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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 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Application of Machine Learning to Predict the Response of the Liquid Mercury Target at the Spallation Neutron Source

The Spallation Neutron Source (SNS) at Oak Ridge National Laboratory is currently the most powerful accelerator-driven neutron source in the world. The intense proton pulses strike on SNS’s mercury target to provide bright neutron beams, which also leads to severe fluid-structure interactions inside the target. Prediction of resultant loading on the target is difficult particularly when helium gas is intentionally injected into mercury to reduce the loading and mitigate the pitting damage on the target’s internal walls. Leveraging the power of machine learning and the measured target strain, we have developed machine learning surrogates for modeling the discrepancy between simulations and experimental strain data. We then employ these surrogates to guide the refinement of the high-fidelity mercury/helium mixture model to predict a better match of target strain response.

Lin, Lianshan↗
DOE OSTI2025

Experiment-modeling studies comparing energy dissipation in the DIII-D SAS and SAS-VW divertors

Recent DIII-D experiments on Small Angle Slot (SAS) divertors have confirmed that a combination of divertor closure and target shaping can enhance cooling across the divertor target and increase energy dissipation, but with significant dependence on B T (toroidal magnetic field) direction. In these novel divertors, the roles of closure, target shaping, drifts, and scale lengths are all interconnected in optimizing dissipation, with the separatrix electron density n eSEP being the key parameter associated with the level of dissipation/detachment. After modifying the original flat-targeted graphite SAS to include a V shape with a tungsten coating on the outer side of the divertor (SAS-VW), matched series of discharges were run to compare to detailed SOLPS-ITER modeling. Experimentally, when run as designed with the outer strike point at the slot vertex, SAS-VW requires nearly identical n eSEP for detachment as the original SAS, with little difference in dissipation for the new geometry. This is in contrast to (1) earlier modeling predictions that a small change of the SAS geometry to a V shape should enhance dissipation at the same n eSEP for magnetic configurations having better H-mode access (ion B × ∇B drift directed into the divertor), and (2) despite the achievement of significantly higher (2-7x) neutral pressures and compression in the SAS-VW slot. Comparisons of experimental density scans to the most recent SOLPS-ITER modeling with ExB drifts show reasonable agreement for dissipation/detachment onset when using separatrix density as the independent parameter. In order to help understand the discrepancy in modeled vs actual performance for the new configuration, additional measurements varying gas injection location and impurity injection were undertaken. In-slot D 2 gas fueling is more effective (5–22 %) in promoting detachment, in accord with modeling. In-slot impurity injection (N 2 or Ne) can yield 30 % lower core Z eff and 15 % less confinement degradation after detachment compared to main chamber puffing, as well as relatively lower tungsten leakage from the divertor. Modeling can also reproduce the improved detachment seen as the strike point moves inboard of the slot vertex. While we can explain the effects of the most important parameters causing energy dissipation in these slot divertors, it remains that many aspects of their behavior cannot be accurately modeled using state-of-art codes such as SOLPS-ITER. This is of concern for future model-driven designs utilizing similar V-shaped geometries.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2015

RECONCILING THE INFRARED CATASTROPHE AND OBSERVATIONS OF SN 2011fe

The observational effects of the “Infrared Catastrophe” are discussed in view of the very late observations of the Type Ia SN 2011fe. Our model spectra at 1000 days take non-local radiative transfer into account and find that this has a crucial impact on the spectral formation. Although rapid cooling of the ejecta to a few 100 K occurs also in these models, the late-time optical/NIR flux is brighter by 1–2 mag due to redistribution of UV emissivity, resulting from non-thermal excitation and ionization. This effect brings models into better agreement with late-time observations of SN 2011fe, and other SNe Ia, and offers a solution to the long-standing discrepancy between models and observations. The models show that spectral formation shifts from Fe ii and Fe iii at 300 days to Fe i at 1000 days, which explains the apparent wavelength shifts seen in SN 2011fe. We discuss the effects of time dependence and energy input from {sup 57}Co, finding both to be important at 1000 days.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

Consistency of the Planck CMB data and ΛCDM cosmology

We test the consistency between Planck temperature and polarization power spectra and the concordance model of Λ Cold Dark Matter cosmology (ΛCDM) within the framework of Crossing statistics. We find that Planck TT best fit ΛCDM power spectrum is completely consistent with EE power spectrum data while EE best fit ΛCDM power spectrum is not consistent with TT data. However, this does not point to any systematic or model-data discrepancy since in the Planck EE data, uncertainties are much larger compared to the TT data. We also investigate the possibility of any deviation from ΛCDM model analyzing the Planck 2015 data. Results from TT, TE and EE data analysis indicate that no deviation is required beyond the flexibility of the concordance ΛCDM model. Our analysis thus rules out any strong evidence for beyond the concordance model in the Planck spectra data. We also report a mild amplitude difference comparing temperature and polarization data, where temperature data seems to have slightly lower amplitude than expected (consistently at all multiples), as we assume both temperature and polarization data are realizations of the same underlying cosmology.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Constraining the low-cloud optical depth feedback at middle and high latitudes using satellite observations

