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At least 91 records · Page 5

G-Mapper: Learning a Cover in the Mapper Construction

The Mapper algorithm is a visualization technique in topological data analysis (TDA) that outputs a graph reflecting the structure of a given dataset. However, the Mapper algorithm requires tuning several parameters in order to generate a “nice” Mapper graph. This paper focuses on selecting the cover parameter. We present an algorithm that optimizes the cover of a Mapper graph by splitting a cover repeatedly according to a statistical test for normality. Our algorithm is based on G-means clustering, which searches for the optimal number of clusters in 𝑘-means by iteratively applying the Anderson–Darling test. Our splitting procedure employs a Gaussian mixture model to carefully choose the cover according to the distribution of the given data. In conclusion, experiments for synthetic and real-world datasets demonstrate that our algorithm generates covers so that the Mapper graphs retain the essence of the datasets, while also running significantly faster than a previous iterative method.

G-means clustering↗

Q-BEEP: Quantum Bayesian Error Mitigation Employing Poisson Modeling over the Hamming Spectrum

Quantum computing technology has grown rapidly in recent years, with new technologies being explored, error rates being reduced, and quantum processor’s qubit capacity growing. However, near-term quantum algorithms are still unable to be induced without compounding consequential levels of noise, leading to non-trivial erroneous results. Quantum Error Correction (in-situ error mitigation) and Quantum Error Mitigation (post-induction error mitigation) are promising fields of research within the quantum algorithm scene, aiming to alleviate quantum errors, increasing the overall fidelity and hence the overall quality of circuit induction. Earlier this year, a pioneering work, namely HAMMER, published in ASPLOS-22 demonstrated the existence of a latent structure regarding post-circuit induction errors when mapping to the Hamming spectrum. However, they intuitively assumed that errors occur in local clusters, and that at higher average Hamming distances this structure falls away. In this work, we show that such a correlation structure is not only local but extends certain non-local clustering patterns which can be precisely described by a Poisson distribution model taking the input circuit, the device run time status (i.e., calibration statistics) and qubit topology into consideration. Using this quantum error characterizing model, we developed an iterative algorithm over the generated Bayesian network state-graph for post-induction error mitigation. Thanks to more precise modeling of the error distribution latent structure and the new iterative method, our Q-Beep approach provides state of the art performance and can boost circuit execution fidelity by up to 234.6% on Bernstein-Vazirani circuits and on average 71.0% on QAOA solution quality, using 16 practical IBMQ quantum processors. For other benchmarks such as those in QASMBench, the fidelity improvement is up to 17.8%. Q-Beep is a light-weight post-processing technique that can be performed offline and remotely, making it a useful tool for quantum vendors to integrate and provide more reliable circuit induction results.

Stein, Samuel A.↗

PNNL-CompBio/bayesian-inference

Bayesian Metabolic Inference is a platform to integrate genome-scale multi-omics data into an iterative method to obtain optimized recommendations for optimal target compound production. This method uses a variational inference approach to approximate new posterior data to analyze in order to obtain useful correlations and control coefficients.

Kumar, Neeraj↗

Extension of the PINN diffusion model to k-eigenvalue problems

This paper extends our recent work on the Physics-Informed Neural Networks (PINN) approach for the fixed source diffusion models and applies it to the diffusion theory based k-eigenvalue problems. To make the PINN equitable for the eigenvalue problems, we introduce a novel integral regularization term to the loss function in the framework, and allow the direct inference of the principal eigenvalue and the associated eigenfunction. The regularization term enforces a pre-defined value on the integration of the model predictions, and this value can be directly related to a physical property of the system. We also introduce an additional learnable parameter to approximate the principal eigenvalue. As a proof of principle, we solve the one-group two-dimensional k-eigenvalue neutron diffusion equation in this work. We then provide two numerical examples to demonstrate the applicability of the PINN approach. In each example, we solve the k-eigenvalue diffusion equation in a multi-region configuration constrained with a set of Robin boundary conditions for generality. We use a FEM solution based on the power-iteration method to verify the results of the PINN solution. The results showed relative percentage error in the predicted eigenvalue of about 0.77% and about 1.2% for example 1 and example 2, respectively. The mean absolute error in the predicted flux for example 1 is ∼ 0.002 and for example 2 is ∼ 0.0024. These results indicate some preliminary successes of the PINN application to k-eigenvalue problems. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Self-supervised physics-informed generative networks for phase retrieval from a single X-ray hologram

