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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 1,477 records · Page 82
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

Integrated Cyber/Physical Grid Resiliency Modeling

This project explored coupling modeling and analysis methods from multiple domains to address complex hybrid (cyber and physical) attacks on mission critical infrastructure. Robust methods to integrate these complex systems are necessary to enable large trade-space exploration including dynamic and evolving cyber threats and mitigations. Reinforcement learning employing deep neural networks, as in the AlphaGo Zero solution, was used to identify "best" (or approximately optimal) resilience strategies for operation of a cyber/physical grid model. A prototype platform was developed and the machine learning (ML) algorithm was made to play itself in a game of 'Hurt the Grid'. This proof of concept shows that machine learning optimization can help us understand and control complex, multi-dimensional grid space. A simple, yet high-fidelity model proves that the data have spatial correlation which is necessary for any optimization or control. Our prototype analysis showed that the reinforcement learning successfully improved adversary and defender knowledge to manipulate the grid. When expanded to more representative models, this exact type of machine learning will inform grid operations and defense - supporting mitigation development to defend the grid from complex cyber attacks! This same research can be expanded to similar complex domains.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Characterizing vertical upper ocean temperature structures in the European Arctic through unsupervised machine learning

In-situ observations of subsurface ocean temperatures are, in many regions, inconsistently distributed in time and space. These spatio-temporal inconsistencies in the observational network lead to difficulties in utilizing those observations effectively for ocean model evaluation or understanding larger-scale ocean characteristics. Model accuracy of subsurface ocean characteristics is especially important within regions that contain complex ocean structures. One such region is the European Arctic which not only contains several types of water masses with unique characteristics, but also wintertime sea ice coverage and complex bathymetry. This study presents an unsupervised neural networking technique that can be used in combination with traditional ocean model evaluation techniques to provide additional information on the accuracy of modeled vertical ocean temperature profiles. Self-organizing maps is an unsupervised machine learning technique that we apply to approximately twenty thousand Argo and CTD temperature profiles from 2012 to 2020 in the European Arctic to categorize the observed vertical ocean temperature structures in the top 150 m. The observed ocean profile categories, or neurons, defined by the self-organizing map show strong spatial and temporal dependencies. We then use the neuron weights, or the learned temperature profile structure of each neuron, to validate the spatial and temporal variability of modeled vertical temperature structures. This analysis gives us new insights about the model’s capabilities to reproduce specific vertical structures of the top-most ocean layer within different regions and seasons. Mapping modeled ocean temperature profiles onto the neuron-space of the observationally-defined self organized map highlights the potential of this method to advance our understanding of model deficiencies in that region.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Accelerating Thermochemical Equilibrium Calculations for Nuclear Reactor Applications

Thermochemical properties play a key role in modeling and simulation of several key phenomena in nuclear reactors. There has been an increasing interest in incorporating CALPHAD-based formulations in multiphysics simulations including for Molten Salt Reactors where knowledge of phase evolution of the salt and the chemical potentials of various elements are of utmost importance in source term analyses and redox control. However, the size of such simulations is often limited by the high computational cost of full thermodynamic equilibrium calculations. This work discusses the current efforts aimed at accelerating thermochemical equilibrium calculations for multiphysics simulations performed using the open-source finite element / finite volume code Multiphysics Object Oriented Simulation Environment (MOOSE) [1]. While several methods have been proposed for accelerating phase equilibrium calculations [2], most focus on relatively small systems and often rely on a- priori knowledge of the state-space of the system. Nuclear materials, however, are often multi-component systems owing to the evolution of composition under irradiation and an approach based on a-priori mapping of phase diagram is often not enough. This work is aimed at demonstrating an on-the-fly surrogate modeling framework that uses active learning to reduce the number of full equilibrium calculations that must be performed. By combining with efficient coupling approaches, the surrogate framework helps in reducing the computational cost of thermodynamic equilibrium informed multiphysics simulations of nuclear materials. The performance is benchmarked against full coupling with the thermochemistry library Thermochimica [3]. This work uses a machine learning based approach for constructing surrogate models to predict the stable phases in a multicomponent system. The surrogates were constructed using neural networks and Gaussian process classification. In this work, we compare the relative performance of the two methods. We also demonstrate the use of caching previous calculations by interpolating the values from nearest neighbors. References [1] Lindsay, A.D., et al. "2.0 – MOOSE: Enabling massively parallel multiphysics simulation", SoftwareX, 20 (2022): 101202. [2] Roos, W.A. and Zietsman J.H. "Accelerating complex chemical equilibrium calculations – A Review", Calphad, 77 (2022): 102380. [3] Piro, M.H.A., et al. "The thermochemistry library Thermochimica", Computational Materials Science, 67 (2013): 266-272.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Feature Selection Techniques for a Machine Learning Model to Detect Autonomic Dysreflexia

