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DOE OSTI2025

Constrained or unconstrained? Neural-network-based equation discovery from data

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or the accurate resolution of their solutions may be intractable. Instead, recently developed methods, including those based on parameter estimation, operator subset selection, and neural networks, allow for the data-driven discovery of both ordinary and partial differential equations (PDEs), on a spectrum of interpretability. The success of these strategies is often contingent upon the correct identification of representative equations from noisy observations of state variables and, as importantly and intertwined with that, the mathematical strategies utilized to enforce those equations. Specifically, the latter has been commonly addressed via unconstrained optimization strategies. Representing the PDE as a neural network, we propose to discover the PDE (or the associated operator) by solving a constrained optimization problem and using an intermediate state representation similar to a physics-informed neural network (PINN). The objective function of this constrained optimization problem promotes matching the data, while the constraints require that the discovered PDE is satisfied at a number of spatial collocation points. We present a penalty method and a widely used trust-region barrier method to solve this constrained optimization problem, and we compare these methods on numerical examples. Our results on several example problems demonstrate that the latter constrained method outperforms the penalty method, particularly for higher noise levels or fewer collocation points. This work motivates further exploration into using sophisticated constrained optimization methods in scientific machine learning, as opposed to their commonly used, penalty-method or unconstrained counterparts. For both of these methods, we solve these discovered neural network PDEs with classical methods, such as finite difference methods, as opposed to PINNs-type methods relying on automatic differentiation. Here, we briefly highlight how simultaneously fitting the data while discovering the PDE improves the robustness to noise and other small, yet crucial, implementation details.

Data-driven discovery↗

DOE OSTI2025

Gradient flow based phase-field modeling using separable neural networks

Allen–Cahn equation is a reaction–diffusion equation and is widely used for modeling phase separation. Machine learning methods for solving the Allen–Cahn equation in its strong form suffer from inaccuracies in collocation techniques, errors in computing higher-order spatial derivatives, and the large system size required by the space–time approach. To overcome these challenges, we propose solving the gradient flow of the Ginzburg–Landau free energy functional, which is equivalent to the Allen–Cahn equation, thereby avoiding the second-order spatial derivatives associated with the Allen–Cahn equation. A minimizing movement scheme is employed to solve the gradient flow problem, eliminating the complexities of a space–time approach. We utilize a separable neural network that efficiently represents the phase field through low-rank tensor decomposition. As we use the minimizing movement scheme to numerically solve the gradient flow problem, we thus, refer to the proposed method as the Separable Deep Minimizing Movement (SDMM) method. The evaluation of the functional in the minimizing movement scheme using the Gauss quadrature technique bypasses the inaccuracies associated with collocation techniques traditionally used to solve partial differential equations. A hyperbolic tangent transformation is introduced on the phase field prior to the evaluation of the functional to ensure that it remains strictly bounded within the values of the two phases. For this transformation, theoretical guarantee for energy stability of the minimizing movement scheme is established. Our results suggest that this transformation helps to improve the accuracy and efficiency significantly. The proposed method resolves the challenges faced by state-of-the-art machine learning techniques, outperforming them in both accuracy and efficiency. It is also the first machine learning method to achieve an order of magnitude speed improvement over the finite element method. In addition to its formulation and computational implementation, several case studies illustrate the applicability of the proposed method.

