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DOE OSTI2019

Contrasting advantages of learning with random weights and backpropagation in non-volatile memory neural networks

Recently, a Cambrian explosion of novel, non-volatile memory (NVM) devices known as memristive devices have inspired effort in building hardware neural networks that learn like the brain. Early experimental prototypes built simple perceptrons from nanosynapses, and recently, fully-connected multi-layer perceptron (MLP) learning systems have been realized. However, while backpropagating learning systems pair well with high-precision computer memories and achieve state-of-the-art performances, this typically comes with a massive energy budget. For future Internet of Things/peripheral use cases, system energy footprint will be a major constraint, and emerging NVM devices may fill the gap by sacrificing high bit precision for lower energy. In this work, we contrast the well known MLP approach with the Extreme Learning Machine (ELM) or NoProp approach, which uses a large layer of random weights to improve the separability of high-dimensional tasks, and is usually considered inferior in a software context. However, we find that when taking device non-linearity into account, NoProp manages to equal hardware MLP system in terms of accuracy. While also using a sign-based adaptation of the delta rule for energy-savings, we find that NoProp can learn effectively with four to six ’bits’ of device analog capacity, while MLP requires eight bit capacity with the same rule. This may allow the requirements for memristive devices to be relaxed in the context of online learning. By comparing the energy footprint of these systems for several candidate nanosynapses, as well as realistic peripherals, we confirm that memristive NoProp systems save energy compared to MLP systems. Lastly, we show that ELM/NoProp systems can achieve better generalization abilities than nanosynaptic MLP systems when paired with pre-processing layers (which do not require backpropagated error). Collectively, these advantages make such systems worthy of consideration in future accelerators or embedded hardware.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Effect of Microstructure and Properties of Ni-WC Composite Coatings on Their Solid Particle Erosion Behavior

Metal matrix composites (MMCs) of WC in Co or Ni matrices are widely used due to their high wear resistance. Previous studies showed that tailoring the mechanical properties through optimum WC content can significantly extend the life cycle of MMCs. In this study, composite coatings of Ni with various WC content ranging from 10 to 55 vol.% were cold sprayed using agglomerated WCNi powders and two different sizes of Ni powders, − 10 + 4 (d{sub 50} = 7 µm) referred to as Ni(7) and − 35 + 15 (d{sub 50} = 25 µm) to as Ni(25). Microstructural characterization, including the deformed structure of Ni splats and retention and distribution of WC, was performed by scanning electron microscopy (SEM). Composite coatings were subjected to solid particle erosion following ASTM G76-13 standard. At low WC content coatings, erosion occurred primarily by plastic deformation of Ni matrices and knocking out of WC particles. At high WC concentrations, network distribution of compacted WCNi particles in the matrices led to a significant drop in erosion rates of coatings, due to a change in erosion mechanism from severe cutting and plowing of Ni and WC dislodging to WCNi splats spalling. Composite coatings sprayed using finer size Ni powder were usually more erosion resistant.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Machine Learning of All Mycobacterium tuberculosis H37Rv RNA-seq Data Reveals a Structured Interplay between Metabolism, Stress Response, and Infection

Mycobacterium tuberculosis is one of the most consequential human bacterial pathogens, posing a serious challenge to 21st century medicine. A key feature of its pathogenicity is its ability to adapt its transcriptional response to environmental stresses through its transcriptional regulatory network (TRN). While many studies have sought to characterize specific portions of the M. tuberculosis TRN, and some studies have performed system-level analysis, few have been able to provide a network-based model of the TRN that also provides the relative shifts in transcriptional regulator activity triggered by changing environments. Here, we compiled a compendium of nearly 650 publicly available, high quality M. tuberculosis RNA-sequencing data sets and applied an unsupervised machine learning method to obtain a quantitative, top-down TRN. It consists of 80 independently modulated gene sets known as “iModulons,” 41 of which correspond to known regulons. These iModulons explain 61% of the variance in the organism’s transcriptional response. We show that iModulons (i) reveal the function of poorly characterized regulons, (ii) describe the transcriptional shifts that occur during environmental changes such as shifting carbon sources, oxidative stress, and infection events, and (iii) identify intrinsic clusters of regulons that link several important metabolic systems, including lipid, cholesterol, and sulfur metabolism. This transcriptome-wide analysis of the M. tuberculosis TRN informs future research on effective ways to study and manipulate its transcriptional regulation and presents a knowledge-enhanced database of all published high-quality RNA-seq data for this organism to date.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2025