Feature selection plays a crucial role in the development of machine learning algorithms. Understanding the impact of the features on a model, and their physiological relevance can improve the performance. This is particularly helpful in the healthcare domain wherein disease states need to be identified with relatively small quantities of data. Autonomic Dysreflexia (AD) is one such example, wherein mismanagement of this neurological condition could lead to severe consequences for individuals with spinal cord injuries. We explore different methods of feature selection needed to improve the performance of a machine learning model in the detection of the onset of AD. We present different techniques used as well as the ideal metrics using a dataset of thirty-six features extracted from electrocardiograms, skin nerve activity, blood pressure and temperature. The best performing algorithm was a 5-layer neural network with five relevant features, which resulted in 93.4% accuracy in the detection of AD. The techniques in this paper can be applied to a myriad of healthcare datasets allowing forays into deeper exploration and improved machine learning model development. Through critical feature selection, it is possible to design better machine learning algorithms for detection of niche disease states using smaller datasets.

electrocardiography↗
DOE OSTI2023

A review on recent machine learning applications for imaging mass spectrometry studies

Imaging mass spectrometry (IMS) is a powerful analytical technique widely used in biology, chemistry, and materials science fields that continue to expand. IMS provides a qualitative compositional analysis and spatial mapping with high chemical specificity. The spatial mapping information can be 2D or 3D depending on the analysis technique employed. Due to the combination of complex mass spectra coupled with spatial information, large high-dimensional datasets (hyperspectral) are often produced. Therefore, the use of automated computational methods for an exploratory analysis is highly beneficial. The fast-paced development of artificial intelligence (AI) and machine learning (ML) tools has received significant attention in recent years. These tools, in principle, can enable the unification of data collection and analysis into a single pipeline to make sampling and analysis decisions on the go. There are various ML approaches that have been applied to IMS data over the last decade. Here, in this review, we discuss recent examples of the common unsupervised (principal component analysis, non-negative matrix factorization, k-means clustering, uniform manifold approximation and projection), supervised (random forest, logistic regression, XGboost, support vector machine), and other methods applied to various IMS datasets in the past five years. The information from this review will be useful for specialists from both IMS and ML fields since it summarizes current and representative studies of computational ML-based exploratory methods for IMS.

47 OTHER INSTRUMENTATION↗
DOE OSTI2020

Deep learning approaches for neural decoding across architectures and recording modalities

Decoding behavior, perception or cognitive state directly from neural signals is critical for brain–computer interface research and an important tool for systems neuroscience. In the last decade, deep learning has become the state-of-the-art method in many machine learning tasks ranging from speech recognition to image segmentation. The success of deep networks in other domains has led to a new wave of applications in neuroscience. In this article, we review deep learning approaches to neural decoding. Here, we describe the architectures used for extracting useful features from neural recording modalities ranging from spikes to functional magnetic resonance imaging. Furthermore, we explore how deep learning has been leveraged to predict common outputs including movement, speech and vision, with a focus on how pretrained deep networks can be incorporated as priors for complex decoding targets like acoustic speech or images. Deep learning has been shown to be a useful tool for improving the accuracy and flexibility of neural decoding across a wide range of tasks, and we point out areas for future scientific development.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2023

Temporal Subsampling Diminishes Small Spatial Scales in Recurrent Neural Network Emulators of Geophysical Turbulence

