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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 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Dynamic load balancing with enhanced shared-memory parallelism for particle-in-cell codes

Furthering our understanding of many of today’s interesting problems in plasma physics – including plasma based acceleration and magnetic reconnection with pair production due to quantum electrodynamic effects – requires large-scale kinetic simulations using particle-in-cell (PIC) codes. However, these simulations are extremely demanding, requiring that contemporary PIC codes be designed to efficiently use a new fleet of exascale computing architectures. To this end, the key issue of parallel load balance across computational nodes must be addressed. We discuss the implementation of dynamic load balancing by dividing the simulation space into many small, self-contained regions or ‘‘tiles,’’ along with shared-memory (e.g., OpenMP) parallelism both over many tiles and within single tiles. The load balancing algorithm can be used with three different topologies, including two space-filling curves. Here, we tested this implementation in the code Osiris and show low overhead and improved scalability with OpenMP thread number on simulations with both uniform load and severe load imbalance. Compared to other load-balancing techniques, our algorithm gives order-of-magnitude improvement in parallel scalability for simulations with severe load imbalance issues.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Ultralow Voltage GaN Vacuum Nanodiodes in Air

The III-nitride semiconductors have many attractive properties for field-emission vacuum electronics, including high thermal and chemical stability, low electron affinity, and high breakdown fields. In this work, we report top-down fabricated gallium nitride (GaN)-based nanoscale vacuum electron diodes operable in air, with record ultralow turn-on voltages down to ~0.24 V and stable high field-emission currents, tested up to several microamps for single-emitter devices. We leverage a scalable, top-down GaN nanofabrication method leading to damage-free and smooth surfaces. Gap-dependent and pressure-dependent studies provide new insights into the design of future, integrated nanogap vacuum electron devices. The results show promise for a new class of high-performance and robust, on-chip, III-nitride-based vacuum nanoelectronics operable in air or reduced vacuum.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2021

Bayesian Poroelastic Aquifer Characterization From InSAR Surface Deformation Data. 2. Quantifying the Uncertainty

Uncertainty quantification of groundwater (GW) aquifer parameters is critical for efficient management and sustainable extraction of GW resources. These uncertainties are introduced by the data, model, and prior information on the parameters. We develop a Bayesian inversion framework that uses Interferometric Synthetic Aperture Radar (InSAR) surface deformation data to infer the laterally heterogeneous permeability of a transient linear poroelastic model of a confined GW aquifer. The Bayesian solution of this inverse problem takes the form of a posterior probability density of the permeability. Exploring this posterior using classical Markov chain Monte Carlo (MCMC) methods is computationally prohibitive due to the large dimension of the discretized permeability field and the expense of solving the poroelastic forward problem. However, in many partial differential equation (PDE)-based Bayesian inversion problems, the data are only informative in a few directions in parameter space. For the poroelasticity problem, we prove this property theoretically for a one-dimensional problem and demonstrate it numerically for a three-dimensional aquifer model. Here we design a generalized preconditioned Crank-Nicolson (gpCN) MCMC method that exploits this intrinsic low dimensionality by using a low-rank-based Laplace approximation of the posterior as a proposal, which we build scalably. The feasibility of our approach is demonstrated through a real GW aquifer test in Nevada. The inherently two-dimensional nature of InSAR surface deformation data informs a sufficient number of modes of the permeability field to allow detection of major structures within the aquifer, significantly reducing the uncertainty in the pressure and the displacement quantities of interest.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

HydraGNN 2.0

HydraGNN is an Oak Ridge National Laboratory (ORNL)-branded implementation of distributed multi-tasking graph neural networks that supports several scientific applications within the ORNL portfolio to support the US-DOE mission. New or improved capabilities included in v2.0.0 release are as follows: 1) Enhancement in message passing layers through class inheritance 2) Adding transformation to ensure translation and rotation invariance 3) Supporting various optimizers 4) Atomic descriptors 5) Integration with continuous CI test 6) Distributed printouts and timers 7) Profiling 8) Support of ADIOS2 for scalable data loading 9) Large-scale system support, including Summit (ORNL) and Perlmutter (NERSC)