Flavor Classification in ICARUS Using Convolutional Visual Networks

In this work, we adapt the Convolutional Visual Network (CVN) approach [1] to the ICARUS detector by incorporating TPC stitching methods inspired by NuGraph[8]. The stitching technique used here is unique to ICARUS, designed specifically to handle its distinct detector segmentations. We then retrain the network using ICARUS-specific data. This study underscores the flexibility of deep learning models in high-energy physics and the importance of accounting for detector-specific features when transferring machine learning techniques between experiments. This poster presents the methods, classification performance, and insights gained from applying CVN to ICARUS data.

Wieler, Felipe [Tech. Fed. Parana U.]↗

DOE OSTI2026

Neutrino Flavor Classification in ICARUS Experiment Using Convolutional Visual Networks

In this work, I adapt the Convolutional Visual Network (CVN) approach to the ICARUS detector by incorporating TPC stitching methods inspired by NuGraph. The stitching technique used here is unique to ICARUS, designed specifically to handle its distinct detector segmentation s. I then retrain the network using ICARUS-specific data. This study underscores the flexibility of deep learning models in high-energy physics and the importance of accounting for detector-specific features when transferring machine learning techniques between experiments. This dissertation presents the methods, classification performance, and insights gained from applying CVN to ICARUS data.

Wieler, Felipe Andre [Parana Tech. Fed. U., Toledo↗

DOE OSTI2023

Deep learning for morphological identification of extended radio galaxies using weak labels

Abstract The present work discusses the use of a weakly-supervised deep learning algorithm that reduces the cost of labelling pixel-level masks for complex radio galaxies with multiple components. The algorithm is trained on weak class-level labels of radio galaxies to get class activation maps (CAMs). The CAMs are further refined using an inter-pixel relations network (IRNet) to get instance segmentation masks over radio galaxies and the positions of their infrared hosts. We use data from the Australian Square Kilometre Array Pathfinder (ASKAP) telescope, specifically the Evolutionary Map of the Universe (EMU) Pilot Survey, which covered a sky area of 270 square degrees with an RMS sensitivity of 25–35 $\mu$ Jy beam $^{-1}$ . We demonstrate that weakly-supervised deep learning algorithms can achieve high accuracy in predicting pixel-level information, including masks for the extended radio emission encapsulating all galaxy components and the positions of the infrared host galaxies. We evaluate the performance of our method using mean Average Precision (mAP) across multiple classes at a standard intersection over union (IoU) threshold of 0.5. We show that the model achieves a mAP $_{50}$ of 67.5% and 76.8% for radio masks and infrared host positions, respectively. The network architecture can be found at the following link: https://github.com/Nikhel1/Gal-CAM

Astronomy & Astrophysics↗

DOE OSTI2023

Multi-module-based CVAE to predict HVCM faults in the SNS accelerator

We present a multi-module framework based on Conditional Variational Autoencoder (CVAE) to detect anomalies in the power signals coming from multiple High Voltage Converter Modulators (HVCMs). We condition the model with the specific modulator type to capture different representations of the $\mathcal{normal}$ waveforms and to improve the sensitivity of the model to identify a specific type of fault when we have limited samples for a given module type. We studied several Artificial Neural Network (ANN) architectures for our CVAE model and evaluated the model performance by looking at their loss landscape for stability and generalization. Our results for the Spallation Neutron Source (SNS) experimental data show that the trained model generalizes well to detecting multiple fault types for several HVCM module types. The results of this study can be used to improve the HVCM reliability and overall SNS uptime.

