, using a combined density functional theory (DFT), neural network interatomic potential-assisted global optimization, and ensemble learning study, we evaluate dynamic surface changes associated with syngas (H and CO) interactions. For this purpose, we have developed a general procedure that we use to model an experimentally relevant 270-atom Fe 182 C 88 NP using the neural network-assisted stochastic surface walk global optimization algorithm (SSW-NN). Once generated, the Fe 182 C 88 NP active sites and particle morphology are thoroughly characterized before the effects of syngas adsorbate interactions are explored by using DFT and molecular dynamics simulations. Lastly, we explore correlations between geometric and electronic features of the active sites and the adsorption of H (H ads ), using a regularized random forest machine learning algorithm. In doing so, we identified the Fe–C coordination number and p orbital occupancy as the most important descriptors affecting H ads . Furthermore, using a combined ML and quantum chemistry approach, our work demonstrates a general and efficient procedure for generating and probing complex surface phenomena on binary nanoparticles.

Adsorption↗

Confronting Domain Shift in Trained Neural Networks

Neural networks (NNs) are known as universal function approximators and can interpolate nonlinear functions between observed data points. However, when the target domain for deployment shifts from the training domain and NNs must extrapolate, the results are notoriously poor. Prior work Martinez et al. (2019) has shown that NN uncertainty estimates can be used to correct binary predictions in shifted domains without retraining the model. We hypothesize that this approach can be extended to correct real-valued time series predictions. As an exemplar, we consider two mechanical systems with nonlinear dynamics. The first system consists of a spring-mass system where the stiffness changes abruptly, and the second is a real experimental system with a frictional joint that is an open challenge for structural dynamicists to model efficiently. Our experiments will test whether 1) NN uncertainty estimates can identify when the input domain has shifted from the training domain and 2) whether the information used to calculate uncertainty estimates can be used to correct the NN’s time series predictions. While the method as proposed did not significantly improve predictions, our results did show potential for modifications that could improve models’ predictions and play a role in structural health monitoring systems that directly impact public safety.

97 MATHEMATICS AND COMPUTING↗

Synergistic learning with multi-task DeepONet for efficient PDE problem solving

Multi-task learning (MTL) is an inductive transfer mechanism designed to leverage useful information from multiple tasks to improve generalization performance compared to single-task learning. It has been extensively explored in traditional machine learning to address issues such as data sparsity and overfitting in neural networks. In this work, we apply MTL to problems in science and engineering governed by partial differential equations (PDEs). However, implementing MTL in this context is complex, as it requires task-specific modifications to accommodate various scenarios representing different physical processes. To this end, we present a multi-task deep operator network (MT-DeepONet) to learn solutions across various functional forms of source terms in a PDE and multiple geometries in a single concurrent training session. We introduce modifications in the branch network of the vanilla DeepONet to account for various functional forms of a parameterized coefficient in a PDE. Additionally, we handle parameterized geometries by introducing a binary mask in the branch network and incorporating it into the loss term to improve convergence and generalization to new geometry tasks. Our approach is demonstrated on three benchmark problems: (1) learning different functional forms of the source term in the Fisher equation; (2) learning multiple geometries in a 2D Darcy Flow problem and showcasing better transfer learning capabilities to new geometries; and (3) learning 3D parameterized geometries for a heat transfer problem and demonstrate the ability to predict on new but similar geometries. Finally, our MT-DeepONet framework offers a novel approach to solving PDE problems in engineering and science under a unified umbrella based on synergistic learning that reduces the overall training cost for neural operators.

42 ENGINEERING↗

Leveraging Artificial Intelligence to Predict Novel Eutectic Alloys

The goal of this project was to train an artificial neural network (ANN) to predict the fractional composition and melting point of eutectic alloys using fundamental atomic properties as inputs. The fundamental properties considered include atomic number, atomic weight, atomic radius, valence electron concentration, electronegativity, and electron affinity. The project involved several phases, starting with data preparation, where phase diagram data was harvested from the ASM International database. Approximately 1300 binary eutectics were collected and cleaned to ensure relevance and accuracy. A regression model was selected for training, utilizing a rectified linear unit as the activation function. Various model configurations were evaluated for predictive accuracy, with validation techniques employed to ensure robustness. The model demonstrated predictive capabilities above random guessing and was able to achieve up to 11% accuracy under certain conditions. An ablative test identified atomic radius and valence electron concentration as critical inputs for model performance. Incorporating the melting point of atomic constituents improved accuracy significantly, although ultimately the model’s predictive capability still fell short of the 80% target. This report details the methodology, results, and implications of the research, contributing to the understanding of employing artificial intelligence to predict the phase transition behavior of eutectic alloys.