X-ray phase contrast imaging significantly improves the visualization of structures with weak or uniform absorption, broadening its applications across a wide range of scientific disciplines. Propagation-based phase contrast is particularly suitable for time- or dose-critical in vivo/in situ/operando (tomography) experiments because it requires only a single intensity measurement. However, the phase information of the wave field is lost during the measurement and must be recovered. Conventional algebraic and iterative methods often rely on specific approximations or boundary conditions that may not be met by many samples or experimental setups. In addition, they require manual tuning of reconstruction parameters by experts, making them less adaptable for complex or variable conditions. Here we present a self-learning approach for solving the inverse problem of phase retrieval in the near-field regime of Fresnel theory using a single intensity measurement (hologram). A physics-informed generative adversarial network is employed to reconstruct both the phase and absorbance of the unpropagated wave field in the sample plane from a single hologram. Unlike most state-of-the-art deep learning approaches for phase retrieval, our approach does not require paired, unpaired, or simulated training data. This significantly broadens the applicability of our approach, as acquiring or generating suitable training data remains a major challenge due to the wide variability in sample types and experimental configurations. The algorithm demonstrates robust and consistent performance across diverse imaging conditions and sample types, delivering quantitative, high-quality reconstructions for both simulated data and experimental datasets acquired at beamline P05 at PETRA III (DESY, Hamburg), operated by Helmholtz-Zentrum Hereon. Furthermore, it enables the simultaneous retrieval of both phase and absorption information.

36 MATERIALS SCIENCE↗

Asymptotic Expansion of the Impedance Per Unit Length for Rectangular Conductors

An iteration method is introduced to obtain the asymptotic form of the impedance per unit length of a rectangular conductor when the half side lengths are large compared to the skin depth. The first terms of the asymptotic expansion are extracted in closed form. The manner in which the corner corrections fit into the expansion are illustrated. The asymptotic results are compared to a numerical solution in the square limit. The odd corner correction for a right angle edge is also discussed.

42 ENGINEERING↗

Impact of Time-Dependent Reactor and Sensor Physics on Core Power Synthesis (Rev.1)

Online synthesis of power distribution is critical in the operation and control of nuclear power reactors to ensure that the core is operating within safety margins and to provide essential knowledge associated with the burnup of the fuel. In light-water reactors, power synthesis is achieved by using some a priori knowledge of the state of the reactor core and updating based on the signals coming from in-core sensors—namely, self-powered neutron detectors (SPNDs). This report examines the effects of fuel burnup and sensor degradation on the ability to accurately synthesize the power distribution in a pressurized water reactor (PWR), considering the typical low-enriched uranium (LEU, 3%-5% enrichment) fuel cycle as well as the higher enrichment LEU+ (5%-8% enrichment) fuel cycle. Several modeling tools were used to simulate power synthesis based on the responses of SPNDs, with emitters made out of Rh or V. A representative PWR LEU core was modeled using the Polaris/Purdue Advanced Reactor Core Simulator (PARCS) approach. The Monte Carlo N-Particle Transport 6 (MCNP6) code was used as well to calculate response functions between different segments of fuel to individual SPNDs; this is a crucial parameter for power synthesis. The Oak Ridge Isotope GENeration (ORIGEN) package in the Standardized Computer Analyses for Licensing Evaluation (SCALE) code was used to model the time-dependent isotopic transmutation in the SPND emitters. All these data were fed into a custom code that enacted the point-based iterative method to simulate power synthesis. Developmental work was also performed on high-fidelity SPND models in the GEometry ANd Tracking 4 (Geant4) code, which enables higher-accuracy modeling of the current responses from SPNDs. In this work, five sets of time-dependent power synthesis test cases were conducted. In these test cases, systematic changes in the input conditions enabled an analysis of the effect of (1) slightly inaccurate a priori power distribution assumptions with respect to fuel burnup, (2) highly inaccurate a priori power distribution assumptions with respect to fuel burnup (such that burnup is not included in the a priori assumed distribution), and (3) differences between Rh and V SPNDs in terms of downstream consequences of the transmutation in the emitters and the extended nature of the LEU+ fuel cycle in comparison with LEU. The authors discovered that one may permissibly have slightly inaccurate a priori assumptions of the fuel burnup (such that the level of burnup may be slightly underapproximated or overapproximated by the accumulated burnup in approximately 9.3 full power days), but to not account for burnup at all in the a priori assumptions leads to severe levels of error, approaching 25% at maximum (for LEU). The authors also discovered that V SPNDs are extraordinarily robust in both the LEU and LEU+ fuel cycles considered in this modeling work, whereas Rh SPNDs undergo significant transmutation that can result in large errors in the synthesized power distribution.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