43 PARTICLE ACCELERATORS↗

DOE OSTI2021

Physics-informed machine learning of the Lagrangian dynamics of velocity gradient tensor

Reduced models describing the Lagrangian dynamics of the velocity gradient tensor (VGT) in homogeneous isotropic turbulence (HIT) are developed under the physics-informed machine learning (PIML) framework. We consider the VGT at both Kolmogorov scale and coarse-grained scale within the inertial range of HIT. Building reduced models requires resolving the pressure Hessian and subfilter contributions, which is accomplished by constructing them using the integrity bases and invariants of the VGT. The developed models can be expressed using the extended tensor basis neural network (TBNN) introduced by Ling et al. [J. Fluid Mech. 807, 155 (2016)]. Physical constraints, such as Galilean invariance, rotational invariance, and incompressibility condition, are thus embedded in the models explicitly. Our PIML models are trained on the Lagrangian data from a high-Reynolds number direct numerical simulation (DNS). To validate the results, we perform a comprehensive out-of-sample test. We observe that the PIML model provides an improved representation for the magnitude and orientation of the small-scale pressure Hessian contributions. Statistics of the flow, as indicated by the joint PDF of second and third invariants of the VGT, show good agreement with the “ground-truth” DNS data. A number of other important features describing the structure of HIT are reproduced by the model successfully. We have also identified challenges in modeling inertial range dynamics, which indicates that a richer modeling strategy is required. This helps us identify important directions for future research, in particular towards including inertial range geometry into the TBNN.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2017

A faster and more reliable data acquisition system for the full performance of the SciCRT

The SciBar Cosmic Ray Telescope (SciCRT) is a massive scintillator tracker to observe cosmic rays at a very high-altitude environment in Mexico. The fully active tracker is based on the Scintillator Bar (SciBar) detector developed as a near detector for the KEK-to-Kamioka long-baseline neutrino oscillation experiment (K2K) in Japan. Since the data acquisition (DAQ) system was developed for the accelerator experiment, we determined to develop a new robust DAQ system to optimize it to our cosmic-ray experiment needs at the top of Mt. Sierra Negra (4600 m). One of our special requirements is to achieve a 10 times faster readout rate. We started to develop a new fast readout back-end board (BEB) based on 100 Mbps SiTCP, a hardware network processor developed for DAQ systems for high energy physics experiments. Then we developed the new BEB which has a potential of 20 times faster than the current one in the case of observing neutrons. Lastly, we installed the new DAQ system including the new BEBs to a part of the SciCRT in July 2015. The system has been operating since then. In this article, we describe the development, the basic performance of the new BEB, the status after the installation in the SciCRT, and the future performance.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2024

Development of message passing-based graph convolutional networks for classifying cancer pathology reports

Abstract Background Applying graph convolutional networks (GCN) to the classification of free-form natural language texts leveraged by graph-of-words features (TextGCN) was studied and confirmed to be an effective means of describing complex natural language texts. However, the text classification models based on the TextGCN possess weaknesses in terms of memory consumption and model dissemination and distribution. In this paper, we present a fast message passing network (FastMPN), implementing a GCN with message passing architecture that provides versatility and flexibility by allowing trainable node embedding and edge weights, helping the GCN model find the better solution. We applied the FastMPN model to the task of clinical information extraction from cancer pathology reports, extracting the following six properties: main site, subsite, laterality, histology, behavior, and grade. Results We evaluated the clinical task performance of the FastMPN models in terms of micro- and macro-averaged F1 scores. A comparison was performed with the multi-task convolutional neural network (MT-CNN) model. Results show that the FastMPN model is equivalent to or better than the MT-CNN. Conclusions Our implementation revealed that our FastMPN model, which is based on the PyTorch platform, can train a large corpus (667,290 training samples) with 202,373 unique words in less than 3 minutes per epoch using one NVIDIA V100 hardware accelerator. Our experiments demonstrated that using this implementation, the clinical task performance scores of information extraction related to tumors from cancer pathology reports were highly competitive.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2025

Topotactic Phase Transformation of Lithiated Spinel to Layered LiMn0.5Ni0.5O2: The Interaction of 3-D and 2-D Li-ion Diffusion

