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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 487 records · Page 27
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Time-resolved RIXS experiment with pulse-by-pulse parallel readout data collection using X-ray free electron laser

Abstract Time-resolved resonant inelastic X-ray scattering (RIXS) is one of the developing techniques enabled by the advent of X-ray free electron laser (FEL). It is important to evaluate how the FEL jitter, which is inherent in the self-amplified spontaneous emission process, influences the RIXS measurement. Here, we use a microchannel plate (MCP) based Timepix soft X-ray detector to conduct a time-resolved RIXS measurement at the Ti L 3 -edge on a charge-density-wave material TiSe 2 . The fast parallel Timepix readout and single photon sensitivity enable pulse-by-pulse data acquisition and analysis. Due to the FEL jitter, low detection efficiency of spectrometer, and low quantum yield of RIXS process, we find that less than 2% of the X-ray FEL pulses produce signals, preventing acquiring sufficient data statistics while maintaining temporal and energy resolution in this measurement. These limitations can be mitigated by using future X-ray FELs with high repetition rates, approaching MHz such as the European XFEL in Germany and LCLS-II in the USA, as well as by utilizing advanced detectors, such as the prototype used in this study.

36 MATERIALS SCIENCE↗
DOE OSTI2020

HIDES: Hybrid Intrusion Detector for Energy Systems

The establishment of a resilient electric grid accompanied by a secure communications network is an ongoing battle as advanced persistent threats continue to exploit existing vulnerabilities in legacy supervisory control and data acquisition system (SCADA) infrastructure. Traditional intrusion detection systems (IDSs) lack consistent performance because of the continuously evolving attack surface of SCADA systems. These shortcomings can be overcome by integrating logical system behavior, protocol-specific knowledge, and data-based learning to develop a comprehensive IDS solution. In this paper, we present a Hybrid Intrusion Detector for Energy Systems by integrating a network-based IDS, state-of-the-art machine learning-based IDS, and model-based IDS to detect unknown and stealthy cyberattacks targeting the SCADA networks. The proposed IDS uses synchrophasor measurements and cyber logs to learn patterns of different scenarios based on spatiotemporal behaviors of power systems. As a proof of concept, we implement and validate the proposed IDS by leveraging resources available at the National Renewable Energy Laboratory's Energy Systems Integration Facility test bed. Experimental results show promising performance in detecting cyberattacks while providing realtime visualization of power system measurements and cyber logs.

machine-learning intrusion detection system↗
DOE OSTI2024

Report on Initial Sodium Testing on the Thermal Hydraulic Experimental Test Article (THETA) (Fiscal Year 2024 Final Report)

The Thermal Hydraulic Experimental Test Article (THETA) is a facility that is used to develop sodium components and instrumentation as well as to acquire experimental data for validation of reactor thermal hydraulic and safety analysis codes. The facility simulates nominal thermal hydraulic conditions as well as protected/unprotected loss of flow accidents in a sodium-cooled fast reactor (SFR). High fidelity distributed temperature profiles of the developed flow field may be acquired with Rayleigh backscatter based optical fiber temperature sensors. The facility was designed in partnership with systems code experts to tailor the experiment to ensure the most relevant and highest quality data for code validation. THETA is comprised of a traditional primary coolant and secondary coolant system. The primary system is submerged in the pool of sodium and consists of a pump, electrically heated core, intermediate heat exchanger, and connected piping and thermal barriers (redan). The secondary system, located outside of the sodium pool, consists of a pump, sodium to air heat exchanger, and connected piping and valves. In fiscal year 2023, thermal stratification tests were completed with the primary system online, while the secondary system was being constructed [1]. These tests had shown that the core barrel and intermediate heat exchanger (IHX) outlet required increased thermal insulation. The THETA primary system was removed from METL, cleaned, thermal insulators installed, and then inserted into METL Test Vessel 4. At the time of this writing the THETA primary and secondary system are operational. During this fiscal year 100+ hours of testing was completed to characterize thermal hydraulic phenomena associated with steady state and transient conditions in a pool type liquid metal cooled reactor. A majority of the testing campaign was completed to satisfy the experimental data acquisition requirements for the GAIN Voucher with Oklo, CRADA 2021-21121. THETA is still operational at the time of this publication and future testing is planned for fiscal year 2025. Work is underway to publish existing and future data to an online database to facilitate collaboration with SFR engineers looking to validate their systems code or computational fluid dynamics models.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2026