Lupo Pasini, Massimiliano [Oak Ridge National Labo↗
DOE OSTI2025

Hiperclust

This software leverages transfer learning to analyze atom probe tomography (APT) data. It is trained on synthetic data and then applies this knowledge to predict the optimal number of clusters for a given APT dataset. Initially, the software used preliminary clustering to estimate the general structure of the data. Based on this, it provides suggestions for key parameters like minimum cluster size and minimum number of points. These parameters are critical for algorithms like HDBSCAN, ensuring accurate cluster formation without the need for trial-and-error testing. The software runs on High-Performance computing (HPC) systems, enabling fast, scalable analysis of large APT datasets, ultimately saving time and improving the reliability of clustering outcomes.

Tang, Yalei [Idaho National Laboratory (INL), Idah↗
DOE OSTI2022

A Real-Time Data Dashboard for Monitoring Travel Behavior

When running a smartphone app-based travel behavior study, it is important to keep participants engaged in contributing data. Long study durations and/or large samples make it difficult to engage participants in reporting their travel behavior. We developed a dashboard web application called 'emdash' to facilitate and ease the data collection maintenance task. It allows deployers to view trip trajectories, track which participants are uploading data, and generate report ready plots. Emdash can also be useful for deployers to identify ways to best encourage a particular transportation mode shift and to provide feedback and reporting to participants. Using the dashboard, a single admin was able to support a classic travel study with 80 participants. The dashboard is currently deployed in a two-year electric bike (e-bike) pilot study called CanBikeCO, conducted in six locations across Colorado, USA. CanBikeCO targets carbon emissions reduction and equity improvement by evaluating changes to participants' travel behavior when they are offered free e-bikes. Each CanBikeCO location has access to their own version of emdash to help them monitor and support participants. After testing and gathering feedback, we made improvements to the dashboard's usability and scalability, and proposed a method to make detecting data issues easier. Improvements include configurable options and limiting the volume of data sent to the dashboard as study size and duration increase.

ADVANCED PROPULSION SYSTEMS,MATHEMATICS AND COMPUT↗
DOE OSTI2021

Scalable, Physical Effects Measureable Microgridfor Cyber Resilience Analysis

SPEMMCRA provides a low-cost simulated method to provide for testing of control theory and cybersecurity approaches to microgrid implementations. Specifically, this physics-based approach provides realistic responses to compare control strategies in simulation, but also to demonstrate actions that can or cannot be taken in taking cybersecurity actions . As communications interfaces or other actions are taken that affect the flow of data, SPEMMCRA can show whether the microgrid operation is adversely affected by what is intended to mitigate a cybersecurity event.

Rieger, CraigG↗
DOE OSTI2020

Dyno-in-the-Loop: An Innovative Hardware-in-the-Loop Development and Testing Platform for Emerging Mobility Technologies

Today’s transportation is quickly transforming with the nascent advent of connectivity, automation, shared-mobility, and electrification. These technologies will not only affect our safety and mobility, but also our energy consumption, and environment. As a result, it is of unprecedented importance to understand the overall system impacts due to the introduction of these emerging technologies and concepts. Existing modeling tools are not able to effectively capture the implications of these technologies, not to mention accurately and reliably evaluating their effectiveness with a reasonable scope. To address these gaps, a dynamometer-in-the-loop (DiL) development and testing approach is proposed which integrates test vehicle(s), chassis dynamometer, and high fidelity traffic simulation tools, in order to achieve a balance between the model accuracy and scalability of environmental analysis for the next generation of transportation systems. With this DiL platform, a connected eco-operation system for the plug-in hybrid electric bus (PHEB) has been developed and tested, which can optimize the vehicle dynamics (and potentially powertrain control via smart energy management) to reduce the operational energy consumption as well as tailpipe emissions of the target PHEB. The system performance has been evaluated on the DiL platform with respect to a variety of traffic congestion levels. The results have shown that the developed system can save fuel by more than 13% while reducing the electricity consumption by 2% in the test scenarios.