This study investigates the structural evolution of LiMn0.5Ni0.5O2 cathode materials for Li-ion batteries as a function of synthesis temperature and its effect on electrochemical performance. It is demonstrated that, as the synthesis temperature increases from 400 to 900 ?C, a gradual topotactic transformation occurs between a lithiated spinel structure, denoted herein as “lithium-excess spinel” LxS-LiMn0.5Ni0.5O2 (or LxS-LMNO), and the well-known layered LiMn0.5Ni0.5O2 structure prepared at high temperature, HT-LiMn0.5Ni0.5O2 (HT-LMNO). The electrochemical capacity of the LiMn0.5Ni0.5O2 electrodes follows a parabolic trend with increasing synthesis temperature, which is attributed primarily to the gradual transformation of 3-dimensional (3-D) to 2-dimensional (2-D) diffusion pathways for the Li ions. When synthesized at 400 °C, LxS-LiMn0.5Ni0.5O2 electrodes perform well, benefitting from the 3-D network of channels within the LxS structure. By contrast, when prepared at 500-700 °C, LiMn0.5Ni0.5O2 electrodes operate poorly, which is attributed to the formation of locally disordered structural arrangements that impede Li-ion diffusion. Such an increase in local disorder in the mid-temperature synthesis range is attributed to the structural frustration between the lithium-excess spinal and layered end-members. The transformation from the locally disordered to more ordered layered components between 700 °C and 900 °C enhances electrochemical performance. The study opens new avenues for designing next-generation Mn-rich cathode materials by fine-tuning the synthesis conditions as well as the composition and structure of LxS-LMNO electrodes.

energy storage↗

DOE OSTI2017

Controlled Assembly of Lignocellulosic Biomass Components and Properties of Reformed Materials

Reforming whole lignocellulosic biomass into value-added materials has yet to be achieved mainly due to the infusible nature of biomass and its recalcitrance to dissolve in common organic solvents. Recently, the solubility of biomass in ionic liquids (ILs) has been explored to develop all-lignocellulosic materials; however, efficient dissolution and therefore production of value-added materials with desired mechanical properties remain a challenge. This article presents an approach to producing high-performance lignocellulosic films from hybrid poplar wood. An autohydrolysis step that removes ≤50% of the hemicellulose fraction is performed to enhance biomass solvation in 1-ethyl-3-methyl imidazolium acetate ([C2mim][OAc]). The resulting biomass–IL solution is then cast into free-standing films using different coagulating solvents, yet preserving the polymeric nature of the biomass constituents. Methanol coagulated films exhibit a cocontinuous 3D-network structure with dispersed domains of less than 100 nm. The consolidated films with controllable morphology and structural order demonstrate tensile properties better than those of quasi-isotropic wood. Here, the methods for producing these biomass derivatives have potential for fabricating novel green materials with superior performance from woody and grassy biomass.

09 BIOMASS FUELS↗

DOE OSTI2026

Differentiable multiphase flow model for physics-informed machine learning in reservoir pressure management

Accurate subsurface reservoir pressure control is extremely challenging due to geological heterogeneity and multiphase fluid-flow dynamics. Predicting behavior in this setting relies on high-fidelity physics-based simulations that are computationally expensive. Yet, the uncertain, heterogeneous properties that control these flows make it necessary to perform many of these expensive simulations, which is often prohibitive. To address these challenges, we introduce a physics-informed machine learning workflow that couples a fully differentiable multiphase flow simulator, which is implemented in the DPFEHM framework with a convolutional neural network (CNN). The CNN learns to predict fluid extraction rates from heterogeneous permeability fields to enforce pressure limits at critical reservoir locations. By incorporating transient multiphase flow physics into the training process, our method enables more practical and accurate predictions for realistic injection-extraction scenarios compared to previous works. To speed up training, we pretrain the model on single-phase, steady-state simulations and then finetune it on full multiphase scenarios, which dramatically reduces the computational cost. We demonstrate that high-accuracy training can be achieved with fewer than three thousand full-physics multiphase flow simulations – compared to previous estimates requiring up to ten million. This drastic reduction in the number of simulations is achieved by leveraging transfer learning from much less expensive single phase simulations.