An iterative dynamic chemical stiffness removal method for reacting flow simulations

Abstract An iterative dynamic chemical stiffness removal method (IDCSR) based on quasi-steady-state approximation (QSSA) is proposed. The IDCSR method is built on a previously developed non-iterative method which has proved to work well for small timestep sizes. A novel iterative procedure is designed in IDCSR to enable explicit time integration of stiff chemistry at relatively large timestep sizes relevant to practical reacting flow simulations. The effectiveness of the iterative procedure is first demonstrated with a toy problem and homogeneous auto-ignition with fixed integration step sizes, showing that larger timestep sizes can be allowed for explicit time integration using IDCSR compared with the previous non-iterative method. IDCSR is then compared with existing explicit chemistry solvers for simulations of homogeneous auto-ignition and shows similar or lower computational cost but significantly higher accuracy across a wide range of timestep sizes. IDCSR is further combined with an automatic adaptive time-stepping scheme for simulations of 0-D homogeneous auto-ignition and a 2-D laminar lifted n -dodecane jet flame. For the 0-D auto-ignition simulations, IDCSR is shown to reduce both the error (by 43%–90%) and computational cost (by 6–15 times) compared with existing explicit solvers, while achieving speed-up factors of up to 400 compared with VODE for a wide range of timestep sizes and reaction mechanisms. For the 2-D jet flame simulations, speed-up factors of 15 and 31 for chemistry integration, and 5 and 9 for overall simulation, are achieved by IDCSR compared with CVODE with and without analytic Jacobian, respectively.

Xu, Chao (ORCID:0000000153074159)↗

Efficient and robust phase-split computations in the internal energy, volume, and moles ( UVN ) space

Phase-split computations in an isolated system, which in general may be defined as one where the total internal energy (U), volume (V), and the number of moles (N = N 1 , N 2 ,..., N n ) of the components are fixed at some specified set of values, involves the determination of the temperature (T), pressure (P), plus the amount and composition of the various phases that constitute the system. A simpler, but analogous problem is one where T, P, and N are specified instead. In TPN space, one may first perform a stability analysis to determine whether the system is stable, meaning whether at equilibrium it will split up into multiple phases or remain in single phase. If the single-phase state is unstable, the stability analysis reliably provides a good set of initial guesses in the subsequent phase-split computations. In UVN space, however, we demonstrate that the stability analysis (which is the main subject of our earlier study [1]) may not in general provide good enough initial guesses; we offer alternative strategies for setting up good initial guesses. Furthermore, we show that a combination of successive substitution iteration (SSI) and Newton's method---two iterative methods that are prevalent in the literature in TPN space---facilitates a robust and efficient algorithm for phase-split computations in isolated systems. Finally, this combination has so far not been applied in UVN space.

02 PETROLEUM↗

AutoPhaseNN: unsupervised physics-aware deep learning of 3D nanoscale Bragg coherent diffraction imaging

Abstract The problem of phase retrieval underlies various imaging methods from astronomy to nanoscale imaging. Traditional phase retrieval methods are iterative and are therefore computationally expensive. Deep learning (DL) models have been developed to either provide learned priors or completely replace phase retrieval. However, such models require vast amounts of labeled data, which can only be obtained through simulation or performing computationally prohibitive phase retrieval on experimental datasets. Using 3D X-ray Bragg coherent diffraction imaging (BCDI) as a representative technique, we demonstrate AutoPhaseNN, a DL-based approach which learns to solve the phase problem without labeled data. By incorporating the imaging physics into the DL model during training, AutoPhaseNN learns to invert 3D BCDI data in a single shot without ever being shown real space images. Once trained, AutoPhaseNN can be effectively used in the 3D BCDI data inversion about 100× faster than iterative phase retrieval methods while providing comparable image quality.