Wu, Guoyuan↗
DOE OSTI2021

Controlled Pyrolysis: A Robust Scalable Composite Recycling Technology

The reinforced composites industry is facing significant challenges in handling the scrap composite material from automobile manufacturing, the wind turbine industry, and others. The fibers in the material, whether they be carbon, glass, or other materials have commercial value if they can be recovered successfully. Successfully means the fibers are clean with no sizing or other binders and have adequate strength and physical properties that would allow them to be economically reprocessed into valuable product. The composites recycling project was an industry-collaborative effort to develop a composite recycling technology using controlled pyrolysis. Through the recycling of scrap and end-of-life (EOL) cured composite materials, this pilot study was intended to create a business case by realizing a cost-effective means for recycling EOL and production scrap composite materials, ultimately reducing the volume of composite materials destined for landfill. The project was led by the Institute for Advanced Composites Manufacturing Innovation (IACMI), the American Composites Manufacturers Association (ACMA), Oak Ridge National Laboratory (ORNL), Continental Structural Plastics (CSP) a Teijin Group Company, CHZ Technologies, and A. Schulman with support from Owens Corning, John Deere, General Electric (GE), Ashland LLC, and Plastics Europe (CEFIC). The team studied and tested CHZ Technologies’ controlled pyrolysis system, known as the Thermolyzer TM , which operates on a scalable basis to convert organic polymer materials into a clean synthesis gas and char containing the recoverable carbon and glass fiber reinforcement. The recoverable energy contained in the input polymers creates the synthesis gas that can be used to provide heat to the Thermolyzer TM primary reactor in a sustainable manner. That is, once the Thermolyzer TM is started with a small amount of external natural gas, the synthesis gas that is created from the polymers will continue to operate the burners so long as feedstock is supplied. The reinforcing fiber materials remaining in the solid phase char were separated and cleaned for re-use in other polymer systems based on the retained properties of the fibers. The study created reports (attached in the appendix) on the Mass and Energy Balances, syngas analytics, VOC assessment, yield analysis and other analytics necessary for a Techno-Economic Analysis (TEA) to quantify the economic impact of the recovery and sustainable re-use of the carbon and glass fibers. The process consisted of 4 steps: Selection of 4 samples of cured composite waste materials from project partners interested in materials recycling and recovering the reinforcing fibers for best case re-use. The materials included glass fiber (GF) polyester/vinyl ester automotive SMC from CSP, GF epoxy balsa/PVC foam wind blades from GE, carbon fiber (CF) epoxy wind blade laminated spar caps from GE, and GF/CF epoxy hybrid assembly from John Deere. Processing the waste composite samples into 1-2” shreds. Packaging the shredded composites into bulk sacks on international shipping pallets for shipment to KUG in Forst (Lausitz), Germany. Pyrolysis of the shredded composites under controlled conditions designed for each polymer system. Collecting samples of the gas and char for analysis. Shipping the char containing the CF/GF back to the US for the next steps of testing the fibers and developing protocols for sustainable re-use of the fibers in composite applications.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Tandem Photovoltaics Core Program Final Technical Report

The Tandem Photovoltaics Core Program was a multi-year initiative aimed at advancing hybrid tandem solar cell technologies to enhance solar module efficiency beyond the limits of single junction devices. This project focused on the development, testing, and scaling of prototype photovoltaic devices, with the goal of achieving commercial relevance and driving industry adoption. The work was divided into three tasks: 1) Comparative Analysis of Tandem Technologies: This task focused on quantifying energy yield under real-world conditions and assessing economic viability of tandems relative to silicon-based modules. The project's modeling framework incorporated performance data, cost of materials, and manufacturing process impacts to optimize tandem designs 2) Tandem Integration and Prototyping: In this task, we developed innovative tandem designs by combining metal halide perovskite (MHP) top cells and silicon (Si) bottom cells. The project focuses on both mechanical integration and direct deposition techniques to enable compatibility with commercially relevant Si technologies, such as passivated contact or PERC cells. 3) Scale-up and Reliability: This task addressed the challenges of large-area fabrication by developing scalable deposition methods and robust interconnection schemes for tandems. The project looked at different accelerated testing such as thermal cycling, damp heat exposure, and potential induced degradation, to ensure long-term stability of devices in field conditions. Tandem solar cells can greatly increase module efficiency beyond conventional single junction (SJ) devices, which are approaching their theoretical limit. There are many ways to fabricate a tandem cell or module in terms of materials used, configuration, and terminal connection. This SETO core project focused critical factors in enabling tandems to enter the market, including hardware integration, technoeconomic analysis (TEA), and energy yield analysis. We focused on MHP/Si hybrid tandem solar cells and modules as a model system for their versatility in module design comparisons, providing valuable insights for other tandem options. While champion cells with areas <1cm2 are regularly demonstrated by groups around the world, it is significantly more challenging to translate these advances into modules, and fewer institutions and companies are working at the module level. This project addressed questions about module fabrication, testing, and reliability that are hard to answer without actually fabricating prototypes. We also performed analysis and road-mapping activities to understand the potential for a wider variety of tandems, including all-perovskite tandems fabricated in collaboration with the Perovskite PV core program. Detailed technical results from this project are described for each task in Section 7.