42 ENGINEERING↗

DOE OSTI2014

An adaptive ANOVA-based PCKF for high-dimensional nonlinear inverse modeling

The probabilistic collocation-based Kalman filter (PCKF) is a recently developed approach for solving inverse problems. It resembles the ensemble Kalman filter (EnKF) in every aspect—except that it represents and propagates model uncertainty by polynomial chaos expansion (PCE) instead of an ensemble of model realizations. Previous studies have shown PCKF is a more efficient alternative to EnKF for many data assimilation problems. However, the accuracy and efficiency of PCKF depends on an appropriate truncation of the PCE series. Having more polynomial chaos basis functions in the expansion helps to capture uncertainty more accurately but increases computational cost. Selection of basis functions is particularly important for high-dimensional stochastic problems because the number of polynomial chaos basis functions required to represent model uncertainty grows dramatically as the number of input parameters (random dimensions) increases. In classic PCKF algorithms, the PCE basis functions are pre-set based on users' experience. Also, for sequential data assimilation problems, the basis functions kept in PCE expression remain unchanged in different Kalman filter loops, which could limit the accuracy and computational efficiency of classic PCKF algorithms. To address this issue, we present a new algorithm that adaptively selects PCE basis functions for different problems and automatically adjusts the number of basis functions in different Kalman filter loops. The algorithm is based on adaptive functional ANOVA (analysis of variance) decomposition, which approximates a high-dimensional function with the summation of a set of low-dimensional functions. Thus, instead of expanding the original model into PCE, we implement the PCE expansion on these low-dimensional functions, which is much less costly. We also propose a new adaptive criterion for ANOVA that is more suited for solving inverse problems. The new algorithm was tested with different examples and demonstrated great effectiveness in comparison with non-adaptive PCKF and EnKF algorithms.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Sparse-grid, reduced-basis Bayesian inversion: Nonaffine-parametric nonlinear equations

We extend the reduced basis (RB) accelerated Bayesian inversion methods for affine-parametric, linear operator equations which are considered in [16,17] to non-affine, nonlinear parametric operator equations. We generalize the analysis of sparsity of parametric forward solution maps in [20] and of Bayesian inversion in [48,49] to the fully discrete setting, including Petrov–Galerkin high-fidelity (“HiFi”) discretization of the forward maps. We develop adaptive, stochastic collocation based reduction methods for the efficient computation of reduced bases on the parametric solution manifold. The nonaffinity and nonlinearity with respect to (w.r.t.) the distributed, uncertain parameters and the unknown solution is collocated; specifically, by the so-called Empirical Interpolation Method (EIM). For the corresponding Bayesian inversion problems, computational efficiency is enhanced in two ways: first, expectations w.r.t. the posterior are computed by adaptive quadratures with dimension-independent convergence rates proposed in [49]; the present work generalizes [49] to account for the impact of the PG discretization in the forward maps on the convergence rates of the Quantities of Interest (QoI for short). Second, we propose to perform the Bayesian estimation only w.r.t. a parsimonious, RB approximation of the posterior density. Based on the approximation results in [49], the infinite-dimensional parametric, deterministic forward map and operator admit N-term RB and EIM approximations which converge at rates which depend only on the sparsity of the parametric forward map. In several numerical experiments, the proposed algorithms exhibit dimension-independent convergence rates which equal, at least, the currently known rate estimates for N-term approximation. We propose to accelerate Bayesian estimation by first offline construction of reduced basis surrogates of the Bayesian posterior density. The parsimonious surrogates can then be employed for online data assimilation and for Bayesian estimation. They also open a perspective for optimal experimental design.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2016

Reduced basis ANOVA methods for partial differential equations with high-dimensional random inputs

In this paper we present a reduced basis ANOVA approach for partial deferential equations (PDEs) with random inputs. The ANOVA method combined with stochastic collocation methods provides model reduction in high-dimensional parameter space through decomposing high-dimensional inputs into unions of low-dimensional inputs. In this work, to further reduce the computational cost, we investigate spatial low-rank structures in the ANOVA-collocation method, and develop efficient spatial model reduction techniques using hierarchically generated reduced bases. We present a general mathematical framework of the methodology, validate its accuracy and demonstrate its efficiency with numerical experiments.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2017

