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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 433 records · Page 24
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

A comparison of surrogate constitutive models for viscoplastic creep simulation of HT-9 steel

Mechanistic microstructure-informed constitutive models for the mechanical response of polycrystals are a cornerstone of computational materials science. However, as these models become increasingly more complex – often involving coupled differential equations describing the effect of specific deformation modes – their associated computational costs can become prohibitive, particularly in optimization or uncertainty quantification tasks that require numerous model evaluations. To address this challenge, surrogate constitutive models that balance accuracy and computational efficiency are highly desirable. Data-driven surrogate models, that learn the constitutive relation directly from data, have emerged as a promising solution. In this work, we develop two local surrogate models for the viscoplastic response of a steel: a piecewise response surface method and a mixture of experts model. These surrogates are designed to adapt to complex material behavior, which may vary with material parameters or operating conditions. The surrogate constitutive models are applied to creep simulations of HT-9 steel, an alloy of considerable interest to the nuclear energy sector due to its high tolerance to radiation damage, using training data generated from viscoplastic self-consistent (VPSC) simulations. In conclusion, we define a set of test metrics to numerically assess the accuracy of our surrogate models for predicting viscoplastic material behavior, and show that the mixture of experts model outperforms the piecewise response surface method in terms of accuracy.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Integrating machine learning interatomic potentials with hybrid reverse Monte Carlo structure refinements in RMCProfile

Structure refinement with reverse Monte Carlo (RMC) is a powerful tool for interpreting experimental diffraction data. To ensure that the under-constrained RMC algorithm yields reasonable results, the hybrid RMC approach applies interatomic potentials to obtain solutions that are both physically sensible and in agreement with experiment. To expand the range of materials that can be studied with hybrid RMC, we have implemented a new interatomic potential constraint in RMCProfile that grants flexibility to apply potentials supported by the Large-scale Atomic/Molecular Massively Parallel Simulator ( LAMMPS ) molecular dynamics code. This includes machine learning interatomic potentials, which provide a pathway to applying hybrid RMC to materials without currently available interatomic potentials. To this end, we present a methodology to use RMC to train machine learning interatomic potentials for hybrid RMC applications.

Cuillier, Paul↗
DOE OSTI2023

Echogenic Segmentation for Ultrasonic Measurements of Spatially Distributed Properties in Solids

An ultrasonic (US) pulse propagating between transducer-receiver responds cumulatively to distributed material properties it encounters, such as temperature distribution. Such convolution makes the unique reconstitution of distributions impossible short of applying tomographic techniques. However, spatial heterogeneity in material properties is readily revealed by segmenting the US propagation path with echogenic features (EF), producing a train of echoes delayed as a function of segmentally localized properties. Unfortunately, in low attenuating waveguides (WGs), primary and trailing echoes (TE) interfere, limiting the number, density, and placement of EFs we can use. We report the design guidelines for creating US-segmented WGs for measuring distributed properties, in which the impact of TEs is minimized or eliminated. Two waveguide designs using these principles have been tested in spatially distributed temperature measurements in high-temperature or restrictive environments.

Walton, Kenneth↗
DOE OSTI2024

Industrial Assessment Center

Established in 1990, San Diego State University’s (SDSU) Industrial Assessment Center (IAC) is proud of its years of service. During this period, it has served over 620 small and medium-sized manufacturing plants in Southern California. SDSU/IAC’s efforts to transfer state-of-the-art technologies to industry have increased revenues, cultivated creativity, improved energy efficiencies, and benefited the environment. The Center has contributed to the region's economic growth and stability by assisting small and medium size companies to better compete in the global market. It has helped mitigate climate change by reducing greenhouse gas emissions. IAC activities have fostered productive relationships between the University and local industry, assisted industrial sectors to improve their energy efficiency and enhance their manufacturing productivity, in turn impacting the material and working conditions of their employees. In addition to financial savings and environmental benefits, we have trained tens of students who became energy specialists in various companies. Thus, a substantial benefit of the IAC has been the ongoing training of engineering faculty and students. All IAC graduates were offered jobs before or within weeks of their graduation. The activities of the SDSU/IAC have expanded the institutional expertise of the College and improved the knowledge base of the faculties involved leading to several related publications, master’s theses, and senior student projects. Significant number of peer-reviewed publications of the IAC director at SDSU have greatly benefitted from the experience of the Center. As a result of this extensive exposure to manufacturing processes, the SDSU/IAC has grown to be an integral component of SDSU’s engineering research and training. We have successfully built upon these established achievements and academic excellence. IAC service to industry is particularly vital in Southern California, a region with one of the highest manufacturing concentrations in the country. SDSU/IAC has understood and implemented the overall objectives of DOE’s IAC program and guidelines except for the pandemic years when the country’s manufacturing sector was put in dire stress. In addition to student training and service to industry, IAC’s contribution to state and local governments as well as utility companies to assess energy policies and design rebate and incentive strategies cannot be undermined.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2024