14 SOLAR ENERGY↗
DOE OSTI2024

Flexible dynamic boundary microgrid operation considering network and load unbalances

Flexible microgrids with dynamic boundaries have recently been introduced in the literature. With the ability to reconfigure the topology of the microgrids dynamically through remotely controlled switches, flexible microgrids with dynamic boundaries can further improve the resiliency and energy efficiency of microgrids with distributed energy resources (DERs). This paper focuses on the optimal operation considering one of the predominant characteristics of microgrids and distribution systems – unbalanced networks and loads. In existing literature, balanced modeling of microgrids is more common due to its attractive simplicity. The three-phase power unbalance has not been considered as a constraint on the generation units in a microgrid. Further, negative sequence constraints have also been neglected. In this article, we propose a set of constraints that is specifically related to the capabilities of inverter interfaced resources to supply unbalanced current/power when the microgrid is islanded from the main distribution grid. We incorporate the new set of constraints into two optimization formulations leveraging two convex relaxations of the three-phase power flow equations: mixed-integer linear programming (MILP) and mixed-integer semidefinite programming (MISDP) that optimize the dispatch of controllable switches and DERs in the microgrid. The algorithms are then extended to networked microgrids with grid-forming sources. We test the algorithms on a realistic community microgrid model in Puerto Rico as well as standardized IEEE distribution test feeders. The testing results demonstrate the performance of the proposed algorithms. The MILP is fast and scalable, and the MISDP enforces the negative sequence voltage constraints.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2026

Stable yet hydrophilic graphene oxide nanomembranes by zwitterionic reduction for dye desalination

Holey graphene oxide (HGO) nanosheets have emerged as a promising membrane platform for dye desalination, and they must be reduced to enhance hydrophobicity and stability for long-term underwater operation, which usually decreases water and salt permeance. Herein, we develop a facile method to stabilize HGO nanosheets while retaining their hydrophilicity by reducing them with sulfobetaine amine (SBAm, a superhydrophilic zwitterion), achieving high water and salt permeance and a high salt/dye separation factor. Specifically, HGO nanosheets react with SBAm via an amine-epoxide reaction, rendering reduced HGO nanosheets containing superhydrophilic zwitterions. The effects of in-plane pores, zwitterion content, and layer thickness on the membrane chemistry, structure, and salt/dye separation properties of the SBAm-reduced HGOs (SHGOs) are thoroughly examined. The membrane achieves a Na 2 SO 4 /Direct red separation factor of up to 850, surpassing the state-of-the-art GO membranes. Moreover, hollow fiber membrane modules based on SHGOs are fabricated and exhibit stable performance in multi-cycle tests over 100 h of operation with mixed dye-salt solutions, demonstrating the scalability of our approach and its potential for practical applications.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Robust scalable initialization for Bayesian variational inference with multi-modal Laplace approximations