Systematic parameter inference in stochastic mesoscopic modeling

We propose a method to efficiently determine the optimal coarse-grained force field in mesoscopic stochastic simulations of Newtonian fluid and polymer melt systems modeled by dissipative particle dynamics (DPD) and energy conserving dissipative particle dynamics (eDPD). The response surfaces of various target properties (viscosity, diffusivity, pressure, etc.) with respect to model parameters are constructed based on the generalized polynomial chaos (gPC) expansion using simulation results on sampling points (e.g., individual parameter sets). To alleviate the computational cost to evaluate the target properties, we employ the compressive sensing method to compute the coefficients of the dominant gPC terms given the prior knowledge that the coefficients are “sparse”. The proposed method shows comparable accuracy with the standard probabilistic collocation method (PCM) while it imposes a much weaker restriction on the number of the simulation samples especially for systems with high dimensional parametric space. Fully access to the response surfaces within the confidence range enables us to infer the optimal force parameters given the desirable values of target properties at the macroscopic scale. Moreover, it enables us to investigate the intrinsic relationship between the model parameters, identify possible degeneracies in the parameter space, and optimize the model by eliminating model redundancies. The proposed method provides an efficient alternative approach for constructing mesoscopic models by inferring model parameters to recover target properties of the physics systems (e.g., from experimental measurements), where those force field parameters and formulation cannot be derived from the microscopic level in a straight forward way.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Ambient Aerosol Is Physically Larger on Cloudy Days in Bondville, Illinois

Particle chemical composition affects aerosol optical and physical properties in ways important for the fate, transport, and impact of atmospheric particulate matter. For example, hygroscopic constituents take up water to increase the physical size of a particle, which can alter the extinction properties and atmospheric lifetime. At the collocated AERosol RObotic NETwork (AERONET) and Interagency Monitoring of PROtected Visual Environments (IMPROVE) network monitoring stations in rural Bondville, Illinois, we employ a novel cloudiness determination method to compare measured aerosol physicochemical properties on predominantly cloudy and clear sky days from 2010 to 2019. On cloudy days, aerosol optical depth (AOD) is significantly higher than on clear sky days in all seasons. Measured angstrom ngstro''m exponents are significantly smaller on cloudy days, indicating physically larger average particle size for the sampled populations in all seasons except winter. Mass concentrations of fine particulate matter that include estimates of aerosol liquid water (ALW) are higher on cloudy days in all seasons but winter. More ALW on cloudy days is consistent with larger particle sizes inferred from angstrom ngstro''m exponent measurements. Aerosol chemical composition that affects hygroscopicity plays a determining impact on cloudy versus clear sky differences in AOD, angstrom ngstro''m exponents, and ALW. This work highlights the need for simultaneous collocated, high-time-resolution measurements of both aerosol chemical and physical properties, in particular at cloudy times when quantitative understanding of tropospheric composition is most uncertain.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2018

Comparison of Daytime Low-Level Cloud Properties Derived From GOES and ARM SGP Measurements

Large-scale satellite data are critical for both verifying and improving general circulation model parameterizations of clouds and radiation for climate prediction. For reliable application of satellite data sets in cloud processes and climate models, it is important to have a reasonable estimate of the errors in the derived cloud properties. The daytime single-layered low-level cloud properties retrieved by the Geostationary Operational Environmental Satellite system (GOES) are compared with ground-based observations and retrievals over the Department of Energy Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP) Central Facility from June 1998 to December 2006. The GOES retrievals are made via the Visible-Infrared Solar-infrared Split-window Technique. They are spatially averaged within a 0.15° × 0.15° box centered on the ARM SGP site, and the ARM surface observations are temporally averaged ±15 min around the GOES scans to produce collocated pairs. Comparisons are made for monthly means, diurnal means, and one-to-one GOES and ARM collocated pairs. GOES T eff is highly correlated with ARM T top cloud temperature, having an R 2 value of 0.75, though GOES exhibits a cold bias. GOES-retrieved τ and liquid water path have very good agreement with ARM retrievals with R 2 s of 0.45 and 0.47, while r e (GOES), on average, is about 2 μm greater than ARM r e . An examination of solar and viewing geometry has shown that GOES-retrieved mean r e and τ values are impacted by solar zenith angle and especially scattering angle, which is not unexpected and needs to be accounted for by users.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Enhance Low Level Temperature and Moisture Profiles Through Combining NUCAPS, ABI Observations, and RTMA Analysis