Machine learning inversion of interatomic force constants from single-crystal inelastic neutron scattering

Atomic vibrations govern many macroscopic properties of materials, but experiments to comprehensively probe them remain challenging. Inelastic neutron scattering (INS) is a powerful technique to map phonon dispersions in crystals, especially when leveraging modern time-of-flight (ToF) spectrometers with large detectors. However, efficiently and robustly extracting interatomic force constants (FCs) parameterizing phonon dynamics from experimental spectra remains a bottleneck due to the complexity and high dimensionality of ToF INS datasets. Here, we present a machine learning approach for the direct inversion of FCs from single-crystal INS measurements. The framework leverages synthetic training data generated using universal machine-learned force fields and an efficient physics-based forward model. We benchmark two neural architectures–one emphasizing structured latent representation learning and the other direct, supervised spectral regression–across simulated datasets for two materials under idealized and noisy conditions. The latent-representation model is subsequently applied to experimental single-crystal INS data on germanium. The model is shown to reproduce FCs derived from both first-principles simulations and from iterative optimization, and furthermore achieves reliable inference even from sparse, single-orientation measurements representing short data acquisitions. Analysis of the learned latent space reveals semantically continuous and physically interpretable encodings that support strong cross-domain generalization. By bridging theoretical and experimental domains, we establish a path toward rapid inversion of experimental spectra and data-driven interpretation of temperature-dependent lattice dynamics.

42 ENGINEERING↗
DOE OSTI2024

SSTDR and FDR Detection of Un-Energized and Energized Cable Anomalies Including Thermal Degradation Using Machine Learning

Historically, cables are initially qualified for nuclear power plant use for 40 years. As plants extend their operating license to 60 and 80 years, continued use of these cables must shift to a performance-based approach since it is cost prohibitive to completely replace cables that are likely still capable of performing their design function. A variety of cable tests are available and are commonly applied during outages when the cables can be taken out of service. Frequency domain reflectometry (FDR) is one of these test methods that is being more broadly accepted and used because it not only detects anomalies along the cable with a low-voltage signal that does not stress the cable insulation, but the technique also locates the anomalies. This supports follow-up local inspection and local repair or partial replacement of a damaged cable segment. Currently, FDR testing is only applied to cables that are taken out of service since the test instrument would be damaged by operational voltages. A related technology that has found some acceptance in the aircraft and rail industry is spread spectrum time domain reflectometry (SSTDR). This technology has been implemented with a custom commercial instrument by LiveWire Innovation that is designed to operate on live cables up to 1000 volts and with a bandwidth of 48 MHz. Initial evaluation by the Pacific Northwest National Laboratory (PNNL) of the Live Wire system indicated that a broader bandwidth (BW) SSTDR may be better for many kinds of flaws. This led PNNL to develop an SSTDR laboratory instrument suitable for tests up to 500 MHz bandwidth. Testing on energized cables is also desirable for online monitoring systems so an inductive clamshell coupler was developed that allows energized cables to be tested up to at least 5 kV and likely higher voltage levels. Dielectric spectroscopy and tan delta testing plus various laboratory destructive tests were included in this data acquisition campaign directed to feed a machine learning (ML) study. With these kinds of developments, online energized cable tests may be possible with industrial adoption of such hardware advances but it will be completely impractical to have highly skilled data analysts continually examine these complex signals for indications of damage or compromised conditions. If online testing is to be implemented in new test hardware, it must be accompanied by software that can interpret the signals and alert plant operators of changing or degraded conditions. The thermally aged, shielded cable investigated here was separately treated for ML analysis. Visual analysis of electrical data showed generally increasing peaks where the cable entered and exited the oven. These peaks were not exactly aligned with expected locations, but these differences were attributed to velocity of propagation calibration errors. Only supervised ML was applied to the thermally aged data as this data was only available shortly before the committed publication date of this report. The supervised ML was structured to divide the 0 to 70-day responses as ‘normal’ from 0 to 35 days or ‘anomalous’ from 36 to 70 days, based on cable tensile elongation at break (EAB) insulation characterization. Using 80% of the data for training and 20% for testing, the supervised ML predicted normal versus anomalous was 70% accurate. Important conclusions include: • Accuracy to predict the presence of cable damage is improved from the 2023 effort by more training data. Weighted accuracies for comparisons among the instruments ranged from 67 to 89 % for unsupervised ML and 71 to 99% for supervised ML. • Based on the synthetic data tests, the unsupervised models are more generalizable to unseen anomalies. The Multi-Layer Perceptron classifier (MLP) model reported as high as 99.7% accuracy on the test data, but this dropped to 58.3% when tested on the synthetic data. In contrast, the unsupervised Pointwise model only achieved 89.7% accuracy on the experimental data but reported 78.3% accuracy on the synthetic data. • The best anomaly indicators are higher frequency (400 MHz BW) FDR data. Other tests may be interesting but for this study, this was the best predicter.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2022