36 MATERIALS SCIENCE↗

Learning and tuning fuzzy logic controllers through reinforcements

This paper presents a new method for learning and tuning a fuzzy logic controller based on reinforcements from a dynamic system. In particular, our generalized approximate reasoning-based intelligent control (GARIC) architecture (1) learns and tunes a fuzzy logic controller even when only weak reinforcement, such as a binary failure signal, is available; (2) introduces a new conjunction operator in computing the rule strengths of fuzzy control rules; (3) introduces a new localized mean of maximum (LMOM) method in combining the conclusions of several firing control rules; and (4) learns to produce real-valued control actions. Learning is achieved by integrating fuzzy inference into a feedforward neural network, which can then adaptively improve performance by using gradient descent methods. We extend the AHC algorithm of Barto et al. (1983) to include the prior control knowledge of human operators. The GARIC architecture is applied to a cart-pole balancing system and demonstrates significant improvements in terms of the speed of learning and robustness to changes in the dynamic system's parameters over previous schemes for cart-pole balancing.

Berenji, Hamid R.↗

Data supporting manuscript from L. Sheneman, G. Stephanopoulos, A.E. Vasdekis titled "Deep learning classification of lipid droplets in quantitative phase images" as currently under review at PLOS ONE. This includes: 1) raw and binary labeled Quantitative Phase Images (QPI) of Y. lipolytica cells used in the analyses described within the manuscript. 2) various derived data including classifier scores, etc.

Data supporting manuscript from L. Sheneman, G. Stephanopoulos, A.E. Vasdekis titled "Deep learning classification of lipid droplets in quantitative phase images" as currently under review at PLOS ONE. This includes: 1) raw and binary labeled Quantitative Phase Images (QPI) of Y. lipolytica cells used in the analyses described within the manuscript. 2) various derived data including classifier scores, etc.

ANN↗

Inductive Learning Approaches for Improving Pilot Awareness of Aircraft Faults

Neural network flight controllers are able to accommodate a variety of aircraft control surface faults without detectable degradation of aircraft handling qualities. Under some faults, however, the effective flight envelope is reduced; this can lead to unexpected behavior if a pilot performs an action that exceeds the remaining control authority of the damaged aircraft. The goal of our work is to increase the pilot s situational awareness by informing him of the type of damage and resulting reduction in flight envelope. Our methodology integrates two inductive learning systems with novel visualization techniques. One learning system, the Inductive Monitoring System (IMS), learns to detect when a simulation includes faulty controls, while two others, Inductive Classification System (INCLASS) and multiple binary decision tree system (utilizing C4.5), determine the type of fault. In off-line training using only non-failure data, IMS constructs a characterization of nominal flight control performance based on control signals issued by the neural net flight controller. This characterization can be used to determine the degree of control augmentation required in the pitch, roll, and yaw command channels to counteract control surface failures. This derived information is typically sufficient to distinguish between the various control surface failures and is used to train both INCLASS and C4.5. Using data from failed control surface flight simulations, INCLASS and C4.5 independently discover and amplify features in IMS results that can be used to differentiate each distinct control surface failure situation. In real-time flight simulations, distinguishing features learned during training are used to classify control surface failures. Knowledge about the type of failure can be used by an additional automated system to alter its approach for planning tactical and strategic maneuvers. The knowledge can also be used directly to increase the pilot s situational awareness and inform manual maneuver decisions. Our multi-modal display of this information provides speech output to issue control surface failure warnings to a lesser-used communication channel and provides graphical displays with pilot-selectable !eve!s of details to issues additional information about the failure. We also describe a potential presentation for flight envelope reduction that can be viewed separately or integrated with an existing attitude indicator instrument. Preliminary results suggest that the inductive approach is capable of detecting that a control surface has failed and determining the type of fault. Furthermore, preliminary evaluations suggest that the interface discloses a concise summary of this information to the pilot.

Spikovska, Lilly↗

Inferring topological transitions in pattern-forming processes with self-supervised learning

Abstract The identification of transitions in pattern-forming processes are critical to understand and fabricate microstructurally precise materials in many application domains. While supervised methods can be useful to identify transition regimes, they need labels, which require prior knowledge of order parameters or relevant microstructures describing these transitions. Instead, we develop a self-supervised, neural-network-based approach that does not require predefined labels about microstructure classes to predict process parameters from observed microstructures. We show that assessing the difficulty of solving this inverse problem can be used to uncover microstructural transitions. We demonstrate our approach by automatically discovering microstructural transitions in two distinct pattern-forming processes: the spinodal decomposition of a two-phase mixture and the formation of binary-alloy microstructures during physical vapor deposition of thin films. This approach opens a path forward for discovering unseen or hard-to-discern transitions and ultimately controlling complex pattern-forming processes.

