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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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NASA NTRS2001

Three Dimensional Atmospheric Radiative Transfer-Applications and Methods Comparison

We review applications of 3D radiative transfer in the atmosphere, emphasizing the wide spectrum of scales important to remote sensing and modeling of cloud fields, and the characteristic scales introduced into observed radiances and fluxes by the distribution of photon pathlengths at conservative and absorbing wavelengths. We define the "plane-parallel bias", which is a measure of the importance of 3D cloud structure in large-scale models, and the "independent pixel errors" that quantify the significance of 3D effects in remote sensing, and emphasize their relative magnitude and scale dependence. A variety of approaches in current use in 3D radiative transfer, and issues of speed, accuracy, and flexibility are summarized. We also describe a recently initiated "International Intercomparison of 3-Dimensional Radiation Codes", or I3RC. I3RC is a 3-phase effort that has as its goals to: (1) understand the errors and limits of 3D methods; (2) provide "baseline" cases for future 3D code development; (3) promote sharing of 3D tools; (4) derive guidelines for 3D tool selection; and (5) improve atmospheric science education in 3D radiative transfer. Selected results from Phases 1 and 2 of I3RC are discussed. These are taken from five cloud fields: a 1D field of bar clouds, a 2D radar-derived field, a 3D Landsat-derived field, a stratiform cloud from the model of C. Moeng, and a convective cloud from the model of B. Stevens. Computations have been carried out for three monochromatic wavelengths (one conservative, one absorptive, and one thermal) and two solar zenith angles (0, 60 degrees).

Cahalan, Robert F.↗
NASA NTRS2014

Towards Test Driven Development for Computational Science with pFUnit

Developers working in Computational Science & Engineering (CSE)/High Performance Computing (HPC) must contend with constant change due to advances in computing technology and science. Test Driven Development (TDD) is a methodology that mitigates software development risks due to change at the cost of adding comprehensive and continuous testing to the development process. Testing frameworks tailored for CSE/HPC, like pFUnit, can lower the barriers to such testing, yet CSE software faces unique constraints foreign to the broader software engineering community. Effective testing of numerical software requires a comprehensive suite of oracles, i.e., use cases with known answers, as well as robust estimates for the unavoidable numerical errors associated with implementation with finite-precision arithmetic. At first glance these concerns often seem exceedingly challenging or even insurmountable for real-world scientific applications. However, we argue that this common perception is incorrect and driven by (1) a conflation between model validation and software verification and (2) the general tendency in the scientific community to develop relatively coarse-grained, large procedures that compound numerous algorithmic steps.We believe TDD can be applied routinely to numerical software if developers pursue fine-grained implementations that permit testing, neatly side-stepping concerns about needing nontrivial oracles as well as the accumulation of errors. We present an example of a successful, complex legacy CSE/HPC code whose development process shares some aspects with TDD, which we contrast with current and potential capabilities. A mix of our proposed methodology and framework support should enable everyday use of TDD by CSE-expert developers.

pFUnit↗
NASA NTRS1985

Encoding Scheme Reduces Hardware Requirements

Shift register time-shared in proposed error-correcting code for tape-storage systems. Error-correcting code logic for blocks of 256-bit data words implemented by 14-stage-time-shared shift register, two 4,096bit RAM's and logic gates. Encoding and decoding logic implement generator polynomial that defines error-correcting code, and error correction based on algorithm also implemented by logic.