Thermodynamic information from low levels in the atmosphere is crucial for operational weather forecasts and meteorological researchers. The NOAA Unique Combined Atmospheric Processing System (NUCAPS) sounding products have been proven beneficial to fill the data gap between synoptic radiosonde observations (RAOBs). However, compared with the upper troposphere, the accuracy of NUCAPS soundings in the low levels still needs improvement. In this study, a deep neural network (DNN) is applied to fuse multiple data sources to enhance the NUCAPS temperature and moisture profiles in the lower atmosphere. The network is developed by combining satellite observations, including NUCAPS sounding retrievals and high resolution geostationary satellite observations from the Advanced Baseline Imager, and surface analysis from the Real-Time Mesoscale Analysis (RTMA) as inputs, while collocated soundings from ECMWF re-analysis version 5 are used as the benchmark for the training. The performance of the model is evaluated by using the independent testing data set, data from a different year, as well as collocated RAOBs, showing improvement to the temperature and moisture profiles by reducing the root-mean-squared-error (RMSE) by more than 30% in the lower atmosphere (from 700 hPa to surface) in both clear sky and partially cloudy conditions. A convective event from June 18, 2017 is presented to illustrate the application of the enhanced low level soundings on high impact weather events. The enhanced soundings from fused data capture the large surface-based convective available potential energy structures in the preconvection environment, which is very useful for severe storm nowcasting and forecasting applications.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2023

Evaluating EAMv2 Simulated High Latitude Clouds Using ARM Measurements in the Northern and Southern Hemispheres

Here, this study evaluates high latitude stratiform mixed-phase clouds (SMPC) in the atmosphere model of the Energy Exascale Earth System Model version 2 (EAMv2) by utilizing one-year-long ground-based remote sensing measurements from the Atmospheric Radiation and Measurement (ARM) program. A nudging approach is applied to model simulations for a constrained comparison with the ARM observations. Observed and modeled SMPCs are sampled and collocated to address the difference in data resolution, so that we can consistently evaluate their macrophysical properties at the North Slope of Alaska (NSA) site in the Arctic and the McMurdo (AWR) site in the Antarctic. We found that EAMv2 overestimates SMPC frequency of occurrence at both sites. However, the model captures the observed larger cloud frequency of occurrence at the NSA site. For collocated SMPCs, the annual statistics of observed cloud macrophysics are generally reproduced at the NSA site, while at the AWR site, there are larger biases. Compared to the AWR site, the lower cloud top and cloud base and the warmer cloud top temperature observed at NSA are well simulated. On the other hand, simulated cloud phase is substantially biased. The model largely overestimates liquid water path, and the ice water path is underestimated at NSA, but at AWR, the liquid water path is frequently underestimated due to the dominance of snow in Antarctic SMPCs. As a result, the observed hemispheric difference in cloud phase partitioning is misrepresented in EAMv2. This study implies that additional model development is needed for high latitude mixed-phase clouds.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Spanning the Gap From Bulk to Bin: A Novel Spectral Microphysics Method

Microphysics methods for climate models and numerical weather prediction typically track one, two, or three moments of a droplet size distribution for various categories of liquid, ice, and aerosol. Such methods rely on conversion parameters between these categories, which introduces uncertainty into predictions. While higher-fidelity options such as bin and Lagrangian schemes exist, they require too many degrees of freedom for climate modeling applications and introduce numerical challenges. Here we introduce a flexible spectral microphysics method based on collocation of basis functions. This method generalizes to a linear bulk scheme when using few basis functions and to a smoothed bin scheme with more degrees of freedom. Tested in an idealized box setting, the method improves spectral accuracy for droplet collision-coalescence and may eliminate the need for precipitation autoconversion rates required by bulk methods; furthermore, it generalizes well to multimodal distributions with less complexity than a bin method. The potential to extend this collocation representation to multiple hydrometeor classes suggests a path forward to unify liquid, ice, and aerosol microphysics in a single, flexible, computational framework for climate modeling.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

High resolution atomic force microscopy with an active piezoelectric microcantilever

Active microcantilevers with on-chip sensing and actuation provide significant advantages in tapping mode Atomic Force Microscopy (AFM). Collocated transduction allows for effective manipulation of cantilever dynamics through feedback control, enabling higher scan rates. However, the adjacency of the sensing and actuation electrodes is known to result in a high level of feedthrough, leading to a low imaging resolution. Readout circuit noise further deteriorates the imaging precision. Here, we investigate the noise sources that affect AFM microcantilevers with collocated aluminum nitride (AlN) actuator–sensor pairs. We reported these cantilevers in earlier work and demonstrated that they display a very low level of feedthrough between the actuation and sensing electrodes. We present a high signal-to-noise ratio (SNR) sensing method that enables us to demonstrate high-resolution AFM on a calibration grating with nm-step silicon carbide (SiC) terraces. In conclusion, measuring the Lorentzian response of the cantilever’s Brownian motion with the on-chip active sensor at resonance enables us to calibrate the dynamic stiffness at the first fundamental resonance mode, without utilizing an optical sensor.