2022 AI Testbed Expeditions Report

By exploiting the coherent properties of a light source, coherent diffraction imaging (CDI) is able to obtain the sample image at a nanoscale resolution using the measured diffraction pattern. Bragg Coherent Diffraction Imaging (BCDI) has become valuable for recovering the displacement and strain field of crystals, providing a valuable tool in material science and solid-state physics. X-ray ptychography is another emerging CDI technique that can produce a high-resolution image of the extended sample and has become popular in many research areas (e.g., materials science, biology, electronics, and optics characterization). CDI including BCDI and ptychography has become an established technique in Synchrotron Facilities including the Advanced Photon Source (APS) and will greatly benefit from the 100x coherent flux increase of the upcoming APS Upgrade (APSU). The current image formation process in CDI employs iterative phase retrieval algorithms, which is a time-consuming and computationally expensive process. Especially after APSU, the traditional iterative methods will not be able to match the experimental data acquisition speed. We employ deep learning (DL) approach to replace the iterative approaches, therefore allowing hundreds of times faster recovery of the object. We developed AutoPhaseNN, a DL-based approach which learns to solve the inverse problem without labeled data. Taking 3D BCDI as a representative technique, AutoPhaseNN has been demonstrated to be one hundred times faster than traditional iterative phase retrieval methods while providing comparable image quality. The current network is trained with 64 x 64 x 64 data size, to achieve higher resolution imaging, we will need to scale the network to input and train/infer 3D arrays of size 256 x 256 x 256 (today) and of size 2560x2560x2560 (APSU). However, the scalability of the network is restricted due to the memory-intensive training process. To perform the training for a 256 x 256 x 256 data size, the required memory exceeds the capacity of the current machine. In this project, we explore using Sambanova system to train the network for the direct data inversion for CDI.

36 MATERIALS SCIENCE↗
DOE OSTI2021

The ATLAS Fast TracKer system

The ATLAS Fast TracKer (FTK) was designed to provide full tracking for the ATLAS high-level trigger by using pattern recognition based on Associative Memory (AM) chips and fitting in high-speed field programmable gate arrays. The tracks found by the FTK are based on inputs from all modules of the pixel and silicon microstrip trackers. The as-built FTK system and components are described, as is the online software used to control them while running in the ATLAS data acquisition system. Also described is the simulation of the FTK hardware and the optimization of the AM pattern banks. An optimization for long-lived particles with large impact parameter values is included. A test of the FTK system with the data playback facility that allowed the FTK to be commissioned during the shutdown between Run 2 and Run 3 of the LHC is reported. The resulting tracks from part of the FTK system covering a limited $\eta$-$\phi$ region of the detector are compared with the output from the FTK simulation. It is shown that FTK performance is in good agreement with the simulation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Chatbots can guide experimentalists to avoid many common mistakes in measuring fluorescence spectra