Sasnowitz, H.↗
NASA NTRS2000

Performance Assessment of OVERFLOW on Distributed Computing Environment

The aerodynamic computer code, OVERFLOW, with a multi-zone overset grid feature, has been parallelized to enhance its performance on distributed and shared memory paradigms. Practical application benchmarks have been set to assess the efficiency of code's parallelism on high-performance architectures. The code's performance has also been experimented with in the context of the distributed computing paradigm on distant computer resources using the Information Power Grid (IPG) toolkit, Globus. Two parallel versions of the code, namely OVERFLOW-MPI and -MLP, have developed around the natural coarse grained parallelism inherent in a multi-zonal domain decomposition paradigm. The algorithm invokes a strategy that forms a number of groups, each consisting of a zone, a cluster of zones and/or a partition of a large zone. Each group can be thought of as a process with one or multithreads assigned to it and that all groups run in parallel. The -MPI version of the code uses explicit message-passing based on the standard MPI library for sending and receiving interzonal boundary data across processors. The -MLP version employs no message-passing paradigm; the boundary data is transferred through the shared memory. The -MPI code is suited for both distributed and shared memory architectures, while the -MLP code can only be used on shared memory platforms. The IPG applications are implemented by the -MPI code using the Globus toolkit. While a computational task is distributed across multiple computer resources, the parallelism can be explored on each resource alone. Performance studies are achieved with some practical aerodynamic problems with complex geometries, consisting of 2.5 up to 33 million grid points and a large number of zonal blocks. The computations were executed primarily on SGI Origin 2000 multiprocessors and on the Cray T3E. OVERFLOW's IPG applications are carried out on NASA homogeneous metacomputing machines located at three sites, Ames, Langley and Glenn. Plans for the future will exploit the distributed parallel computing capability on various homogeneous and heterogeneous resources and large scale benchmarks. Alternative IPG toolkits will be used along with sophisticated zonal grouping strategies to minimize the communication time across the computer resources.

Djomehri, M. Jahed↗
NASA NTRS1997

OPAD-EDIFIS Real-Time Processing

The Optical Plume Anomaly Detection (OPAD) detects engine hardware degradation of flight vehicles through identification and quantification of elemental species found in the plume by analyzing the plume emission spectra in a real-time mode. Real-time performance of OPAD relies on extensive software which must report metal amounts in the plume faster than once every 0.5 sec. OPAD software previously written by NASA scientists performed most necessary functions at speeds which were far below what is needed for real-time operation. The research presented in this report improved the execution speed of the software by optimizing the code without changing the algorithms and converting it into a parallelized form which is executed in a shared-memory multiprocessor system. The resulting code was subjected to extensive timing analysis. The report also provides suggestions for further performance improvement by (1) identifying areas of algorithm optimization, (2) recommending commercially available multiprocessor architectures and operating systems to support real-time execution and (3) presenting an initial study of fault-tolerance requirements.

Katsinis, Constantine↗
NASA NTRS1998

Navier-Stokes Computations on Commodity Computers

In this paper we discuss and demonstrate the feasibility of solving high-fidelity, nonlinear computational fluid dynamics (CFD) problems of practical interest on commodity machines, namely Pentium Pro PC's. Such calculations have now become possible due to the progress in computational power and memory of the off-the-shelf commodity computers, along with the growth in bandwidth and communication speeds of networks. A widely used CFD code known as TLNS3D, which was developed originally on large shared memory computers was selected for this effort. This code has recently been ported to massively parallel processor (MPP) type machines, where natural partitioning along grid blocks is adopted in which one or more blocks are distributed to each of the available processors. In this paper, a similar approach is adapted to port this code to a cluster of Pentium Pro computers. The message passing among the processors is accomplished through the use of standard message passing interface (MPI) libraries. Scaling studies indicate fairly high level of parallelism on such clusters of commodity machines, thus making solutions to Navier-Stokes equations for practical problems more affordable.

Vatsa, Veer N.↗
NASA NTRSDate not supplied

Developing Open-Source Training Materials for AI/ML and Space Biological Sciences Using NASA Cloud-Based Data

Artificial Intelligence (AI) and Machine Learning (ML) has gained significant traction in the biological and biomedical research fields in the last two decades, in part thanks to an increasing culture of open data sharing and reuse. Due to its capability for identifying complex relationships and patterns, AI/ML methodology is particularly well suited to recognize and predict biological patterns from high-dimensional next-generation sequencing data (e.g. whole genome sequencing, transcriptomic sequencing), as well as from biological or medical imaging data (e.g. microscopy, computed tomography, ultrasound, magnetic resonance imaging, radiography). These methodologies hold particular promise for space biosciences research and automated space health monitoring systems. However, there are many key considerations for properly training, validating, and testing a machine learning model in biological research or clinical application. Even with the positive culture of Open Science and data sharing, inexperienced researchers working quickly without proper checks can produce models that perform poorly outside of the immediate training dataset. Lessons learned from biological AI/ML research indicate that Open Science principles such as data sharing and open-source code must go hand-in-hand with publicly available, high-quality training curricula in best practices, with modules centered on real-life scientific use cases and data so future AI/ML practitioners gain experience on real problems. Here we present the development of open-source training materials for AI/ML and space biosciences, as part of the NASA Transform to Open Science Training (TOPST) initiative. We develop 4 independent training programs, focused on the following topics: 1) Fundamentals of Machine Learning and Space Biosciences Domain, 2) Open Science, Artificial Intelligence, and Ethical Best Practices for Data Sharing and Analysis, 3) Using AI/ML Classification to Identify Gene Networks Affected By Space Exposure in Mouse Liver, and 4) Using Neural Networks to Find DNA Damage Patterns in Immune Cells after Radiation. All programs leverage cloud-based NASA biological datasets. The curriculum we present will enable worldwide access to training in AI/ML and scientific analysis.