The immense computational cost of traditional numerical weather and climate models has sparked the development of machine learning (ML) based emulators. Because ML methods benefit from long records of training data, it is common to use data sets that are temporally subsampled relative to the time steps required for the numerical integration of differential equations. Here, we investigate how this often overlooked processing step affects the quality of an emulator's predictions. We implement two ML architectures from a class of methods called reservoir computing: (a) a form of Nonlinear Vector Autoregression (NVAR), and (b) an Echo State Network (ESN). Despite their simplicity, it is well documented that these architectures excel at predicting low dimensional chaotic dynamics. We are therefore motivated to test these architectures in an idealized setting of predicting high dimensional geophysical turbulence as represented by Surface Quasi-Geostrophic dynamics. In all cases, subsampling the training data consistently leads to an increased bias at small spatial scales that resembles numerical diffusion. Interestingly, the NVAR architecture becomes unstable when the temporal resolution is increased, indicating that the polynomial based interactions are insufficient at capturing the detailed nonlinearities of the turbulent flow. The ESN architecture is found to be more robust, suggesting a benefit to the more expensive but more general structure. Spectral errors are reduced by including a penalty on the kinetic energy density spectrum during training, although the subsampling related errors persist. Future work is warranted to understand how the temporal resolution of training data affects other ML architectures.

58 GEOSCIENCES↗
DOE OSTI2019

Fast and interpretable classification of small X-ray diffraction datasets using data augmentation and deep neural networks

X-ray diffraction (XRD) data acquisition and analysis is among the most time-consuming steps in the development cycle of novel thin-film materials. We propose a machine-learning-enabled approach to predict crystallographic dimensionality and space group from a limited number of thin-film XRD patterns. We overcome the scarce-data problem intrinsic to novel materials development by coupling a supervised machine learning approach with a model-agnostic, physics-informed data augmentation strategy using simulated data from the Inorganic Crystal Structure Database (ICSD) and experimental data. As a test case, 115 thin-film metal-halides spanning 3 dimensionalities and 7 space-groups are synthesized and classified. After testing various algorithms, we develop and implement an all convolutional neural network, with cross-validated accuracies for dimensionality and space-group classification of 93% and 89%, respectively. We propose average class activation maps, computed from a global average pooling layer, to allow high model interpretability by human experimentalists, elucidating the root-causes of misclassification. Finally, we systematically evaluate the maximum XRD pattern step size (data acquisition rate) before loss of predictive accuracy occurs, and determine it to be 0.16° 2θ, which enables an XRD pattern to be obtained and classified in 5.5 minutes or less.

36 MATERIALS SCIENCE↗
DOE OSTI2024

UNNT: A novel Utility for comparing Neural Net and Tree-based models

The use of deep learning (DL) is steadily gaining traction in scientific challenges such as cancer research. Advances in enhanced data generation, machine learning algorithms, and compute infrastructure have led to an acceleration in the use of deep learning in various domains of cancer research such as drug response problems. In our study, we explored tree-based models to improve the accuracy of a single drug response model and demonstrate that tree-based models such as XGBoost (eXtreme Gradient Boosting) have advantages over deep learning models, such as a convolutional neural network (CNN), for single drug response problems. However, comparing models is not a trivial task. To make training and comparing CNNs and XGBoost more accessible to users, we developed an open-source library called UNNT (A novel Utility for comparing Neural Net and Tree-based models). The case studies, in this manuscript, focus on cancer drug response datasets however the application can be used on datasets from other domains, such as chemistry.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2023

Machine Learning Automated Analysis of Enormous Synchrotron X-ray Diffraction Datasets

X-ray diffraction (XRD) data analysis can be a time-consuming and laborious task. Deep neural network (DNN) based models trained with synthetic XRD patterns have been proven to be a highly efficient, accurate, and automated method for analyzing common XRD data collected from solid samples in ambient environments. However, it remains unclear whether synthetic XRD-based models can be effective in solving micro(μ)-XRD mapping data for in situ experiments involving liquid phases, which always have lower quality and significant artifacts. In this study, we collected μ-XRD mapping data from a LaCl 3 -calcite hydrothermal fluid system and trained two categories of models to analyze the experimental XRD patterns. Here, the models trained solely with synthetic XRD patterns showed low accuracy (as low as 64%) when solving experimental μ-XRD mapping data. However, the accuracy of the DNN models significantly improved (90% or above) when we trained them with a data set containing both synthetic and a small number of labeled experimental μ-XRD patterns. This study highlights the importance of labeled experimental patterns in training DNN models to solve μ-XRD mapping data from in situ experiments involving liquid phases.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRSDate not supplied