The concepts of fluorescence spectroscopy are well established, yet the experimental collection of a spectrum is susceptible to a range of errors. A recent study showed that chatbots – derived from substantial advances in artificial intelligence – can provide practical assistance to the experimentalist during collection of absorption spectra and thereby achieve improved quality. Fluorescence, while the complement of absorption, is prone to a far richer array of experimental errors encompassing environmental factors, instrumental settings, and experimentalist mistakes. Here, 10 chatbots (ChatGPT 5, Gemini 2.5 Pro, Microsoft Copilot, Meta AI Llama 4, Claude Sonnet 4.5, X.ai Grok 4, Perplexity Pro, DeepSeek V3.1, Z.ai GLM-4.6, and KIMI K2) have been evaluated for the suitability in providing quality-improvement advice to the experimentalist during acquisition of fluorescence spectra. Significant opportunities for practical assistance are noted, although intervention by a domain expert is often required. Here, if combined with image recognition and/or signal processing, chatbots embedded in instruments may enable real-time guidance during fluorescence data acquisition.

Artificial intelligence↗
DOE OSTI2021

Predictive Data Analytics Framework Using Advanced Test Reactor Acoustic Data

Although a nuclear reactor is a hostile environment for sensing and electrical communications, the reactor core is amenable to acoustic communication. An acoustic measurement infrastructure (AMI) has been installed at the Advanced Test Reactor (ATR) nozzle trench area to record acoustic signals that has the ability to capture different operating regime of the reactor. This AMI includes ATR in-pile structural components, coolant, acoustic receivers, primary coolant pumps (PCP) as signal sources, a data acquisition system, and signal-processing algorithms, enabling real-time. This report will discusses development of recursive Fast Fourier Transform approach to process in real-time acoustic signals, application of short time Fast Fourier Transform to the ATR brush data to understand the vibration level and to develop spectrograms for different primary coolant pump combinations. The combination of primary coolant pumps for normal and power axial locator mechanism of ATR are different and generates different signatures. These acoustic signatures were used to develop machine learning approaches to automatically classify different operating regimes. This lay the foundation for predictive analytic framework that can be leverage by ATR to optimize their operation and maintenance. The path forward involves continued engagement with ATR and expanded implementation of AMI and predictive framework at ATR and other facilities within INL and at other experimental reactors.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2026

Toward Unified Autonomous Scattering Experiments: A Cross-Facility Case Study at ALS and PETRA III

Autonomous experiments rely on the integration of control, data acquisition, analysis, and decision-making frameworks. While such systems have been demonstrated at individual facilities, adapting them to additional instruments remains challenging due to differences in local infrastructure. We present a modular workflow that connects existing open-source tools for data access (Tiled), workflow orchestration (Prefect), analysis and visualization (pyFAI, Plotly Dash), and Gaussian-process-based adaptive sampling (gpCAM) into a unified framework for autonomous scattering experiments. The same configuration operates across two synchrotron beamlines (ALS 7.3.3 and PETRA III P03) with only minimal facility-specific adjustments, as shown in proof-of-concept demonstrations. This validates that a consistent design emphasizing modularity and shared interfaces can ease deployment across diverse experimental environments. The resulting framework provides a flexible foundation for extending autonomous control and analysis capabilities beyond a single beamline or instrument.

47 OTHER INSTRUMENTATION↗
DOE OSTI2025

Detecting anomalous SRF cavity behavior with unsupervised learning

We present an unsupervised learning framework for detecting anomalous superconducting radio-frequency (SRF) cavity behavior at the Continuous Electron Beam Accelerator Facility (CEBAF), emphasizing its initial performance and effectiveness. Key to the system’s success was the development of data acquisition systems (DAQs) that capture fast-sampled, information-rich signals, essential for detecting transient effects. The approach involves creating daily cavity-specific models using principal component analysis to handle variations in rf signal behavior and mitigate performance degradation from data drift. This unsupervised method eliminates the need for expensive labeling by continuously updating models with recent data. Deployed and operational for 3 months before a scheduled shutdown, the system successfully identified several issues with DAQ signals, confirming its effectiveness. Despite access to only a fraction of CEBAF’s SRF cavity signals, the framework efficiently detected several instances requiring intervention, demonstrating a significant improvement over traditional, labor-intensive methods of manual plot inspection. Published by the American Physical Society 2025