James Andrew Casaletto↗
NASA NTRSDate not supplied

Developing Open-Source Training Materials for AI/ML and Space Biological Sciences Using NASA Cloud-Based Data

Artificial Intelligence (AI) and Machine Learning (ML) has gained significant traction in the biological and biomedical research fields, in part due to a culture of open data sharing and reuse. AI/ML methodology is well-suited to recognize and predict biological patterns from high-dimensional next-generation sequencing data (e.g. whole genome sequencing, transcriptomic sequencing), as well as from biological or medical imaging data (e.g. microscopy, computed tomography, ultrasound, magnetic resonance imaging, radiography). These methodologies hold particular promise for space biosciences research and automated space health monitoring systems. However, there are key considerations for properly training, validating, and testing a machine learning model in biological research or clinical application. Inexperienced researchers can produce models that perform poorly outside of the training dataset. Open Science principles such as data sharing and open-source code must go hand-in-hand with publicly available, high-quality training curricula in best practices, with modules centered on real-life scientific use cases and data so future AI/ML practitioners gain experience on real problems. Here we present the development of open-source training materials for AI/ML and space biosciences, as part of the NASA Transform to Open Science Training (TOPST) initiative. We develop 4 independent training programs, focused on the following topics: 1) Fundamentals of Machine Learning and Space Biosciences Domain, 2) Open Science, Artificial Intelligence, and Ethical Best Practices for Data Sharing and Analysis, 3) Using AI/ML Classification to Identify Gene Networks Affected By Space Exposure in Mouse Liver, and 4) Using Neural Networks to Find DNA Damage Patterns in Immune Cells after Radiation. All programs leverage cloud-based NASA biological datasets. The curriculum we present will enable worldwide access to training in AI/ML and scientific analysis.

James Casaletto↗
NASA NTRS1998

Interfacing with Legacy using Remote Method Invocation

The assignment described was enough to make a neophyte Java developer bolt for the door: provide a remote method for use by an applet which invokes a native method that wraps a function in an existing legacy library. The purpose of the remote method is to return an instance of a class object whose contents reflect the data structure returned by the legacy function. While embroiled in implementation, I would have spent the time wading through their JNI use group archive as well, but I couldn't seem to locate one. Subsequently, I made the decision to try to document my findings in order to assist others. Before we start on the class design, let's look at what the existing legacy code does. The C function to be called, Get-Legacy-Data, consists of two steps: an ASII file is read from the local disk and its contents are parsed into a Legacy_Type structure whose address is passed as an argument by the caller. The legacy code was compiled into a shared object library, legacy. so, using the IRIX 6.2 compiler and then loaded onto the Web server, a Silicon Graphics Indy station loaded with the IRIX 6.4 operating system. As far as the class design is concerned, the first thing required is a class to act as a template for the data structure returned by the legacy function. This class, JLegacy, declares a series of public instance variables which correspond to the members of Legacy_Type and provides a parameterless constructor. This constructor is never called, not even by the native method which allocates the object for return to the remote method. Next, the remote interface declaration for the remote object must be defined. In order for JLegacyRO to implement getJLegacy, JLegacyRO must interface with the existing legacy code through a native method, getn. getn is declared in the JLegacyRO class but implemented in C, just like the legacy code. getn returns a JLegacy instance and is declared static since its implementation is the same for all instances of the JLegacyRO class.

Howard, Scott M.↗
NASA NTRS1983

On Why It Is Impossible to Prove that the BDX90 Dispatcher Implements a Time-sharing System

The Software Implemented Fault Tolerance SIFT system, is written in PASCAL except for about a page of machine code. The SIFT system implements a small time sharing system in which PASCAL programs for separate application tasks are executed according to a schedule with real time constraints. The PASCAL language has no provision for handling the notion of an interrupt such as the B930 clock interrupt. The PASCAL language also lacks the notion of running a PASCAL subroutine for a given amount of time, suspending it, saving away the suspension, and later activating the suspension. Machine code was used to overcome these inadequacies of PASCAL. Code which handles clock interrupts and suspends processes is called a dispatcher. The time sharing/virtual machine idea is completely destroyed by the reconfiguration task. After termination of the reconfiguration task, the tasks run by the dispatcher have no relation to those run before reconfiguration. It is impossible to view the dispatcher as a time-sharing system implementing virtual BDX930s running concurrently when one process can wipe out the others.