Voltage Mining for (De)lithiation-Stabilized Cathodes and a Machine Learning Model for Li-Ion Cathode Voltage

Advances in lithium-metal anodes have inspired interest in discovery of Li-free cathodes, most of which are natively found in their charged state. This is in contrast to today's commercial lithium-ion battery cathodes, which are more stable in their discharged state. In this study, we combine calculated cathode voltage information from both categories of cathode materials, covering 5577 and 2423 total unique structure pairs, respectively. The resulting voltage distributions with respect to the redox pairs and anion types for both classes of compounds emphasize design principles for high-voltage cathodes, which favor later Period 4 transition metals in their higher oxidation states and more electronegative anions like fluorine or polyanion groups. Generally, cathodes that are found in their charged, delithiated state are shown to exhibit voltages lower than those that are most stable in their lithiated state, in agreement with thermodynamic expectations. Deviations from this trend are found to originate from different anion distributions between redox pairs. In addition, a machine learning model for voltage prediction based on chemical formulas is trained and shows state-of-the-art performance when compared to two established composition-based ML models for material properties predictions, Roost and CrabNet.

25 ENERGY STORAGE↗
DOE OSTI2020

Featureless adaptive optimization accelerates functional electronic materials design

Electronic materials that exhibit phase transitions between metastable states (e.g., metal-insulator transition materials with abrupt electrical resistivity transformations) are challenging to decode. For these materials, conventional machine learning methods display limited predictive capability due to data scarcity and the absence of features that impede model training. In this article, we demonstrate a discovery strategy based on multi-objective Bayesian optimization to directly circumvent these bottlenecks by utilizing latent variable Gaussian processes combined with high-fidelity electronic structure calculations for validation in the chalcogenide lacunar spinel family. We directly and simultaneously learn phase stability and bandgap tunability from chemical composition alone to efficiently discover all superior compositions on the design Pareto front. Previously unidentified electronic transitions also emerge from our featureless adaptive optimization engine. Our methodology readily generalizes to optimization of multiple properties, enabling co-design of complex multifunctional materials, especially where prior data is sparse.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Learning viscoelasticity models from indirect data using deep neural networks

In this study, we propose a novel approach to model viscoelasticity materials, where rate-dependent and non-linear constitutive relationships are approximated with deep neural networks. We assume that inputs and outputs of the neural networks are not directly observable, and therefore common training techniques with input–output pairs for the neural networks are inapplicable. To that end, we develop a novel computational approach to both calibrate parametric and learn neural-network-based constitutive relations of viscoelasticity materials from indirect displacement data in the context of multiple-physics systems. We show that limited displacement data holds sufficient information to quantify the viscoelasticity behavior. We formulate the inverse computation – modeling viscoelasticity properties from observed displacement data – as a PDE-constrained optimization problem and minimize the error functional using a gradient-based optimization method. The gradients are computed by a combination of automatic differentiation and implicit function differentiation rules. The effectiveness of our method is demonstrated through numerous benchmark problems in geomechanics and porous media transport.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Monotonic Gaussian Process for Physics-Constrained Machine Learning With Materials Science Applications

Physics-constrained machine learning is emerging as an important topic in the field of machine learning for physics. One of the most significant advantages of incorporating physics constraints into machine learning methods is that the resulting model requires significantly less data to train. By incorporating physical rules into the machine learning formulation itself, the predictions are expected to be physically plausible. Gaussian process (GP) is perhaps one of the most common methods in machine learning for small datasets. In this paper, we investigate the possibility of constraining a GP formulation with monotonicity on three different material datasets, where one experimental and two computational datasets are used. The monotonic GP is compared against the regular GP, where a significant reduction in the posterior variance is observed. The monotonic GP is strictly monotonic in the interpolation regime, but in the extrapolation regime, the monotonic effect starts fading away as one goes beyond the training dataset. Imposing monotonicity on the GP comes at a small accuracy cost, compared to the regular GP. The monotonic GP is perhaps most useful in applications where data are scarce and noisy, and monotonicity is supported by strong physical evidence.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Deep Learning Predicts Stress–Strain Relations of Granular Materials Based on Triaxial Testing Data