25 ENERGY STORAGE↗

DOE OSTI2013

Application of a fuzzy neural network model in predicting polycyclic aromatic hydrocarbon-mediated perturbations of the Cyp1b1 transcriptional regulatory network in mouse skin

Polycyclic aromatic hydrocarbons (PAHs) are present in the environment as complex mixtures with components that have diverse carcinogenic potencies and mostly unknown interactive effects. Non-additive PAH interactions have been observed in regulation of cytochrome P450 (CYP) gene expression in the CYP1 family. To better understand and predict biological effects of complex mixtures, such as environmental PAHs, an 11 gene input-1 gene output fuzzy neural network (FNN) was developed for predicting PAH-mediated perturbations of dermal Cyp1b1 transcription in mice. Input values were generalized using fuzzy logic into low, medium, and high fuzzy subsets, and sorted using k-means clustering to create Mamdani logic functions for predicting Cyp1b1 mRNA expression. Model testing was performed with data from microarray analysis of skin samples from FVB/N mice treated with toluene (vehicle control), dibenzo[def,p]chrysene (DBC), benzo[a]pyrene (BaP), or 1 of 3 combinations of diesel particulate extract (DPE), coal tar extract (CTE) and cigarette smoke condensate (CSC) using leave-one-out cross-validation. Predictions were within 1 log{sub 2} fold change unit of microarray data, with the exception of the DBC treatment group, where the unexpected down-regulation of Cyp1b1 expression was predicted but did not reach statistical significance on the microarrays. Adding CTE to DPE was predicted to increase Cyp1b1 expression, whereas adding CSC to CTE and DPE was predicted to have no effect, in agreement with microarray results. The aryl hydrocarbon receptor repressor (Ahrr) was determined to be the most significant input variable for model predictions using back-propagation and normalization of FNN weights. - Highlights: ► Tested a model to predict PAH mixture-mediated changes in Cyp1b1 expression ► Quantitative predictions in agreement with microarrays for Cyp1b1 induction ► Unexpected difference in expression between DBC and other treatments predicted ► Model predictions for combining PAH mixtures in agreement with microarrays ► Predictions highly dependent on aryl hydrocarbon receptor repressor expression.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2022

Data-driven wind turbine wake modeling via probabilistic machine learning

Wind farm design primarily depends on the variability of the wind turbine wake flows to the atmospheric wind conditions and the interaction between wakes. Physics-based models that capture the wake flow field with high-fidelity are computationally very expensive to perform layout optimization of wind farms, and, thus, data-driven reduced-order models can represent an efficient alternative for simulating wind farms. In this work, we use real-world light detection and ranging (LiDAR) measurements of wind-turbine wakes to construct predictive surrogate models using machine learning. Specifically, we first demonstrate the use of deep autoencoders to find a low-dimensional latent space that gives a computationally tractable approximation of the wake LiDAR measurements. Then, we learn the mapping between the parameter space and the (latent space) wake flow fields using a deep neural network. Additionally, we also demonstrate the use of a probabilistic machine learning technique, namely, Gaussian process modeling, to learn the parameter-space-latent-space mapping in addition to the epistemic and aleatoric uncertainty in the data. Finally, to cope with training large datasets, we demonstrate the use of variational Gaussian process models that provide a tractable alternative to the conventional Gaussian process models for large datasets. Furthermore, we introduce the use of active learning to adaptively build and improve a conventional Gaussian process model predictive capability. Overall, we find that our approach provides accurate approximations of the wind-turbine wake flow field that can be queried at an orders-of-magnitude cheaper cost than those generated with high-fidelity physics-based simulations.

Deep neural networks↗

DOE OSTI2024

Mo 3 S 13 Chalcogel: A High-Capacity Electrode for Conversion-Based Li-Ion Batteries

Despite large theoretical energy densities, metal-sulfide electrodes for energy storage systems face several limitations that impact the practical realization. Here, we present the solution-processable, room temperature (RT) synthesis, local structures, and application of a sulfur-rich Mo 3 S 13 chalcogel as a conversion-based electrode for lithium-sulfide batteries (LiSBs). The structure of the amorphous Mo 3 S 13 chalcogel is derived through operando Raman spectroscopy, synchrotron X-ray pair distribution function (PDF), X-ray absorption near edge structure (XANES), and extended X-ray absorption fine structure (EXAFS) analysis, along with ab initio molecular dynamics (AIMD) simulations. A key feature of the three-dimensional (3D) network is the connection of Mo 3 S 13 units through S–S bonds. Li/Mo 3 S 13 half-cells deliver initial capacity of 1013 mAh g –1 during the first discharge. After the activation cycles, the capacity stabilizes and maintains 312 mAh g –1 at a C/3 rate after 140 cycles, demonstrating sustained performance over subsequent cycling. Such high-capacity and stability are attributed to the high density of (poly)sulfide bonds and the stable Mo–S coordination in Mo 3 S 13 chalcogel. Importantly, these findings showcase the potential of Mo 3 S 13 chalcogels as metal-sulfide electrode materials for LiSBs.