Boyer, R. S.↗
NASA NTRS1995

MODTRAN3: An update and recent validations against airborne high resolution interferometer measurements

MODTRAN, the Moderate Resolution Atmospheric Radiance and Transmittance Model, encompasses all the capabilities of LOWTRAN 7, the widely used 20 cm(exp -1) resolution radiance code, but incorporates a much more sensitive molecular band model with 2 cm(exp -1) resolution. MODTRAN contains many important elements that other band model based radiative transfer codes do not incorporate. It shares with FASCODE: spherical geometry, single and multiple scattering default atmospheric profile descriptors (gases, aerosols, clouds, fogs, and rain), and molecular continua (H2O, CO2, O3, O2, N2). In addition, it can calculate the solar/lunar direct and scattered radiation. MODTRAN3 was released to the general public in November 1994. It has several important features that the previous version, MODTRAN2, does not have. Chloro-fluorocarbon (CFC) and related heavy molecules (whose spectroscopic properties first appear on the HITRAN92 data base as temperature-dependent cross sections) have been incorporated into pseudo-band models, with provision for using both default and user supplied profiles. The addition of SO2 and O2 in the UV, along with upgraded ozone Chappuis bands in the visible is also part of MODTRAN3. An improved multiple scattering algorithm, the DIScrete Ordinate Radiative Transfer (DISORT) has also been incorporated into MODTRAN3. MODTRAN is very fast: simple timing runs of MODTRAN3 vs. FASCOD3 show an improvement of more than a factor of 100 for a typical 500 cm(exp -1) spectral interval and comparable vertical layering. Speed is an important consideration in heating/cooling rates calculations, where a large number of radiative transfer calculations are needed. The MODTRAN3 used in this study is based on HITRAN92, but as mentioned, above, it will be upgraded to HITRAN94 upon its release at the end of 1994. MODTRAN has been adopted by: some researchers in the AVIRIS program as one radiative transfer code to derive surface reflectance from AVIRIS measurements. The accuracy of the code is very important because any errors in the radiative transfer calculation will directly translate into errors in the derived surface reflectance. In this paper, the new solar irradiance calculated by Kurucz, which is adopted in MODTRAN3, will be presented. Recent validations of MODTRAN3 with airborne high resolution interferometer measurements over ocean will be discussed. Good agreeement between model calculations and measurements was achieved.

Anderson, Gail P.↗
NASA NTRS2012

Production Level CFD Code Acceleration for Hybrid Many-Core Architectures

In this work, a novel graphics processing unit (GPU) distributed sharing model for hybrid many-core architectures is introduced and employed in the acceleration of a production-level computational fluid dynamics (CFD) code. The latest generation graphics hardware allows multiple processor cores to simultaneously share a single GPU through concurrent kernel execution. This feature has allowed the NASA FUN3D code to be accelerated in parallel with up to four processor cores sharing a single GPU. For codes to scale and fully use resources on these and the next generation machines, codes will need to employ some type of GPU sharing model, as presented in this work. Findings include the effects of GPU sharing on overall performance. A discussion of the inherent challenges that parallel unstructured CFD codes face in accelerator-based computing environments is included, with considerations for future generation architectures. This work was completed by the author in August 2010, and reflects the analysis and results of the time.

Duffy, Austen C.↗
NASA NTRS1994

DTS: Building custom, intelligent schedulers

DTS is a decision-theoretic scheduler, built on top of a flexible toolkit -- this paper focuses on how the toolkit might be reused in future NASA mission schedulers. The toolkit includes a user-customizable scheduling interface, and a 'Just-For-You' optimization engine. The customizable interface is built on two metaphors: objects and dynamic graphs. Objects help to structure problem specifications and related data, while dynamic graphs simplify the specification of graphical schedule editors (such as Gantt charts). The interface can be used with any 'back-end' scheduler, through dynamically-loaded code, interprocess communication, or a shared database. The 'Just-For-You' optimization engine includes user-specific utility functions, automatically compiled heuristic evaluations, and a postprocessing facility for enforcing scheduling policies. The optimization engine is based on BPS, the Bayesian Problem-Solver (1,2), which introduced a similar approach to solving single-agent and adversarial graph search problems.