This study presents an AI-based constitutive modelling framework wherein the prediction model directly learns from triaxial testing data by combining discrete element modelling (DEM) and deep learning. A constitutive learning strategy is proposed based on the generally accepted frame-indifference assumption in constructing material constitutive models. The low-dimensional principal stress-strain sequence pairs, measured from discrete element modelling of triaxial testing, are used to train recurrent neural networks, and then the predicted principal stress sequence is augmented to other high-dimensional or general stress tensor via coordinate transformation. Through detailed hyperparameter investigations, it is found that long short-term memory (LSTM) and gated recurrent unit (GRU) networks have similar prediction performance in constitutive modelling problems, and both satisfactorily predict the stress responses of granular materials subjected to a given unseen strain path. Furthermore, the unique merits and ongoing challenges of data-driven constitutive models for granular materials are discussed.

42 ENGINEERING↗
DOE OSTI2020

GHEP WIBT (Guidelines for Home Energy Professionals Weatherization Installer Badges Toolkit) [SWR-20-69]

The National Renewable Energy Laboratory (NREL) and the U.S. Department of Energy's (DOE) Weatherization Assistance Program (WAP) are collaborating with the home energy retrofit industry to support the development of skilled workers. The Installer Badges Toolkit provides a flexible, customizable, and voluntary approach to training and skills recognition for WAP implementers, utility programs, private-sector workers, and contractors. In Fiscal Year 2018, DOE, along with the Crew Leader scheme committee, determined the Retrofit Installer Technician (RIT) Job Task Analysis (JTA) could be eliminated and its tasks inserted in the Crew Leader JTA. The RIT tasks became the basis of the Installer Badges. The Badges Toolkit was updated in 2020 to add licensing and copyright agreements. Users must cite the copyright information when using or distributing copies of the Badges Toolkit in any way. The technical content has remained unchanged from previous versions of the Badges Toolkit. The Toolkit for home retrofits consists of 25 Badges, each representing different energy efficiency tasks that an installer could perform on a home. Each Badge defines the desired outcome, criteria to verify, applicable material requirements, and references to Standard Work Specifications (SWS) or other relevant standards. The Badges provide a consistent approach to training by ensuring that installers in different regions are learning the same skills nationwide. Organizations can also customize the Toolkit by choosing only those Badges that are relevant to their program. The Badges Toolkit includes five pieces: "How to Use the Badges Toolkit" provides a brief overview of how a Grantee, Subgrantee, or training provider may approach using the toolkit. The "Badges Toolkit: Worksheet" includes what to consider when determining whether and how to best incorporate the Badges into a weatherization assistance program. The "Crew Leader Job Task Analysis Spreadsheet" indicates how the Badges align with specific areas of the JTA. The "Installer Badges Passport" features separate pages for each Badge, which include places for the installer and the supervisor/trainer to record the number of times a task has been successfully completed. The "Installer Badges Verification Criteria" includes sample inspection checklists for each Badge. These can be modified as needed based on approved variance requests or more stringent requirements. They also provide a basis for consistent inspections and awarding of Badges. Note: All elements of the Badges Toolkit are designed for use by potential program implementers. They are not off-the-shelf products intended for immediate deployment. Program implementers must review and modify as needed, revise verification criteria to match local requirements, complete worksheets, and otherwise define the parameters of their own badging program.