A Neural Network Parametrization of Volumetric Cloud Fraction Profiles Using Satellite Observations and MERRA-2 Reanalysis Meteorological Data

Clouds play a crucial role in regulating the hydrologic cycle and Earth's radiative energy budget, yet they are often poorly represented in global climate models (GCMs). This study applies deep machine learning techniques to develop a physical parameterization of volumetric cloud fraction (VCF), the fraction of a 3-D grid volume occupied by clouds using satellite lidar-radar measurements. The neural network (NN) captures the complicated relationships between observed VCF profiles and collocated meteorological variables from MERRA-2 reanalysis data. Our results show that the NN model, particularly a sequence-to-sequence long short-term memory (LSTM) network with a sixfactor loss function, effectively learns the underlying cloud physical processes. The NN model outperforms MERRA-2 reanalysis in representing low-level clouds in tropical and subtropical regions and low- and middle-level clouds over midlatitude storm-track regions, and also improves VCF histograms. These improvements are reflected in the vertical distributions of zonally, meridionally, and globally averaged VCFs, geographic distributions of low-, middle-, and high-level clouds, and seasonal variations in monthly-mean VCF. Furthermore, the NN predictions effectively capture the El Niño-Southern Oscillation (ENSO) effects and other interannual variations. The NN parameterization is further evaluated through a sensitivity analysis, in which a single predictor is perturbed at a time. This reveals that relative humidity (RH) is the dominant factor influencing variations in globally averaged VCF at low and middle altitudes, followed by temperature. At higher altitudes, temperature becomes the primary driver of VCF through its effect on RH. Changes in wind components had minimal impact on globally averaged VCF.

Shan Zeng↗
DOE OSTI2020

Deeply uncertain: comparing methods of uncertainty quantification in deep learning algorithms

We present a comparison of methods for uncertainty quantification (UQ) in deep learning algorithms in the context of a simple physical system. Three of the most common uncertainty quantification methods - Bayesian Neural Networks (BNN), Concrete Dropout (CD), and Deep Ensembles (DE) - are compared to the standard analytic error propagation. We discuss this comparison in terms endemic to both machine learning ("epistemic" and "aleatoric") and the physical sciences ("statistical" and "systematic"). The comparisons are presented in terms of simulated experimental measurements of a single pendulum - a prototypical physical system for studying measurement and analysis techniques. Our results highlight some pitfalls that may occur when using these UQ methods. For example, when the variation of noise in the training set is small, all methods predicted the same relative uncertainty independently of the inputs. This issue is particularly hard to avoid in BNN. On the other hand, when the test set contains samples far from the training distribution, we found that no methods sufficiently increased the uncertainties associated to their predictions. This problem was particularly clear for CD. In light of these results, we make some recommendations for usage and interpretation of UQ methods.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2025

Novel artificial neural network model for instantaneous power losses and operational efficiency mapping of MW-scale vanadium redox flow battery for improved technoeconomic analysis

A novel data-driven, machine-learning-based method for modeling the instantaneous power losses of a distribution-sited 2 MW/8MWh vanadium redox flow battery (VRFB), a grid-scale electrochemical storage technology, is introduced and compared against benchmark empirical modeling approaches, including symmetric and asymmetric models, as well as a recent convex hull modeling approach. The novel loss modeling method introduces several advantages over the benchmark models and over simplistic efficiency estimates, the most significant of which is that the model can accurately reflect the stepwise and non-linear parasitic losses associated with the duty cycles of mechanical auxiliary systems like pump motor drives and blower fans. Residuals of the models are compared; the proposed data driven model features significantly improved accuracy over the benchmark models. The model's coefficient of determination is also improved relative to that of the benchmark models. Furthermore, a novel method for visualization of operational efficiency of the grid-scale storage technology is introduced. To demonstrate the benefits of the novel data-driven method for modeling the VRFB, the benchmark models and the proposed models are embedded into an Open DSS distribution network model to study two applications of the grid-scale electrical storage system: load leveling for grid support and energy arbitrage. This article demonstrates that the accuracy of the instantaneous power loss model significantly impacts the understanding of the state of charge of the VRFB. In turn, the accuracy of the efficiency modeling of the VRFB impacts the understanding of the potential economic value and technical benefits to the distribution network operators. In conclusion, the presented power loss modeling approach is, therefore, highly relevant for utility-stakeholders, battery asset owners, system engineers, system designers, and financial planners interested in evaluating or optimizing the operation of grid-scale VRFBs.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Neural Networks to Find the Optimal Forcing for Offsetting the Anthropogenic Climate Change Effects