36 MATERIALS SCIENCE↗

Particulate Fuel Modeling of MC 2 -3 using Iterative Local Spatial Self-shielding Method

We report a new spatial self-shielding method for particulate fuels has been developed based on disadvantage factors and implemented in the MC 2 -3 code. This method named the iterative local spatial self-shielding (ILSS) method considers the shadowing effect of randomly distributed particles on spatial self-shielding in particles through a homogenized composition region added outside the particle of interest at the center. The self-shielded cross sections of the central particle are determined iteratively since they are used in determining the cross sections of the homogenized composition region. The ILSS method was verified for infinite stochastic medium problems of single and multiple types of particles, VHTR unit cell problems, and HTTR assembly problems. The verification test results show that the ILSS method accurately predicts the stochastic particle shadowing effect and reaction rates in particles, whereas the regular array model and the stochastic collision probability method underpredict the particle shadowing effect and overestimate reaction rates in particles.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Analysis of SCALE Criticality and Sensitivity Calculations for Reflected HEU Cylinders [Abstract]

The SCALE code package offers several nuclear data libraries to support Monte Carlo (MC) transport, as well as MC-based derivation of $\kappa$ eff sensitivity and uncertainty (S/U) data. The CSAS sequence using the KENO MC code can utilize continuous-energy (CE) cross sections, or pre-generated multigroup (MG) cross section libraries. The use of MG libraries introduces bias into calculations in exchange for faster transport solutions. The TSUNAMI-3D sequence also utilizes KENO MC calculations. TSUNAMI-3D has two CE calculational methods: the Iterated Fission Probability (IFP) method, and the Contribution-Linked eigenvalue sensitivity/Uncertainty estimation via Tracklength importance CHaracterization (CLUTCH) method. Previous work has shown poor agreement between CLUTCH and confirmatory direct perturbation calculations in specific applications, e.g., fissionable and polyethylene reflectors

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Analysis of SCALE Criticality and Sensitivity Calculations for Reflected HEU Cylinders

The SCALE code package offers multiple nuclear data libraries and sensitivity and uncertainty (S/U) methods supporting and derived from Monte Carlo (MC) transport. The CSAS and TSUNAMI-3D sequences use KENO MC, utilizing either continuous-energy (CE) cross sections or multigroup (MG) cross section libraries. TSUNAMI-3D has two CE calculational methods: the iterated fission probability (IFP) method, and the Contribution-Linked eigenvalue sensitivity/Uncertainty estimation via Tracklength importance CHaracterization (CLUTCH) method. Previous work has shown poor agreement between CLUTCH and confirmatory direct perturbation calculations in specific applications (e.g., fissionable and polyethylene reflectors). The HEU-MET-FAST-084 (HMF-084) International Criticality Safety Benchmark Evaluation Project evaluation consists of 27 cylindrical highly enriched uranium metal cores with 14 unique reflector materials of 0.5 and 1 in. thicknesses. Included in this list of reflector materials are natural uranium and polyethylene. This work utilized SCALE 6.2.4 models of the HMF-084 evaluation, with additional non-physical configurations to test both the MG bias and CLUTCH functionality across a variety of reflector material thicknesses. The evaluation’s use of concentric cylinders allowed for examination of several MG self-shielding methods: infinite homogenous, cylindrical, and spherical. The results indicate that the use of polyethylene reflectors with CLUTCH is not fundamentally impossible but sensitive to geometry. The poor performance of CLUTCH with fissionable reflectors was reaffirmed. The 2 in. and greater polyethylene-reflected calculations demonstrate the necessity of using the 302-group library for fast systems. The nickel MG bias was substantial, as discussed in a companion paper, as were cobalt and iron.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Ultrasonic Measurements of Temperature Profile and Heat Fluxes in Coal-Fired Power Plants (Final Report)