Abram, Marcin↗

Uncontrolled Learning: Codesign of Neuromorphic Hardware Topology for Neuromorphic Algorithms

Neuromorphic computing has the potential to revolutionize future technologies and our understanding of intelligence, yet it remains challenging to realize in practice. The learning-from-mistakes algorithm, inspired by the brain's simple learning rules of inhibition and pruning, is one of the few brain-like training methods. This algorithm is implemented in neuromorphic memristive hardware through a codesign process that evaluates essential hardware trade-offs. While the algorithm effectively trains small networks as binary classifiers and perceptrons, performance declines significantly with increasing network size unless the hardware is tailored to the algorithm. This work investigates the trade-offs between depth, controllability, and capacity—the number of learnable patterns—in neuromorphic hardware. This highlights the importance of topology and governing equations, providing theoretical tools to evaluate a device's computational capacity based on its measurements and circuit structure. The findings show that breaking neural network symmetry enhances both controllability and capacity. Additionally, by pruning the circuit, neuromorphic algorithms in all-memristive circuits can utilize stochastic resources to create local contrasts in network weights. Through combined experimental and simulation efforts, the parameters are identified that enable networks to exhibit emergent intelligence from simple rules, advancing the potential of neuromorphic computing.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Development of Machine Learning Algorithms to Segment and Study Images of Astromaterial Samples

Introduction: Micrometer-scale chemical analyses of chondritic meteorites and mission-returned asteroid samples can reveal details of the physical and chemical processes operating in the early solar system, including processes that gave rise to planets, moons, and minor bodies. These primitive astromaterials are comprised of chondrules, calcium- and aluminum-rich inclusions (CAI), and many other silicates, oxides, metals, sulfides, and fine-grained materials. The chemical and mineralogical complexity of these samples, vast populations of different components, and heterogeneity across mm to km scales, all limit our understanding of the origin and evolution of these materials. Here, we describe recent efforts to use machine learning techniques to automate the segmentation of chemical maps of chondritic meteorites, designed to aid studies of asteroid samples returned by spacecraft. By automating the task of segmentation it will become possible to rapidly analyze and interpret the sizes, shapes, mineralogy, chemistry, and other properties of every chondrule, calcium- and aluminum-rich inclusion (CAI) and other clast within and between asteroid samples. Sample return missions significantly accelerate and heighten the need to develop such new data analysis techniques, and associated data repositories. Techniques: Neural networks require abundant training data, i.e. images which have been segmented by a human user. We have manually segmented data available from previous petrologic and chemical work at NASA Johnson Space Center and the American Museum of Natural History [1-4]. These data were derived from energy- and wavelength-dispersive X-ray spectroscopy (EDS, WDS) mapping of samples from many chondrite groups. The Deeplabv3+ [5] neural network architecture was trained on human-labeled masks and used to create machine-labeled masks. Several different algorithms were investigated, with inputs ranging from common RGB image formats through to hyperspectral datasets, with raw data comprising greyscale maps of Mg, Ca, and Al, with or without Si, Fe, Ti for both EDS and WDS data, and extending to other elements in EDS only. Each greyscale image was paired with a binary mask for each labelled particle type. Results: The trained algorithms can segment (Fig 1), classify, and measure the dimensions of thousands of particles in chemical maps of a standard 1-inch round petrographic section in seconds to minutes, rather than many hours needed by a human. Accuracy of the algorithms varied from chondrite to chondrite and across particle types. Further results and details of the algorithms will be presented at the workshop. Future directions: Machine learning has the potential to revolutionize our understanding of complex particle populations contained within primitive astromaterial, with segmentation being a critical first step. Example applications include better understanding of particle transport, nebular reservoirs, parent body accretion, and a deeper understanding of the relationships between particle populations and bulk rock elemental and isotopic compositions. In addition to benefits that machine learning can bring to individual researchers, building a community data repository of thousands to millions of particles across hundreds of samples will open up many other possibilities. For example, with a large enough dataset it will be possible to search for exceptionally closely matching particles across disparate samples. Such a capability would enable a single CAI from OSIRISREx or Hayabusa/II samples to be matched to chondritic CAIs that exhibit near-identical size, texture, and mineralogy, down to the level of similar core phenocrysts, zonation, and rim sequences. Such comparative analyses will help to disentangle precursor chemistry, chronology, gas/dust reservoirs during heating, and accretion. Such an endeavor would be impossible without machine learning and a large community data repository of astromaterial chemical/mineralogic maps.

Machine Learning↗

Lossy compression of statistical data using quantum annealer

Abstract We present a new lossy compression algorithm for statistical floating-point data through a representation learning with binary variables. The algorithm finds a set of basis vectors and their binary coefficients that precisely reconstruct the original data. The optimization for the basis vectors is performed classically, while binary coefficients are retrieved through both simulated and quantum annealing for comparison. A bias correction procedure is also presented to estimate and eliminate the error and bias introduced from the inexact reconstruction of the lossy compression for statistical data analyses. The compression algorithm is demonstrated on two different datasets of lattice quantum chromodynamics simulations. The results obtained using simulated annealing show 3–3.5 times better compression performance than the algorithm based on neural-network autoencoder. Calculations using quantum annealing also show promising results, but performance is limited by the integrated control error of the quantum processing unit, which yields large uncertainties in the biases and coupling parameters. Hardware comparison is further studied between the previous generation D-Wave 2000Q and the current D-Wave Advantage system. Our study shows that the Advantage system is more likely to obtain low-energy solutions for the problems than the 2000Q.

97 MATHEMATICS AND COMPUTING↗