Desai, Jal↗
DOE OSTI2022

Predicting defects in laser powder bed fusion using in-situ thermal imaging data and machine learning

Variation in the local thermal history during the Laser Powder Bed Fusion (LPBF) process in Additive Manufacturing (AM) can cause micropore defects, which add to the uncertainty of the mechanical properties (e.g., fatigue life, tensile strength) of the built materials. In-situ sensing has been proposed for monitoring the AM process to minimize defects, but successful minimization requires establishing a quantitative relationship between the sensing data and the porosity, which is particularly challenging with a large number of variables (e.g., laser speed, power, scan path, powder property). Physics-based modeling can simulate such an in-situ sensing-porosity relationship, but it is computationally costly. In this work, we develop Machine Learning (ML) models that can use in-situ thermographic data to predict the micropore of LPBF stainless steel materials. This work considers two identified key features from the thermal histories: the time above the apparent melting threshold ($\tau$) and the maximum radiance ( T max ). These features are computed, stored for each voxel in the built material, and then used as inputs. The binary state of each voxel, either defective or normal, is the output. Different ML models are trained and tested for the binary classification task. In addition to using the thermal features of each voxel to predict its own state, the thermal features of neighboring voxels are also included as inputs. This is shown to improve the prediction accuracy, which is consistent with thermal transport physics around each voxel contributing to its final state. Among the models trained, the F1 scores on test sets reach above 0.96 for Random Forests. Feature importance analysis based on the ML models shows that T max is more important to the voxel state than $\tau$ . The analysis also finds that the thermal history of the voxels above the present voxel is more influential than those beneath it. Our study significantly extends the capability of using in-situ thermographic data to predict porosity in LPBF materials. Finally, since ML models are fast, they may play integral roles in the optimization and control of such AM technologies.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Federated learning for 2D synchrotron x-ray diffractometry: a cross-institutional approach for phase quantification of Ti–6Al–4V alloy

High-energy Two dimensional (2D) synchrotron x-ray diffractometry provides important insights into the atomistic structure and phase evolution of materials, yet traditional analysis methods remain complex, knowledge-intensive, and computationally demanding. Deep-learning models offer a powerful alternative for automating their analysis. Institutions that hold these datasets may be unwilling to share their data due to privacy and security policies, as well as the challenges associated with large-scale data transfer. As a result, models trained on local datasets often perform well only on their own data but exhibit bias and poor generalization across different instruments or facilities. To overcome these limitations, we explore federated learning (FL) for 2D synchrotron diffractograms, enabling collaborative model training without exchanging raw data. In this study, 2D synchrotron diffractograms of Ti–6Al–4V alloy collected from two independent facilities are used to train convolutional neural networks for predicting the β-phase volume fraction. Experimental results show that federated global models significantly outperform locally trained models in terms of generalization and achieve accuracy comparable to centralized trained models. These findings demonstrate the potential of FL to enable secure, cross-institutional collaboration and enhance the scalability of deep-learning-based materials characterization.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Direct Prediction of Phonon Density of States With Euclidean Neural Networks

Abstract Machine learning has demonstrated great power in materials design, discovery, and property prediction. However, despite the success of machine learning in predicting discrete properties, challenges remain for continuous property prediction. The challenge is aggravated in crystalline solids due to crystallographic symmetry considerations and data scarcity. Here, the direct prediction of phonon density‐of‐states (DOS) is demonstrated using only atomic species and positions as input. Euclidean neural networks are applied, which by construction are equivariant to 3D rotations, translations, and inversion and thereby capture full crystal symmetry, and achieve high‐quality prediction using a small training set of examples with over 64 atom types. The predictive model reproduces key features of experimental data and even generalizes to materials with unseen elements, and is naturally suited to efficiently predict alloy systems without additional computational cost. The potential of the network is demonstrated by predicting a broad number of high phononic specific heat capacity materials. The work indicates an efficient approach to explore materials' phonon structure, and can further enable rapid screening for high‐performance thermal storage materials and phonon‐mediated superconductors.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Use of Graph Theory and Neural Networks for Microstructural Classification

Recent advances in materials data analytics have provided new avenues for determining process-structure-property (PSP) linkages in a variety of materials. Machine learning techniques including few-shot learning have increased the efficiency of classifying microscopy images for the purposes of material characterization. Modifications in segmentation also show potential in improving the accuracy of our current pyCHIP classifier. Replacing previous encoders trained on ImageNet with those trained on microscopy images like MicroNet has initially shown better performance at classifying images of irradiated samples. Additionally, different normalization approaches were tested to show no discernable effect on classification. The Louvain method for community detection is analyzed on a set of irradiated samples with different parameters to determine which proved beneficial under what circumstances. We suggest that microscopy experiments be automated in the future using a combination of these techniques to enable high-throughput analyses.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Automated segmentation of porous thermal spray material CT scans with predictive uncertainty estimation