Many industrial processes are inaccessible or inhospitable to characterization by traditional temperature measurement methods, such as thermocouples, especially over prolonged exposure to harsh environments. Ultrasound is an established characterization technology with diverse applications ranging from medical imaging to therapies to flaw detection to nondestructive evaluation. Ultrasound may characterize solid materials and components noninvasively as a nondestructive evaluation modality and obtain internal measurements of material properties. For example, the speed of ultrasound propagation changes with Young’s modulus and Poisson’s ratio, which can be found from its measurements. Traditional ultrasonic characterization assumes all material properties remain constant with the position. When this assumption holds, a property of interest may be measured by relating it to the speed of ultrasound propagation (or a speed of sound, SOS) and measuring the SOS by timing the ultrasound propagation through a known distance. However, when a property of interest is spatially distributed, the propagation time depends on the SOS changing with the position along the ultrasound propagation path. The multiple temperature distributions may lead to an identical time of flight (TOF). Temperature is one property that impacts the speed of ultrasound and often cannot be assumed to remain constant with the position. Previously, in the context of temperature, we addressed the challenge of ultrasonic characterization of spatially distributed properties by developing a method for measuring segmental temperature distributions (MSTD). This method divides the ultrasonic propagation into segments bound by echogenic features. These features provide ultrasonic interfaces where some energy is reflected toward the receiving transducer, and the rest continues through the medium. The time-of-flight between the echoes reflected from echogenic features characterizes the spatial distribution in the properties of interest in the corresponding segment of the ultrasonic propagation path. This project demonstrated the application of the MSTD method in industrial conditions of the coal-fired power plant. We implemented the MSTD using metals and alloys waveguides, which may be the existing structure for which the temperature distribution is characterized or purposefully designed waveguides added to the structure by welding or other means specifically to quantify thermal properties using the MSTD method. Previous iterations of the MSTD method used ceramic and cementitious waveguides, which significantly attenuate ultrasound. On the other hand, low attenuation in metallic waveguides creates interactions between echogenic features which compilates the signal analysis in the segmental TOF measurements. We have established the WG design principles that minimize the interferences between trailing and primary echoes and, in some cases, eliminate them. The waveguides in which echoes do not interfere improve the timing accuracy and the robustness of ultrasonic measurements of the spatial distributions in material properties. Our emphasis remained on the estimation of the temperature distributions. We have developed general recommendations for designing ultrasonically segmented waveguides with the reduced influence of trailing echoes. Two of our waveguide designs were tested in the industry. The first waveguide was designed for insertion into a combustion zone of the utility-scale coal-fired power plant boiler. The second design allows the characterization of temperature distribution in the direction normal to the boiler’s water wall, a large heat exchanger converting the chemical energy released during combustion to the steam driving the electrical power generation turbines. These waveguides were designed to operate within a restrictive space of thermally insulated water wall and incorporate densely located echogenic features while combatting the influence of trailing echoes. The project has successfully demonstrated the feasibility of using the developed method for accurate, continuous, and robust temperature measurements in extreme environments of power generation and other industrial processes. It, therefore, has achieved its overarching goal of advancing the technology readiness level of the novel Ultrasound Measurements of Segmental Temperature Distribution (US-MSTD) method for real-time measurements of the temperature distribution and heat fluxes closer to commercial availability, developing a prototype multipoint measurement system, and validating its performance on coal-fired utility boilers. The success of this project was achieved in collaboration with the power generator, Rocky Mountain Power, and set the stage for the transfer of this technology from the laboratory to the industry.

01 COAL, LIGNITE, AND PEAT↗

Determining Partial Atomic Charges for Liquid Water: Assessing Electronic Structure and Charge Models

Partial atomic charges provide an intuitive and efficient way to describe the charge distribution and the resulting intermolecular electrostatic interactions in liquid water. Many charge models exist and it is unclear which model provides the best assignment of partial atomic charges in response to the local molecular environment. In this work, we systematically scrutinize various electronic structure methods and charge models (Mulliken, natural population analysis, CHelpG, RESP, Hirshfeld, Iterative Hirshfeld, and Bader) by evaluating their performance in predicting the dipole moments of isolated water, water clusters, and liquid water as well as charge transfer in the water dimer and liquid water. Although none of the seven charge models is capable of fully capturing the dipole moment increase from isolated water (1.85 D) to liquid water (about 2.9 D), the Iterative Hirshfeld method performs best for liquid water, reproducing its experimental average molecular dipole moment, yielding a reasonable amount of intermolecular charge transfer, and showing modest sensitivity to the local water environment. The performance of the charge model is dependent on the choice of the density functional and the quantum treatment of the environment. The computed molecular dipole moment of water generally increases with the percentage of the exact Hartree–Fock exchange in the functional, whereas the amount of charge transfer between molecules decreases. For liquid water, including two full solvation shells of surrounding water molecules (within about 5.5 Å of the central water) in the quantum chemical calculation converges the charges of the central water molecule. Furthermore, our final pragmatic quantum chemical charge-assigning protocol for liquid water is the Iterative Hirshfeld method with M06-HF/aug-cc-pVDZ and a quantum region cutoff radius of 5.5 Å.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