Hansson, Othar↗
NASA NTRS2001

Comparison of 250 MHz R10K Origin 2000 and 400 MHz Origin 2000 Using NAS Parallel Benchmarks

This report describes results of benchmark tests on Steger, a 250 MHz Origin 2000 system with R10K processors, currently installed at the NASA Ames National Advanced Supercomputing (NAS) facility. For comparison purposes, the tests were also run on Lomax, a 400 MHz Origin 2000 with R12K processors. The BT, LU, and SP application benchmarks in the NAS Parallel Benchmark Suite and the kernel benchmark FT were chosen to measure system performance. Having been written to measure performance on Computational Fluid Dynamics applications, these benchmarks are assumed appropriate to represent the NAS workload. Since the NAS runs both message passing (MPI) and shared-memory, compiler directive type codes, both MPI and OpenMP versions of the benchmarks were used. The MPI versions used were the latest official release of the NAS Parallel Benchmarks, version 2.3. The OpenMP versions used were PBN3b2, a beta version that is in the process of being released. NPB 2.3 and PBN3b2 are technically different benchmarks, and NPB results are not directly comparable to PBN results.

Turney, Raymond D.↗
NASA NTRS2001

Comparison of Origin 2000 and Origin 3000 Using NAS Parallel Benchmarks

This report describes results of benchmark tests on the Origin 3000 system currently being installed at the NASA Ames National Advanced Supercomputing facility. This machine will ultimately contain 1024 R14K processors. The first part of the system, installed in November, 2000 and named mendel, is an Origin 3000 with 128 R12K processors. For comparison purposes, the tests were also run on lomax, an Origin 2000 with R12K processors. The BT, LU, and SP application benchmarks in the NAS Parallel Benchmark Suite and the kernel benchmark FT were chosen to determine system performance and measure the impact of changes on the machine as it evolves. Having been written to measure performance on Computational Fluid Dynamics applications, these benchmarks are assumed appropriate to represent the NAS workload. Since the NAS runs both message passing (MPI) and shared-memory, compiler directive type codes, both MPI and OpenMP versions of the benchmarks were used. The MPI versions used were the latest official release of the NAS Parallel Benchmarks, version 2.3. The OpenMP versiqns used were PBN3b2, a beta version that is in the process of being released. NPB 2.3 and PBN 3b2 are technically different benchmarks, and NPB results are not directly comparable to PBN results.

Turney, Raymond D.↗
NASA NTRS2003

Large-Eddy Simulation Code Developed for Propulsion Applications

A large-eddy simulation (LES) code was developed at the NASA Glenn Research Center to provide more accurate and detailed computational analyses of propulsion flow fields. The accuracy of current computational fluid dynamics (CFD) methods is limited primarily by their inability to properly account for the turbulent motion present in virtually all propulsion flows. Because the efficiency and performance of a propulsion system are highly dependent on the details of this turbulent motion, it is critical for CFD to accurately model it. The LES code promises to give new CFD simulations an advantage over older methods by directly computing the large turbulent eddies, to correctly predict their effect on a propulsion system. Turbulent motion is a random, unsteady process whose behavior is difficult to predict through computer simulations. Current methods are based on Reynolds-Averaged Navier- Stokes (RANS) analyses that rely on models to represent the effect of turbulence within a flow field. The quality of the results depends on the quality of the model and its applicability to the type of flow field being studied. LES promises to be more accurate because it drastically reduces the amount of modeling necessary. It is the logical step toward improving turbulent flow predictions. In LES, the large-scale dominant turbulent motion is computed directly, leaving only the less significant small turbulent scales to be modeled. As part of the prediction, the LES method generates detailed information on the turbulence itself, providing important information for other applications, such as aeroacoustics. The LES code developed at Glenn for propulsion flow fields is being used to both analyze propulsion system components and test improved LES algorithms (subgrid-scale models, filters, and numerical schemes). The code solves the compressible Favre-filtered Navier- Stokes equations using an explicit fourth-order accurate numerical scheme, it incorporates a compressible form of Smagorinsky s model for the subgrid-scale turbulence, and it uses generalized curvilinear coordinates to allow analysis of a wide range of geometries. The code runs in parallel on shared memory multiprocessor computers and is written in Fortran 90 with dynamic memory allocation. A sample result for a Mach-1.4 round jet is presented in the figure. Instantaneous Mach number contours in several cross-planes downstream of the nozzle exit are shown, illustrating how an LES captures the large unsteady three-dimensional turbulent structures present in the jet.