The increase in cloud optical depth with warming at middle and high latitudes is a robust cloud feedback response found across all climate models. This study builds on results that suggest the optical depth response to temperature is timescale invariant for low-level clouds. The timescale invariance allows one to use satellite observations to constrain the models' optical depth feedbacks. Three passive-sensor satellite retrievals are compared against simulations from eight models from the Atmosphere Model Intercomparison Project (AMIP) of the 5th Coupled Model Intercomparison Project (CMIP5). This study confirms that the low-cloud optical depth response is timescale invariant in the AMIP simulations, generally at latitudes higher than 40°. Compared to satellite estimates, most models overestimate the increase in optical depth with warming at the monthly and interannual timescales. Many models also do not capture the increase in optical depth with estimated inversion strength that is found in all three satellite observations and in previous studies. The discrepancy between models and satellites exists in both hemispheres and in most months of the year. A simple replacement of the models' optical depth sensitivities with the satellites' sensitivities reduces the negative shortwave cloud feedback by at least 50% in the 40°–70°S latitude band and by at least 65% in the 40°–70°N latitude band. Furthermore, based on this analysis of satellite observations, we conclude that the low-cloud optical depth feedback at middle and high latitudes is likely too negative in climate models.

58 GEOSCIENCES↗
DOE OSTI2015

TOWARD CHARACTERIZATION OF THE TYPE IIP SUPERNOVA PROGENITOR POPULATION: A STATISTICAL SAMPLE OF LIGHT CURVES FROM Pan-STARRS1

In recent years, wide-field sky surveys providing deep multiband imaging have presented a new path for indirectly characterizing the progenitor populations of core-collapse supernovae (SNe): systematic light-curve studies. We assemble a set of 76 grizy-band Type IIP SN light curves from Pan-STARRS1, obtained over a constant survey program of 4 yr and classified using both spectroscopy and machine-learning-based photometric techniques. We develop and apply a new Bayesian model for the full multiband evolution of each light curve in the sample. We find no evidence of a subpopulation of fast-declining explosions (historically referred to as ''Type IIL'' SNe). However, we identify a highly significant relation between the plateau phase decay rate and peak luminosity among our SNe IIP. These results argue in favor of a single parameter, likely determined by initial stellar mass, predominantly controlling the explosions of red supergiants. This relation could also be applied for SN cosmology, offering a standardizable candle good to an intrinsic scatter of ≲ 0.2 mag. We compare each light curve to physical models from hydrodynamic simulations to estimate progenitor initial masses and other properties of the Pan-STARRS1 Type IIP SN sample. We show that correction of systematic discrepancies between modeled and observed SN IIP light-curve properties and an expanded grid of progenitor properties are needed to enable robust progenitor inferences from multiband light-curve samples of this kind. This work will serve as a pathfinder for photometric studies of core-collapse SNe to be conducted through future wide-field transient searches.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Modeling 2p-2h Interactions in Neutrino-Nucleus Scattering: A Comparative Overview of the Valencia and SuSAv2-MEC Models

2p-2h interactions are crucial for describing ν-A scattering, especially in the "dip region" between the Quasi-Elastic (QE) peak and Δ-resonance production. They involve the ejection of two nucleons, leading to two holes in the nuclear ground state. The underlying nuclear dynamics involve short-range correlations (SRC), long-range correlations (Random Phase Approximation, RPA), and the interplay of one- and two-body currents. Several models attempt to describe 2p-2h interactions, including Microscopic Models (e.g., Valencia), Scaling-Based Models (e.g., SuSAv2-MEC), and Transport Models (e.g., GiBUU). Despite progress, discrepancies exist between models and with data. I Will give an overview of 2p2h from the perspective of two Models Valencia and SuSAv2-MEC.

Hassinin, Karim [Houston U.]↗
DOE OSTI2025

A Model Intercomparison Study of Aerosol‐Cloud‐Turbulence Interactions in a Cloud Chamber: 1. Model Results