A family of independent Variable Eddington Factor methods with efficient preconditioned iterative solvers

We present a family of discretizations for the Variable Eddington Factor (VEF) equations that have high-order accuracy on curved meshes and efficient preconditioned iterative solvers. The VEF discretizations are combined with the Discontinuous Galerkin transport discretization from to form effective high-order, linear transport methods. The VEF discretizations are derived by extending the unified analysis of Discontinuous Galerkin methods for elliptic problems presented by Arnold et al. to the VEF equations. This framework is used to define analogs of the interior penalty, second method of Bassi and Rebay, minimal dissipation local Discontinuous Galerkin, and continuous finite element methods. The analysis of subspace correction preconditioners, which use a continuous operator to iteratively precondition the discontinuous discretization, is extended to the case of the non-symmetric VEF system. Numerical results demonstrate that the VEF discretizations have arbitrary-order accuracy on curved meshes, preserve the thick diffusion limit, and are effective on a proxy problem from thermal radiative transfer in both outer transport iterations and inner preconditioned linear solver iterations. We demonstrate that the VEF solution converges to the S N transport solution as the mesh is refined on both problems with smooth and non-smooth behavior in angle. Parallel performance studies show that the interior penalty VEF discretization's linear solve weak scales out to 1024 processors and strong scales well on a single node. Particular attention is paid to the parallel performance of the VEF algorithm when used in combination with a parallel block Jacobi transport sweep.

97 MATHEMATICS AND COMPUTING↗

Atomic resolution coherent x-ray imaging with physics-based phase retrieval

Coherent x-ray imaging and scattering from accelerator based sources such as synchrotrons continue to impact biology, medicine, technology, and materials science. Many synchrotrons around the world are currently undergoing major upgrades to increase their available coherent x-ray flux by approximately two orders of magnitude. The improvement of synchrotrons may enable imaging of materials in operando at the atomic scale which may revolutionize battery and catalysis technologies. Current algorithms used for phase retrieval in coherent x-ray imaging are based on the projection onto sets method. These traditional iterative phase retrieval methods will become more computationally expensive as they push towards atomic resolution and may struggle to converge. Additionally, these methods do not incorporate physical information that may additionally constrain the solution. In this work, we present an algorithm which incorporates molecular dynamics into Bragg coherent diffraction imaging (BCDI). This algorithm, which we call PRAMMol (Phase Retrieval with Atomic Modeling and Molecular Dynamics) combines statistical techniques with molecular dynamics to solve the phase retrieval problem. We present several examples where our algorithm is applied to simulated coherent diffraction from 3D crystals and show convergence to the correct solution at the atomic scale.

47 OTHER INSTRUMENTATION↗

Evaluating the Performance of Random Forest and Iterative Random Forest Based Methods when Applied to Gene Expression Data

Gene-to-gene networks, such as Gene Regulatory Networks (GRN) and Predictive Expression Networks (PEN) capture relationships between genes and are beneficial for use in downstream biological analyses. There exists multiple network inference tools to produce these gene-to-gene networks from matrices of gene expression data. Random Forest-Leave One Out Prediction (RF-LOOP) is a method that has been shown to be efficient at producing these gene-to-gene networks, frequently known as GEne Network Inference with Ensemble of trees (GENIE3). Here we validate that iterative Random Forest-Leave One Out Prediction (iRF-LOOP) produces higher quality networks than GENIE3. We use both synthetic and empirical networks from the Dialogue for Reverse Engineering Assessment and Methods (DREAM) Challenges by Sage Bionetworks, as well as two additional empirical networks created from Arabidopsis thaliana and Populus trichocarpa expression data.

iRF-Loop, expression network, Populus Trichocarpa↗