43 PARTICLE ACCELERATORS↗
DOE OSTI2021

Improved Neutron Lifetime Measurement with UCN$\tau$

In this work, we report an improved measurement of the free neutron lifetime $\tau_n$ using the UCN$\tau$ apparatus at the Los Alamos Neutron Science Center. We count a total of approximately 38×10 6 surviving ultracold neutrons (UCNs) after storing in UCN$\tau$’s magnetogravitational trap over two data acquisition campaigns in 2017 and 2018. We extract $\tau$ n from three blinded, independent analyses by both pairing long and short storage time runs to find a set of replicate $\tau$ n measurements and by performing a global likelihood fit to all data while self-consistently incorporating the β-decay lifetime. Both techniques achieve consistent results and find a value $\tau$ n =877.75±0.28 stat +0.22/–0.16 syst s. With this sensitivity, neutron lifetime experiments now directly address the impact of recent refinements in our understanding of the standard model for neutron decay.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Rapid detection of rare events from in situ X-ray diffraction data using machine learning

High-energy X-ray diffraction methods can non-destructively map the 3D microstructure and associated attributes of metallic polycrystalline engineering materials in their bulk form. These methods are often combined with external stimuli such as thermo-mechanical loading to take snapshots of the evolving microstructure and attributes over time. However, the extreme data volumes and the high costs of traditional data acquisition and reduction approaches pose a barrier to quickly extracting actionable insights and improving the temporal resolution of these snapshots. This article presents a fully automated technique capable of rapidly detecting the onset of plasticity in high-energy X-ray microscopy data. The technique is computationally faster by at least 50 times than the traditional approaches and works for data sets that are up to nine times sparser than a full data set. This new technique leverages self-supervised image representation learning and clustering to transform massive data sets into compact, semantic-rich representations of visually salient characteristics ( e.g. peak shapes). These characteristics can rapidly indicate anomalous events, such as changes in diffraction peak shapes. It is anticipated that this technique will provide just-in-time actionable information to drive smarter experiments that effectively deploy multi-modal X-ray diffraction methods spanning many decades of length scales.

Zheng, Weijian↗
DOE OSTI2022

Streaming readout for next generation electron scattering experiments

Current and future experiments at the high-intensity frontier are expected to produce an enormous amount of data that needs to be collected and stored for offline analysis. Thanks to the continuous progress in computing and networking technology, it is now possible to replace the standard ‘triggered’ data acquisition systems with a new, simplified and outperforming scheme. ‘Streaming readout’ (SRO) DAQ aims to replace the hardware-based trigger with a much more powerful and flexible software-based one, that considers the whole detector information for efficient real-time data tagging and selection. Considering the crucial role of DAQ in an experiment, validation with on-field tests is required to demonstrate SRO performance. In this paper, we report results of the on-beam validation of the Jefferson Lab SRO framework. In this work, we exposed different detectors (PbWO-based electromagnetic calorimeters and a plastic scintillator hodoscope) to the Hall-D electron-positron secondary beam and to the Hall-B production electron beam, with increasingly complex experimental conditions. By comparing the data collected with the SRO system against the traditional DAQ, we demonstrate that the SRO performs as expected. Furthermore, we provide evidence of its superiority in implementing sophisticated AI-supported algorithms for real-time data analysis and reconstruction.

47 OTHER INSTRUMENTATION↗
DOE OSTI2021

Integrating Models with Real-time Field Data for Extreme Events: From Field Sensors to Models and Back with AI in the Loop

Focal Area(s): This whitepaper is responsive to focal area (1) Data acquisition and assimilation enabled by machine learning, AI, and advanced methods including experimental/network design/optimization, unsupervised learning (including deep learning), and hardware-related efforts involving AI (e.g., edge computing). We discuss Artificial Intelligence and Machine Learning (AI/ML) enabled integration of real-time data into the extreme event modeling workflow to improve the predictive capabilities of these models, and deliver real-time feedback to remote sensors, including software and data engineering challenges.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Atmospheric Radiation Measurement (ARM) airborne field campaign data products between 2013 and 2018