Predictive modeling typically relies on Bayesian model calibration to provide uncertainty quantification. Variational inference utilizing fully independent (“mean-field”) Gaussian distributions are often used as approximate probability density functions. This simplification is attractive since the number of variational parameters grows only linearly with the number of unknown model parameters. However, the resulting diagonal covariance structure and unimodal behavior can be too restrictive to provide useful approximations of intractable Bayesian posteriors that exhibit highly non-Gaussian behavior, including multimodality. High-fidelity surrogate posteriors for these problems can be obtained by considering the family of Gaussian mixtures. Gaussian mixtures are capable of capturing multiple modes and approximating any distribution to an arbitrary degree of accuracy, while maintaining some analytical tractability. Unfortunately, variational inference using Gaussian mixtures with full-covariance structures suffers from a quadratic growth in variational parameters with the number of model parameters. The existence of multiple local minima due to strong nonconvex trends in the loss functions often associated with variational inference present additional complications, These challenges motivate the need for robust initialization procedures to improve the performance and computational scalability of variational inference with mixture models. In this work, we propose a method for constructing an initial Gaussian mixture model approximation that can be used to warm-start the iterative solvers for variational inference. The procedure begins with a global optimization stage in model parameter space. In this step, local gradient-based optimization, globalized through multistart, is used to determine a set of local maxima, which we take to approximate the mixture component centers. Around each mode, a local Gaussian approximation is constructed via the Laplace approximation. Finally, the mixture weights are determined through constrained least squares regression. The robustness and scalability of the proposed methodology is demonstrated through application to an ensemble of synthetic tests using high-dimensional, multimodal probability density functions. Here, the practical aspects of the approach are demonstrated with inversion problems in structural dynamics.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

An implicit barotropic mode solver for MPAS-ocean using a modern Fortran solver interface

Here, we demonstrate use of a modern Fortran solver interface to manage solver algorithms for an implicit barotropic mode solver in the Model for Predictions Across Scales-Ocean (MPAS-O). ForTrilinos, a Fortran interface to Trilinos that contains a large collection of solver capabilities written in C++, has been implemented in MPAS-O to provide access to a suite of linear solver options. By virtue of the simplified wrapper and interface generator (SWIG) automation tool that generates modern Fortran interfaces to C++ code, we were able to implement the Fortran solver interface in MPAS-O using a familiar Fortran coding style while minimizing performance degradation. The ForTrilinos solver interface is written within MPAS-O’s time stepping modules as a subroutine in conjunction with MPAS-O code. Applied to an idealized ocean and a high-resolution realistic ocean test case, parallel performance of ForTrilinos solvers is examined. It is found that parallel scalability of the ForTrilinos solvers is highly dependent on the number of global synchronization points per solver iteration in each iterative solver algorithm. ForTrilinos solvers perform best compared to the Fortran hand-crafted (FHC) solver when the amount of work per processor is large enough. However, parallel scalability is better with the FHC solver and so when the work per core is modest FHC outperforms ForTrilinos. The intercomparison between the ForTrilinos and FHC solvers reveals that this performance hit in the ForTrilinos solver mostly comes from the global synchronization process, while suggesting that the matrix-vector multiplication process in the FHC solver needs to be optimized for better performance.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Enabling Parallel Execution of System-level Simulations in SAM

This report summarizes the recent code updates related to “element ghosting” in SAM to enable the parallel execution of system-level simulations using multiple processors/cores. Unlike typical MOOSE-based applications, for system-level simulations, SAM mostly deals with a collection of discrete small pieces of meshes, and the connection of physics on these meshes are realized by using “connector” types of components/code structures, such as conjugate heat transfer and flow junctions. The required code implementation is to correctly mark the necessary ghost elements for each type of such components/code structures; thus, the lower-level libraries can correctly perform the necessary data transfer between processors (CPUs) when executed in parallel mode. After the code updates, SAM can now run system-level simulations in the parallel mode. The parallel execution capability was then tested with an ABTR input model with 23k DOFs. Significant speedup was demonstrated when the optimal number of CPUs were used in parallel mode. Future systematic studies on parallelization performance using additional test cases covering different physics/scenarios will be needed to provide additional insights into the scalability of SAM parallelization.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Knowledge-Based Hazardous Waste Determinations for Solar Photovoltaic (PV) Modules: A Foundational Study