47 OTHER INSTRUMENTATION↗

DOE OSTI2016

Improving Lidar Turbulence Estimates for Wind Energy

Remote sensing devices (e.g., lidars) are quickly becoming a cost-effective and reliable alternative to meteorological towers for wind energy applications. Although lidars can measure mean wind speeds accurately, these devices measure different values of turbulence intensity (TI) than an instrument on a tower. In response to these issues, a lidar TI error reduction model was recently developed for commercially available lidars. The TI error model first applies physics-based corrections to the lidar measurements, then uses machine-learning techniques to further reduce errors in lidar TI estimates. The model was tested at two sites in the Southern Plains where vertically profiling lidars were collocated with meteorological towers. Results indicate that the model works well under stable conditions but cannot fully mitigate the effects of variance contamination under unstable conditions. To understand how variance contamination affects lidar TI estimates, a new set of equations was derived in previous work to characterize the actual variance measured by a lidar. Terms in these equations were quantified using a lidar simulator and modeled wind field, and the new equations were then implemented into the TI error model.

17 WIND ENERGY↗

DOE OSTI2019

METACLUSTER—an R package for context-specific expression analysis of metabolic gene clusters

Abstract Summary Plants and microbes produce numerous compounds to cope with their environments but the biosynthetic pathways for most of these compounds have yet to be elucidated. Some biosynthetic pathways are encoded by enzymes collocated in the chromosome. To facilitate a more comprehensive condition and tissue-specific expression analysis of metabolic gene clusters, we developed METACLUSTER, a probabilistic framework for characterizing metabolic gene clusters using context-specific gene expression information. Availability and implementation METACLUSTER is freely available at https://github.com/mbanf/METACLUSTER. Supplementary information Supplementary data are available at Bioinformatics online.

Biochemistry & Molecular Biology↗

DOE OSTI2023

Staggered scheme for the compressible fluctuating hydrodynamics of multispecies fluid mixtures

Here, we present a numerical formulation for the solution of nonisothermal, compressible Navier-Stokes equations with thermal fluctuations to describe mesoscale transport phenomena in multispecies fluid mixtures. The novelty of our numerical method is the use of staggered grid momenta along with a finite volume discretization of the thermodynamic variables to solve the resulting stochastic partial differential equations. The key advantages of the numerical scheme are that it significantly simplifies the discretization of diffusive and stochastic momentum fluxes into a more compact form, and it provides an unambiguous prescription of boundary conditions involving pressure. The staggered grid scheme more accurately reproduces the equilibrium static structure factor of hydrodynamic fluctuations in gas mixtures compared to a collocated scheme described previously by Balakrishnan et al. [Phys. Rev. E 89, 013017 (2014)1539-375510.1103/PhysRevE.89.013017]. The numerical method is tested for ideal noble gases mixtures under various nonequilibrium conditions, such as applied thermal and concentration gradients, to assess the role of cross-diffusion effects, such as Soret and Dufour, on the long-ranged correlations of hydrodynamic fluctuations, which are also more accurately reproduced compared to the collocated scheme. We numerically study giant nonequilibrium fluctuations driven by concentration gradients and fluctuation-driven Rayleigh-Taylor instability in gas mixtures. Wherever applicable, excellent agreement is observed with theory and measurements from the direct simulation Monte Carlo method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Floodplain nitrifiers harbor the genetic potential for utilizing a wide range of organic nitrogen compounds