Abstract Of great relevance to climate engineering is the systematic relationship between the radiative forcing to the climate system and the response of the system, a relationship often represented by the linear response function (LRF) of the system. However, estimating the LRF often becomes an ill-posed inverse problem due to high-dimensionality and nonunique relationships between the forcing and response. Recent advances in machine learning make it possible to address the ill-posed inverse problem through regularization and sparse system fitting. Here, we develop a convolutional neural network (CNN) for regularized inversion. The CNN is trained using the surface temperature responses from a set of Green’s function perturbation experiments as imagery input data together with data sample densification. The resulting CNN model can infer the forcing pattern responsible for the temperature response from out-of-sample forcing scenarios. This promising proof of concept suggests a possible strategy for estimating the optimal forcing to negate certain undesirable effects of climate change. The limited success of this effort underscores the challenges of solving an inverse problem for a climate system with inherent nonlinearity. Significance Statement Predicting the climate response for a given climate forcing is a direct problem, while inferring the forcing for a given desired climate response is often an inverse, ill-posed, problem, posing a new challenge to the climate community. This study makes the first attempt to infer the radiative forcing for a given target pattern of global surface temperature response using a deep learning approach. The resulting deeply trained convolutional neural network inversion model shows promise in capturing the forcing pattern corresponding to a given surface temperature response, with a significant implication on the design of an optimal solar radiation management strategy for curbing global warming. This study also highlights the technical challenges that future research should prioritize in seeking feasible solutions to the inverse climate problem.

Ren, Huiying↗
NASA NTRSDate not supplied

Physics Informed Neural Nets for Systems Health Management

To facilitate and solve the prediction problem, awareness of the current health state of the system is key, since it is necessary to perform condition-based predictions. To accurately predict the future state of any system, it is required to possess knowledge of its current health state and future operational conditions. Development in data-driven algorithms in regression of complex nonlinear functions and classification tasks have generated a growing interest in artificial intelligence for industrial applications. Complex multi-physics models as well as digital twins, once purely built on physics and corresponding simplified lumped parameter iterations, can now benefit from machine learning algorithms to mitigate the lack of understanding of some complex behavior. The research work presents application of physics-informed neural nets application to a representative electric powertrain for unmanned aerial vehicles. The model is composed of physics-derived and empirical equations, integrated with connected networks that are strategically placed within the model to substitute equations that are subject to large uncertainty. Polynomial fit driven by heuristics or empirical observations can be substituted by more flexible networks that can minimize the error between model predictions and observations without being restricted to a predefined functional form. This modeling strategy allows training of networks deep inside the model and unknown parameters in a single learning stage.