DeBonis, James R.↗
NASA NTRS2000

Performance of the OVERFLOW-MLP and LAURA-MLP CFD Codes on the NASA Ames 512 CPU Origin System

The shared memory Multi-Level Parallelism (MLP) technique, developed last year at NASA Ames has been very successful in dramatically improving the performance of important NASA CFD codes. This new and very simple parallel programming technique was first inserted into the OVERFLOW production CFD code in FY 1998. The OVERFLOW-MLP code's parallel performance scaled linearly to 256 CPUs on the NASA Ames 256 CPU Origin 2000 system (steger). Overall performance exceeded 20.1 GFLOP/s, or about 4.5x the performance of a dedicated 16 CPU C90 system. All of this was achieved without any major modification to the original vector based code. The OVERFLOW-MLP code is now in production on the inhouse Origin systems as well as being used offsite at commercial aerospace companies. Partially as a result of this work, NASA Ames has purchased a new 512 CPU Origin 2000 system to further test the limits of parallel performance for NASA codes of interest. This paper presents the performance obtained from the latest optimization efforts on this machine for the LAURA-MLP and OVERFLOW-MLP codes. The Langley Aerothermodynamics Upwind Relaxation Algorithm (LAURA) code is a key simulation tool in the development of the next generation shuttle, interplanetary reentry vehicles, and nearly all "X" plane development. This code sustains about 4-5 GFLOP/s on a dedicated 16 CPU C90. At this rate, expected workloads would require over 100 C90 CPU years of computing over the next few calendar years. It is not feasible to expect that this would be affordable or available to the user community. Dramatic performance gains on cheaper systems are needed. This code is expected to be perhaps the largest consumer of NASA Ames compute cycles per run in the coming year.The OVERFLOW CFD code is extensively used in the government and commercial aerospace communities to evaluate new aircraft designs. It is one of the largest consumers of NASA supercomputing cycles and large simulations of highly resolved full aircraft are routinely undertaken. Typical large problems might require 100s of Cray C90 CPU hours to complete. The dramatic performance gains with the 256 CPU steger system are exciting. Obtaining results in hours instead of months is revolutionizing the way in which aircraft manufacturers are looking at future aircraft simulation work. Figure 2 below is a current state of the art plot of OVERFLOW-MLP performance on the 512 CPU Lomax system. As can be seen, the chart indicates that OVERFLOW-MLP continues to scale linearly with CPU count up to 512 CPUs on a large 35 million point full aircraft RANS simulation. At this point performance is such that a fully converged simulation of 2500 time steps is completed in less than 2 hours of elapsed time. Further work over the next few weeks will improve the performance of this code even further.The LAURA code has been converted to the MLP format as well. This code is currently being optimized for the 512 CPU system. Performance statistics indicate that the goal of 100 GFLOP/s will be achieved by year's end. This amounts to 20x the 16 CPU C90 result and strongly demonstrates the viability of the new parallel systems rapidly solving very large simulations in a production environment.

Taft, James R.↗
NASA NTRS1989

Ground-state coding in partially connected neural networks

Patterns over (-1,0,1) define, by their outer products, partially connected neural networks, consisting of internally strongly connected, externally weakly connected subnetworks. The connectivity patterns may have highly organized structures, such as lattices and fractal trees or nests. Subpatterns over (-1,1) define the subcodes stored in the subnetwork, that agree in their common bits. It is first shown that the code words are locally stable stares of the network, provided that each of the subcodes consists of mutually orthogonal words or of, at most, two words. Then it is shown that if each of the subcodes consists of two orthogonal words, the code words are the unique ground states (absolute minima) of the Hamiltonian associated with the network. The regions of attraction associated with the code words are shown to grow with the number of subnetworks sharing each of the neurons. Depending on the particular network architecture, the code sizes of partially connected networks can be vastly greater than those of fully connected ones and their error correction capabilities can be significantly greater than those of the disconnected subnetworks. The codes associated with lattice-structured and hierarchical networks are discussed in some detail.

Baram, Yoram↗