This study presents the first model intercomparison of aerosol‐cloud‐turbulence interactions in a controlled cloudy Rayleigh‐Bénard Convection chamber environment, utilizing the Pi Chamber at Michigan Technological University. We analyzed simulated cloud chamber‐averaged statistics of microphysics and thermodynamics in a warm‐phase, cloudy environment under steady‐state conditions at varying aerosol injection rates. Simulation results from seven distinct models (DNS, LES, and a 1D turbulence model) were compared. Our findings demonstrate that while all models qualitatively capture observed trends in droplet number concentration, mean radius, and droplet size distributions at both high and low aerosol injection rates, significant quantitative differences were observed. Notably, droplet number concentrations varied by over two orders of magnitude between models for the same injection rates, indicating sensitivities to the model treatments in droplet activation and removal and wall fluxes. Furthermore, inconsistencies in vertical relative humidity profiles and in achieving steady‐state liquid water content suggest the need for further investigation into the mechanisms driving these variations. Despite these discrepancies, the models generally reproduced consistent power‐law relationships between the microphysical variables. This model intercomparison underscores the importance of controlled cloud chamber experiments for validating and improving cloud microphysical parameterizations. Recommendations for future modeling studies are also highlighted, including constraining wall conditions and processes, investigating droplet/aerosol removal (including sidewall losses), and conducting simplified experiments to isolate specific processes contributing to model divergence and reduce model uncertainties.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Temperature effect on phase state and reactivity controls atmospheric multiphase chemistry and transport of PAHs

Polycyclic aromatic hydrocarbons like benzo(a)pyrene (BaP) in atmospheric particulate matter pose a threat to human health because of their high carcinogenicity. In the atmosphere, BaP is mainly degraded through a multiphase reaction with ozone, but the fate and atmospheric transport of BaP are poorly characterized. Earlier modeling studies used reaction rate coefficients determined in laboratory experiments at room temperature, which may overestimate/underestimate degradation rates when applied under atmospheric conditions. Moreover, the effects of diffusion on the particle bulk are not well constrained, leading to large discrepancies between model results and observations. We show how regional and global distributions and transport of BaP can be explained by a new kinetic scheme that provides a realistic description of the temperature and humidity dependence of phase state, diffusivity, and reactivity of BaP-containing particles. Low temperature and humidity can substantially increase the lifetime of BaP and enhance its atmospheric dispersion through both the planetary boundary layer and the free troposphere. The new scheme greatly improves the performance of multiscale models, leading to better agreement with observed BaP concentrations in both source regions and remote regions (Arctic), which cannot be achieved by less-elaborate degradation schemes (deviations by multiple orders of magnitude). Our results highlight the importance of considering temperature and humidity effects on both the phase state of aerosol particles and the chemical reactivity of particulate air pollutants.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Basic Research Needs for Inverse Methods for Complex Systems under Uncertainty

Inverse problems, which aim to infer unknown properties of a system using experimental and observational data, are central to addressing many of the U.S. Department of Energy’s (DOE) most critical scientific and engineering challenges. Accurate, computationally efficient, and data-efficient solutions to inverse problems are essential for advancing DOE mission-critical science drivers, including analyzing data from large-scale experimental facilities, optimizing fusion reactor performance, accelerating materials discovery, enhancing geophysical imaging, improving wildfire predictions, and enabling autonomous systems and digital twins. However, these problems are becoming increasingly complex, often involving nonlinear, highdimensional, and interconnected systems and models that span multiple physics and scales, while relying on data with varying quantity, quality, and information content. Compounding these challenges is the uncertainty inherent in DOE-relevant systems, where errors in inputs, noise in data, incompleteness of data, and discrepancies between models and reality constrain the accuracy and precision of solutions. At the same time, the convergence of recent scientific computing trends—scientific machine learning, artificial intelligence, and computing advances such as exascale computing—is creating unprecedented opportunities for tackling these challenges. The cross-cutting nature of inverse problems, combined with their growing complexity and rapidly evolving data and algorithmic demands, strongly motivates the formulation of a prioritized research agenda to maximize their capabilities and impact. In response to this need, DOE’s Advanced Scientific Computing Research (ASCR) program in the Office of Science convened the Workshop on Basic Research Needs for Inverse Problems for Complex Systems Under Uncertainty in June 2025. This workshop brought together experts across disciplines to identify grand challenges and major opportunities in the field. Through collaborative discussions, the workshop defined transformative research directions aimed at addressing the mathematical, statistical, and computational challenges posed by inverse problems under uncertainty. As a result of these efforts, four priority research directions (PRDs) were identified to guide future research and development in this area. These PRDs, summarized below, represent a roadmap for advancing the foundational science and mathematics of inverse problems, enabling robust, scalable, and uncertainty-aware solutions that are critical for DOE applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

22 Ne Phase Separation as a Solution to the Ultramassive White Dwarf Cooling Anomaly

The precise astrometric measurements of the Gaia Data Release 2 have opened the door to detailed tests of the predictions of white dwarf cooling models. Significant discrepancies between theory and observations have been identified, the most striking affecting ultramassive white dwarfs. Cheng et al. found that a small fraction of white dwarfs on the so-called Q branch must experience an extra cooling delay of ~8 Gyr not predicted by current models. 22 Ne phase separation in a crystallizing C/O white dwarf can lead to a distillation process that efficiently transports 22 Ne toward its center, thereby releasing a considerable amount of gravitational energy. Here, using state-of-the-art Monte Carlo simulations, we show that this mechanism can largely resolve the ultramassive cooling anomaly if the delayed population consists of white dwarfs with moderately above-average 22 Ne abundances. We also argue that 22 Ne phase separation can account for the smaller cooling delay currently missing for models of white dwarfs with more standard compositions.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Machine learning based algorithms for uncertainty quantification in numerical weather prediction models