Abstract Thermal sprayed metal coatings are used in many industrial applications, and characterizing the structure and performance of these materials is vital to understanding their behavior in the field. X-ray computed tomography (CT) enables volumetric, nondestructive imaging of these materials, but precise segmentation of this grayscale image data into discrete material phases is necessary to calculate quantities of interest related to material structure. In this work, we present a methodology to automate the CT segmentation process as well as quantify uncertainty in segmentations via deep learning. Neural networks (NNs) have been shown to excel at segmentation tasks; however, memory constraints, class imbalance, and lack of sufficient training data often prohibit their deployment in high resolution volumetric domains. Our 3D convolutional NN implementation mitigates these challenges and accurately segments full resolution CT scans of thermal sprayed materials with maps of uncertainty that conservatively bound the predicted geometry. These bounds are propagated through calculations of material properties such as porosity that may provide an understanding of anticipated behavior in the field.

Martinez, Carianne↗
DOE OSTI2026

Boride-based Ceramic Super-high Temperature Thermocouples in Harsh Environments (Final Scientific/Technical Report)

An electromotive force (emf) can be generated along a temperature gradient between the cold end and hot end of a thermoelectric material, termed the Seebeck effect. Based on the Seebeck effect, metallic alloys have been extensively employed to detect temperatures for centuries, named thermocouples. However, commercially available thermocouple alloys suffer from limitations, such as oxidation, chemical degradation, and poor long-term stability under high-temperature harsh environments. This DOE-funded project aimed to develop high-temperature, chemically tolerant thermocouples suitable for operation in extreme environments relevant to semiconducting thermoelectric materials. The research focused on boride-based semiconducting thermoelectric compounds as candidates for next-generation thermocouples with enhanced oxidation resistance, chemical stability, and thermal robustness under conditions representative of charcoal-fired electricity facilities. During the funded years, boride materials were synthesized using an arc-plasma technique under ambient air and argon atmospheres, enabling scalable and cost-effective production compared with conventional boride fabrication methods. The synthesized borides were processed into nanostructured powders, followed by consolidation into dense bulk materials using a spark plasma sintering (SPS) bottom-up approach. Comprehensive characterization was performed, including microstructural analysis, electrical transport measurements, and optical and thermal property evaluation. Both p-type and n-type boride electric legs were fabricated and integrated into boride-based thermocouples. The thermal and irradiation stabilities of the boride nanomaterials and bulk thermoelectric materials were systematically evaluated to assess suitability for long-term operation in harsh environments. Additionally, 12 students were broadly hands-on trained spanning the full research workflow, including word processing and technical editing (e.g., LATEX for manuscript and poster preparation), data collection and analysis (using Python and related libraries and hardware interfaces), sample preparation (including arc-plasma synthesis and spark plasma sintering), and advanced characterization techniques (such as X-ray diffraction, UV–vis spectroscopy, electron microscopy, differential thermal analysis (DTA), and Seebeck coefficient measurements, etc). Overall, this project demonstrated the feasibility of boride-based thermoelectric materials as durable high-temperature thermocouples, providing a promising pathway toward robust temperature sensing technologies aligned with DOE energy infrastructure and extreme-environment monitoring needs.

20 FOSSIL-FUELED POWER PLANTS↗
DOE OSTI2024

CORE-CM in The Greater Green River and Wind River Basins: Transforming and Advancing a National Coal Asset (Final Report)