Airborne measurements are pivotal for providing detailed, spatiotemporally resolved information about atmospheric parameters and aerosol and cloud properties, thereby enhancing our understanding of dynamic atmospheric processes. For 30 years, the US Department of Energy (DOE) Office of Science supported an instrumented Gulfstream 1 (G-1) aircraft for atmospheric field campaigns. Data from the final decade of G-1 operations were archived by the Atmospheric Radiation Measurement (ARM) Data Center and made publicly available at no cost to all registered users. To ensure a consistent data format and to improve the accessibility of the ARM airborne data, an integrated dataset was recently developed covering the final 6 years of G-1 operations (2013 to 2018, https://doi.org/10.5439/1999133; Mei and Gaustad, 2024). The integrated dataset includes data collected from 236 flights (766.4 h), which covered the Arctic, the US Southern Great Plains (SGP), the US West Coast, the eastern North Atlantic (ENA), the Amazon Basin in Brazil, and the Sierras de Córdoba range in Argentina. These comprehensive data streams provide much-needed insight into spatiotemporal variability in the thermodynamic quantities and aerosol and cloud properties for addressing essential science questions in Earth system process studies. This paper describes the DOE ARM merged G-1 datasets, including information on the acquisition, data collection challenges and future potentials, and quality control processes. It further illustrates the usage of this merged dataset to evaluate the Energy Exascale Earth System Model (E3SM) with the Earth System Model Aerosol–Cloud Diagnostics (ESMAC Diags) package.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Design and analysis of a wake model for spatially heterogeneous flow

Abstract. Methods of turbine wake modeling are being developed to more accurately account for spatially variant atmospheric conditions within wind farms. Most current wake modeling utilities are designed to apply a uniform flow field to the entire domain of a wind farm. When this method is used, the accuracy of power prediction and wind farm controls can be compromised depending on the flow-field characteristics of a particular area. In an effort to improve strategies of wind farm wake modeling and power prediction, FLOw Redirection and Induction in Steady State (FLORIS) was developed to implement sophisticated methods of atmospheric characterization and power output calculation. In this paper, we describe an adapted FLORIS model that features spatial heterogeneity in flow-field characterization. This model approximates an observed flow field by interpolating from a set of atmospheric measurements that represent local weather conditions. The objective of this method is to capture heterogeneous atmospheric effects caused by site-specific terrain features, without explicitly modeling the geometry of the wind farm terrain. The implemented adaptations were validated by comparing the simulated power predictions generated from FLORIS to the actual recorded wind farm output from the supervisory control and data acquisition (SCADA) recordings and large eddy simulations (LESs). When comparing the performance of the proposed heterogeneous model to homogeneous FLORIS simulations, the results show a 14.6 % decrease for mean absolute error (MAE) in wind farm power output predictions for cases using wind farm SCADA data and a 18.9 % decrease in LES case studies. The results of these studies also indicate that the efficacy of the proposed modeling techniques may vary with differing site-specific operational conditions. This work quantifies the accuracy of wind plant power predictions under heterogeneous flow conditions and establishes best practices for atmospheric surveying for wake modeling.

17 WIND ENERGY↗
DOE OSTI2025

Does provable absence of barren plateaus imply classical simulability?

A large amount of effort has recently been put into understanding the barren plateau phenomenon. In this perspective article, we face the increasingly loud elephant in the room and ask a question that has been hinted at by many but not explicitly addressed: Can the structure that allows one to avoid barren plateaus also be leveraged to efficiently simulate the loss classically? We collect evidence-on a case-by-case basis-that many commonly used models whose loss landscapes avoid barren plateaus can also admit classical simulation, provided that one can collect some classical data from quantum devices during an initial data acquisition phase. This follows from the observation that barren plateaus result from a curse of dimensionality, and that current approaches for solving them end up encoding the problem into some small, classically simulable, subspaces. Thus, while stressing that quantum computers can be essential for collecting data, our analysis sheds doubt on the information processing capabilities of many parametrized quantum circuits with provably barren plateau-free landscapes. We end by discussing the (many) caveats in our arguments including the limitations of average case arguments, the role of smart initializations, models that fall outside our assumptions, the potential for provably superpolynomial advantages and the possibility that, once larger devices become available, parametrized quantum circuits could heuristically outperform our analytic expectations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