Organic compounds such as urea and cyanate can serve as nitrogen (N) sources for nitrifying microorganisms, including ammonia-oxidizing archaea (AOA) and bacteria (AOB), complete ammonia-oxidizing (comammox) bacteria, and nitrite-oxidizing bacteria (NOB). Here we investigated metagenome-assembled genomes (MAGs) for all four nitrifier guilds generated from hydrologically variable floodplain sediments of the Wind River Basin (WRB; Riverton, WY, USA) for their genetic potential to utilize organic N compounds. A vast majority of WRB nitrifier MAGs harbored urease (ure) and at least one urea transporter ( utp, urt, dur3 ). AOA were the most abundant and phylogenetically diverse nitrifiers in WRB floodplain sediments. Several AOA MAGs encoded cyanase ( cynS ), nitrilase ( nit1 ), omega-amidase ( nit2 ), nitrile hydratase ( nthA ), and genes related to purine degradation, including biuret hydrolase ( biuH ), oxamic transcarbamylase ( allFGH ), and catabolic carbamate kinase ( allK ). AOA often encoded an uncharacterized amidohydrolase collocated with biuH , rather than allophanate hydrolase ( atzF ). A small number of AOA encoded atzF , functioning in an unknown pathway. AOB and comammox were of relatively low abundance and taxonomic diversity and were present only at certain depths in WRB; however, they encoded triuret/biuret degradation genes ( trtA, biuH , and atzH ), and in comammox, these genes were also collocated with allFGHK . The genetic potential of ammonia oxidizers in the WRB floodplain suggests that organic N may support nitrification in this system. The proposed pathways for utilizing purine degradation products other than urea potentially expand the known metabolic capabilities of AOA, AOB, and comammox bacteria and reveal the possibility for cryptic N cycling between microbial community members.

floodplain↗

DOE OSTI2015

Constructing Surrogate Models of Complex Systems with Enhanced Sparsity: Quantifying the influence of conformational uncertainty in biomolecular solvation

Biomolecules exhibit conformational fluctuations near equilibrium states, inducing uncertainty in various biological properties in a dynamic way. We have developed a general method to quantify the uncertainty of target properties induced by conformational fluctuations. Using a generalized polynomial chaos (gPC) expansion, we construct a surrogate model of the target property with respect to varying conformational states. We also propose a method to increase the sparsity of the gPC expansion by defining a set of conformational “active space” random variables. With the increased sparsity, we employ the compressive sensing method to accurately construct the surrogate model. We demonstrate the performance of the surrogate model by evaluating fluctuation-induced uncertainty in solvent-accessible surface area for the bovine trypsin inhibitor protein system and show that the new approach offers more accurate statistical information than standard Monte Carlo approaches. Further more, the constructed surrogate model also enables us to directly evaluate the target property under various conformational states, yielding a more accurate response surface than standard sparse grid collocation methods. In particular, the new method provides higher accuracy in high-dimensional systems, such as biomolecules, where sparse grid performance is limited by the accuracy of the computed quantity of interest. Our new framework is generalizable and can be used to investigate the uncertainty of a wide variety of target properties in biomolecular systems.

coarse graining↗

DOE OSTI2021

Computational Complexity of Neuromorphic Algorithms

Neuromorphic computing has several characteristics that make it an extremely compelling computing paradigm for post Moore computation. Some of these characteristics include intrinsic parallelism, inherent scalability, collocated processing and memory, and event-driven computation. While these characteristics impart energy efficiency to neuromorphic systems, they do come with their own set of challenges. One of the biggest challenges in neuromorphic computing is to establish the theoretical underpinnings of the computational complexity of neuromorphic algorithms. In this paper, we take the first steps towards defining the space and time complexity of neuromorphic algorithms. Specifically, we describe a model of neuromorphic computation and state the assumptions that govern the computational complexity of neuromorphic algorithms. Next, we present a theoretical framework to define the computational complexity of a neuromorphic algorithm. We explicitly define what space and time complexities mean in the context of neuromorphic algorithms based on our model of neuromorphic computation. Finally, we leverage our approach and define the computational complexities of six neuromorphic algorithms: constant function, successor function, predecessor function, projection function, neuromorphic sorting algorithm and neighborhood subgraph extraction algorithm.

Date, Prasanna↗