25 ENERGY STORAGE↗

DOE OSTI2020

Tailoring conductive networks within hollow carbon nanospheres to host phosphorus for advanced sodium ion batteries

The formidable sustainability challenges in advancing energy storage technologies call for game-changing research in battery designs. The previous pursuing of novel cathode materials with high redox potentials impedes the vast applications due to the simultaneous electrolyte decomposition at high potentials, though they are expected to deliver high specific capacities. Eventually, people start thinking in an opposite way, desirable anode materials with low redox potentials can also own high specific capacities. Among all the promising candidates, phosphorus-based anodes in sodium ion batteries (SIBs) have received considerable attention owing to the low cost and relatively high natural abundance of phosphorus. More importantly, phosphorus can store three sodium atoms and enable a high theoretical capacity of 2596 mAh g -1 , which overwhelms any other SIB anode currently available. However, the poor electronic conductivity and large volume change of phosphorus during cycling severely deteriorate battery performance. The most widely used strategy is to confine phosphorus within well-designed carbon hosts. Here, we thereby introduce a new type of porous hollow carbon with conductive-network interior as phosphorus host, which not only improves the electrical conductivity, but also creates enough interior surface for maximizing phosphorus utilization and shortening the ion's diffusion distance, compared to those conventional hollow carbon hosts. Therefore, the as-prepared red phosphorus-carbon spheres composites (RP/CS) exhibit superior rate performance (similar to 1083 mAh g -1 at 4 A g -1 , similar to 837 mAh g -1 even at 8 A g -1 ) and excellent cycle life (1027 mAh g -1 at 4 A g -1 more than 2000 cycles).

25 ENERGY STORAGE↗

DOE OSTI2026

Data Summarization and Inference at Scale

This is the final report for the DOE ASCR grant SC-0022260, Data Summarization and Inference at Scale, PI: Alex Pothen, Purdue University. The goal of the project was to solve data-intensive and compute-intensive problems in the physical sciences, engineering, information science, data science, etc. by designing and implementing new algorithms that could work with a subset of the data. The four subgoals were: (a) The solution of problems where the data is too large to be stored in the memory of a computer. In this streaming model of computation, the data arrives as a stream of elements to the computer, each element is processed as it arrives, and a decision is made to discard the data or to store it; only a small subset of the data proportional to the size of the output solution is stored, and when all the data has been streamed, a solution to the problem is computed from the stored subset. (b) The use of machine learning methods to compute solutions to data-intensive problems. The use of GPUs is critical to obtain high performance on machine learning tasks, but their memory sizes are smaller relative to that of CPUs. For large-scale problems, the data is sampled many times, and small samples are used with repetition, for robustness, to compute solutions to inference tasks. This sampling reduces the memory required to solve the problem, but attention is needed to avoid slow convergence to the solutions, and reduced accuracy of inference. We propose submodular optimization, Large Language Models, and physics-informed neural networks to enable GPU computations here. (c) Modeling and visualization of high-dimensional data using interpretable features. Clinical proteomic data sets from immunology for the detection of cancer and other diseases are temporal and high-dimensional, and algorithms for visualizing these data sets using clinically interpretable features are lacking. We propose methods that compute distances based on the optimal transportation problem and graph edit distances to address this problem. We also propose the use of optimal transport-based distances, spatial statistics, and network structure to classify image data sets, We apply these algorithms to electron micrographs of the peripheral nervous system in the digestive tract. (d) The design of data-intensive algorithms on emerging architectures, specifically, noisy, intermediate-scale quantum (NISQ) devices. Quantum computers offer the possibility of exploring large solution spaces due to the principle of superposition, but current quantum computers are limited by few qubits, short coherence times due to noise, poor interconections among the qubits, etc. We propose the use of the divide and conquer paradigm to solve large-scale problems, wherein collections of small subproblems are solved on the quantum devices, and the solutions to the subproblems are integrated into a solution for the original problem on a classical computer.

97 MATHEMATICS AND COMPUTING↗