Physics Informed↗
DOE OSTI2021

Machine learning accelerated turbulence modeling of transient flashing jets

We report modeling the sudden depressurization of superheated liquids through nozzles is a challenge because the pressure drop causes rapid flash boiling of the liquid. The resulting jet usually demonstrates a wide range of structures, including ligaments and droplets, due to both mechanical and thermodynamic effects. As the simulation comprises increasingly numerous phenomena, the computational cost begins to increase. One way to moderate the additional cost is to use machine learning surrogacy for specific elements of the calculation. This study presents a machine learning-assisted computational fluid dynamics approach for simulating the atomization of flashing liquids accounting for distinct stages, from primary atomization to secondary breakup to small droplets using the Σ - Y model coupled with the homogeneous relaxation model. Notably, the models for thermodynamic non-equilibrium (HRM) and Σ - Y are coupled, for the first time, with a deep neural network that simulates the turbulence quantities, which are then used in the prediction of superheated liquid jet atomization. The data-driven component of this method is used for turbulence modeling, avoiding the solution of the two-equation turbulence model typically used for Reynolds-averaged Navier-Stokes simulations for these problems. Both the accuracy and speed of the hybrid approach are evaluated, demonstrating adequate accuracy and at least 25% faster computational fluid dynamics simulations than the traditional approach. This acceleration suggests that perhaps additional components of the calculation could be replaced for even further benefit. Published under an exclusive license by AIP Publishing.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Accelerating defect predictions in semiconductors using graph neural networks

First-principles computations reliably predict the energetics of point defects in semiconductors but are constrained by the expense of using large supercells and advanced levels of theory. Machine learning models trained on computational data, especially ones that sufficiently encode defect coordination environments, can be used to accelerate defect predictions. Here, we develop a framework for the prediction and screening of native defects and functional impurities in a chemical space of group IV, III–V, and II–VI zinc blende semiconductors, powered by crystal Graph-based Neural Networks (GNNs) trained on high-throughput density functional theory (DFT) data. Using an innovative approach of sampling partially optimized defect configurations from DFT calculations, we generate one of the largest computational defect datasets to date, containing many types of vacancies, self-interstitials, anti-site substitutions, impurity interstitials and substitutions, as well as some defect complexes. We applied three types of established GNN techniques, namely crystal graph convolutional neural network, materials graph network, and Atomistic Line Graph Neural Network (ALIGNN), to rigorously train models for predicting defect formation energy (DFE) in multiple charge states and chemical potential conditions. We find that ALIGNN yields the best DFE predictions with root mean square errors around 0.3 eV, which represents a prediction accuracy of 98% given the range of values within the dataset, improving significantly on the state-of-the-art. We further show that GNN-based defective structure optimization can take us close to DFT-optimized geometries at a fraction of the cost of full DFT. The current models are based on the semi-local generalized gradient approximation-Perdew–Burke–Ernzerhof (PBE) functional but are highly promising because of the correlation of computed energetics and defect levels with higher levels of theory and experimental data, the accuracy and necessity of discovering novel metastable and low energy defect structures at the PBE level of theory before advanced methods could be applied, and the ability to train multi-fidelity models in the future with new data from non-local functionals. The DFT-GNN models enable prediction and screening across thousands of hypothetical defects based on both unoptimized and partially optimized defective structures, helping identify electronically active defects in technologically important semiconductors.

Rahman, Md Habibur (ORCID:000000027705984X)↗
DOE OSTI2022

Dynamic calibration of differential equations using machine learning, with application to turbulence models

We present a methodology for calibration of parametric ordinary and partial differential equation models, using off-the-shelf software for back-propagation in Neural Networks (NN). As a prototypical example, we consider calibration of a Reynolds-averaged Navier-Stokes (RANS) turbulence closure model, against ground truth data from direct numerical simulations (DNS) of two different turbulent flows. Numerical time integration is represented as a custom NN, where only the RANS model parameters are trainable. A loss function is defined to quantify the mismatch between the NN prediction and the ground truth over a predefined, finite time integration window. This loss function is then minimized using a gradient descent method utilizing the back-propagation algorithm. Furthermore, this dynamic approach to training is to be contrasted with a static approach, wherein a least square regression estimate for parameters is obtained in the limit of an infinitesimal time integration window. In a first test of static and dynamic approaches against ground truth data generated by the model, the former proves to be significantly faster and more accurate than the latter at recovering the parameters. When both calibration approaches are tested against DNS data, for which it is known that the model cannot achieve a perfect fit, the static approach yields a good prediction only for short times, while the dynamic approach results in physical and stable predictions over the entire integration window. After optimization of the dynamic approach for time step, spatial resolution, stability, and physics-based constraints, we obtain a 50% improvement of outcomes over those obtained from the existing, manually calibrated set of parameters, demonstrating the merits of this systematic and automated procedure.

97 MATHEMATICS AND COMPUTING↗