The following document summarizes project results from “CORE-CM in the Greater Green River and Wind River Basins: Transforming and Advancing a National Coal Asset”. This project is part of the U.S. Department of Energy’s (“DOE”) National Energy Technology Laboratory’s (“NETL”) Carbon Ore, Rare Earth Elements, and Critical Minerals (CORE-CM) Initiative. This report concludes that the Greater Green River and Wind River Basins (GGRB-WRB) Area-of-Interest (AOI 9) is the ideal region for continued research and development in progressing the broader CORE-CM goals outlined by the DOE. Based upon the extensive analyses of technical, social, and community criteria, this report illustrates that the GGRB-WRB hosts numerous potential CORE-CM feedstocks (both coal- and non-coal based), diverse opportunities for utilizing existing industrial waste streams, ample infrastructure and industry to support new CORE-CM-focused technologies, and a highly motivated, well educated, and adaptable workforce to further develop the regional and national CORE-CM supply chain. Additionally, some potential solutions for technological gaps suggest that the GGRB-WRB's diverse resources can play a significant role in achieving the national goal of critical materials independence. With full community participation, meaningful involvement of regional Tribal Nations, and building upon the stakeholder engagement demonstrated here, the GGRB-WRB region presents a unique opportunity for advancing the CORE-CM Initiative. This project was designed to bring together coal-based communities and stakeholders from across the GGRB-WRB to advance new industries for CORE-CM resources. The University of Wyoming (UWyo) School of Energy Resources (SER) led a project team of experts from the Colorado Geological Survey (CGS), Colorado School of Mines (CSM), Los Alamos National Lab (LANL), and local community colleges. Input from basinal, regional, and national experts bolstered the coalition in order to advance the mission of DOE’s CORE-CM initiative and develop the domestic CORE-CM supply chain. Phase I of this project was designed to address the goal of developing and catalyzing economic growth, job creation, and technology innovation in the GGRB-WRB of Wyoming and Colorado, by increasing the supply of CORE-CM to manufacturers of non-fuel Carbon Based Products (CBP) and products reliant upon CM. The GGRB-WRB CORE-CM project worked toward providing benefit through several avenues of performance and research. • Develop a coalition team to achieve project objectives • Complete detailed assessments, including State-of-the-Art (SOTA) Data acquisition of potential CORE-CM materials across the AOI, and meaningfully contributes to DOE’s CORE-CM goals nationally. • Strategic planning for regional economic growth, job creation, and associated technology innovation around coal materials, including plans to maximize the development of potential CORE-CM resources and technology by creating regional public-private partnerships. • Define regional economic growth potential around existing strengths, energy infrastructure, business and industry, including planning for the leveraging of highly trained workforces, existing and novel coal technologies, and energy infrastructure in development of CORE-CM supply chains. • Develop a preliminary strategic plan for increasing the supply of CORE-CM materials to manufacturers of non-fuel Carbon Based Products (CBP) and products reliant upon CM, focusing on regional strengths that result in an emerging diversified CORE-CM economy. • Assemble a committed network of stakeholders and communities that learn about, accept, and grow new energy technologies within coal regions. Additionally, the project team significantly contributed to the CORE-CM Initiative’s national goals, through cross-regional scoping, collaborating with CORE-CM projects in other AOIs, and including parallel regional project experts. In addition to active inclusion and meaningful engagement and contribution to DOE-led working groups, the project team focused on engaging with regional communities including Tribal Nations, economic development groups, and regional government organizations. The project’s CORE-CM development and commercialization plan identified diverse CORECM feedstocks, potential routes towards integration with existing industries, methods for supply-chain development that leverage existing infrastructure and businesses considering the regional economy, identified entry barriers for incorporating traditional and new technologies in those supply chains, recognized opportunities for public-private partnerships to develop technology innovation centers, identified diverse workforces, and conducted stakeholder outreach and education to build a community of understanding on CORE-CM potential in the GGRB-WRB region. Detailed task descriptions can be found in each chapter.

01 COAL, LIGNITE, AND PEAT↗
DOE OSTI2024

Statistical Complexity of Quantum Learning

Abstract Learning problems involve settings in which an algorithm has to make decisions based on data, and possibly side information such as expert knowledge. This study has two main goals. First, it reviews and generalizes different results on the data and model complexity of quantum learning, where the data and/or the algorithm can be quantum, focusing on information‐theoretic techniques. Second, it introduces the notion of copy complexity, which quantifies the number of copies of a quantum state required to achieve a target accuracy level. Copy complexity arises from the destructive nature of quantum measurements, which irreversibly alter the state to be processed, limiting the information that can be extracted about quantum data. As a result, empirical risk minimization is generally inapplicable. The paper presents novel results on the copy complexity for both training and testing. To make the paper self‐contained and approachable by different research communities, an extensive background material is provided on classical results from statistical learning theory, as well as on the distinguishability of quantum states. Throughout, the differences between quantum and classical learning are highlighted by addressing both supervised and unsupervised learning, and extensive pointers are provided to the literature.

97 MATHEMATICS AND COMPUTING↗