This report explores using "generator knowledge" to determine whether a solar photovoltaic (PV) module must be managed as hazardous waste prior to recycling or landfilling. Generator knowledge is a legal term and existing regulatory pathway for making a hazardous waste determination that has been used by other industries but is a relatively unknown option to the PV industry. In the United States, a hazardous waste determination often acts as a pre-requisite to recycle or landfill a PV module. The results of the hazardous waste determination dictate whether the PV module must be managed as hazardous waste or nonhazardous solid waste. Managing a PV module as hazardous waste requires compliance with stringent U.S. federal and state hazardous waste law. In addition to increased management costs, which can be ten times higher, legal liability for PV modules regulated as hazardous waste is also heightened with both civil and criminal penalties for noncompliance which includes making an inaccurate or faulty hazardous waste determination. The most common reason a PV module would be regulated as hazardous is if it contains a regulated metal in an amount that equals or exceeds the toxicity characteristic limits. To determine whether a PV module exhibits a hazardous characteristic, the regulated person/entity must "apply knowledge...in light of the materials and processes used." In the absence of adequate knowledge to determine whether the PV module is hazardous, it must be tested using Test Method 1311 Toxicity Characteristic Leaching Procedure (TCLP) or an equivalent EPA-approved method. Although TCLP is the predominant method used to today to make a hazardous waste determination for PV modules, evidence from this study concludes it is not practical to TCLP test every PV module even in a single utility-scale installation, and a scalable solution is needed. This study finds that knowledge-based hazardous waste determinations may allow a regulated person/entity to make a hazardous waste determination for more than one PV module at one time - making this regulatory pathway a potential scalable solution. Through legal analysis and interviews with 44 experts, the authors explore what it means to make an accurate knowledge-based hazardous determination for PV modules considering sources and forms of information as well as potential limitations. The work aims to provide a foundation for building consensus on whether knowledge-based hazardous determinations are a viable, scalable industry approach for solar.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2021

A Scalable Semi-Implicit Barotropic Mode Solver for the MPAS-Ocean

A scalable semi-implicit barotropic mode solver for the ocean component of the model for prediction across scales has been implemented as a competitor to an existing explicit-subcycling scheme to allow faster and more stable simulations while not sacrificing accuracy. The semi-implicit solver adopts the pipelined preconditioned bi-conjugate gradient stabilization algorithm as an iterative solver in conjunction with the restricted additive Schwarz preconditioner that accelerates the convergence rate of the iterative solver. The preconditioner is constructed from a linearized barotropic system that also reorders the system for optimal performance, while the semi-implicit solver deals with the fully nonlinear barotropic system that requires reassembly of the coefficient matrix for every time step. Several numerical experiments, from simple one-dimensional tests to three-dimensional real-world tests, demonstrate that the semi-implicit solver has almost the same accuracy and better parallel scalability compared with the existing scheme while allowing faster and more stable simulations. Furthermore, the semi-implicit solver accelerates the barotropic mode up to 2.9 times faster than the existing scheme on 16,320 processors, leading to an overall runtime speedup of 1.9.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

An FPGA-Based Neuromorphic Processor with All-to-All Connectivity

Neuromorphic computing is a promising paradigm for future energy-efficient computing. At present, however, it is in its nascent stages—most hardware implementations are research-grade, commercial products are not available, and the software tools are not production-ready. The lack of hardware and software tools makes neuromorphic computing inaccessible to researchers around the globe. To this extent, we intend to build a low-cost, open-source, FPGA-based digital neuromorphic processor that can be used by researchers worldwide. In this paper, we present a preliminary implementation of the processor on a Xilinx Artix-7 FPGA using SystemVerilog. Our implementation supports the integrate-and-fire neuron with two parameters each for neurons and synapses. It also features all-to-all connectivity among neurons on the hardware. We test our implementation on four cases: bars and stripes datasets, shortest path algorithm, logic gates, and 8-3 encoder. We also perform a scalability study to understand the resource utilization of the FPGA as the number of all-to-all connected neurons increases. With our implementation, the Artix- 7 supports 65 neurons with all-to-all connectivity. Moreover, all the test cases mentioned above achieve 100% accuracy.

Maheshwari, Disha↗