Complex numerical weather prediction models incorporate a variety of physical processes, each described by multiple alternative physical schemes with specific parameters. The selection of the physical schemes and the choice of the corresponding physical parameters during model configuration can significantly impact the accuracy of model forecasts. There is no combination of physical schemes that works best for all times, at all locations, and under all conditions. It is therefore of considerable interest to understand the interplay between the choice of physics and the accuracy of the resulting forecasts under different conditions. This paper demonstrates the use of machine learning techniques to study the uncertainty in numerical weather prediction models due to the interaction of multiple physical processes. The first problem addressed herein is the estimation of systematic model errors in output quantities of interest at future times, and the use of this information to improve the model forecasts. The second problem considered is the identification of those specific physical processes that contribute most to the forecast uncertainty in the quantity of interest under specified meteorological conditions. In order to address these questions we employ two machine learning approaches, random forests and artificial neural networks. The discrepancies between model results and observations at past times are used to learn the relationships between the choice of physical processes and the resulting forecast errors. Numerical experiments are carried out with the Weather Research and Forecasting (WRF) model. The output quantity of interest is the model precipitation, a variable that is both extremely important and very challenging to forecast. The physical processes under consideration include various micro-physics schemes, cumulus parameterizations, short wave, and long wave radiation schemes. The experiments demonstrate the strong potential of machine learning approaches to aid the study of model errors.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Significant inconsistency of vegetation carbon density in CMIP5 Earth system models against observational data

Abstract Earth system models (ESMs) have been widely used for projecting global vegetation carbon dynamics, yet how well ESMs performed for simulating vegetation carbon density remains untested. We compiled observational data of vegetation carbon density from literature and existing data sets to evaluate nine ESMs at site, biome, latitude, and global scales. Three variables—root (including fine and coarse roots), total vegetation carbon density, and the root:total vegetation carbon ratios (R/T ratios), were chosen for ESM evaluation. ESM models performed well in simulating the spatial distribution of carbon densities in root ( r = 0.71) and total vegetation ( r = 0.62). However, ESM models had significant biases in simulating absolute carbon densities in root and total vegetation biomass across the majority of land ecosystems, especially in tropical and arctic ecosystems. Particularly, ESMs significantly overestimated carbon density in root (183%) and total vegetation biomass (167%) in climate zones of 10°S–10°N. Substantial discrepancies between modeled and observed R/T ratios were found: the R/T ratios from ESMs were relatively constant, approximately 0.2 across all ecosystems, along latitudinal gradients, and in tropic, temperate, and arctic climatic zones, which was significantly different from the observed large variations in the R/T ratios (0.1–0.8). There were substantial inconsistencies between ESM‐derived carbon density in root and total vegetation biomass and the R/T ratio at multiple scales, indicating urgent needs for model improvements on carbon allocation algorithms and more intensive field campaigns targeting carbon density in all key vegetation components.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

The Role of Internal Variability in Springtime Arctic Amplification from 1980 to 2022

Arctic amplification (AA) refers to the enhanced warming of the Arctic relative to the global average due to rising greenhouse gases, measured as the ratio of Arctic-mean to global-mean surface air temperature (SAT) trends. From 1980 to 2022, annual-mean AA reached 4.2 (Arctic defined as north of 70°N). Climate models simulate AA but fail to reproduce its magnitude. Sweeney et al. attributed much of this model–observation discrepancy to internal variability. AA shows seasonality and so does the discrepancy. Spring (March–May) shows the largest gap: Observed AA is 4.2, while the multimodel mean is 2.7. This raises several questions: 1) What role does internal variability play in observed spring AA? 2) How does simulated spring AA compare to observations when internal variability is removed? 3) If internal variability is significant, what mechanisms drive it? To address these, we adapted the machine learning algorithm from Sweeney et al., training on simulated multidecadal spring SAT and sea level pressure (SLP) trend maps. Our results show that internal variability enhanced spring Arctic warming by 37% and reduced global warming by 10%. Removing internal variability reconciles the spring AA discrepancy. The estimated internal contribution to Arctic spring warming is supported by an independent dynamical adjustment approach. We identify an atmospheric circulation pattern in observations associated with this internal warming. Observed internal Siberian SAT and SLP trends follow the simulated SAT–SLP relationship but lie at the distribution’s extreme, suggesting models generally underestimate internal variability unless the observed configuration reflects a rare real-world realization.

Arctic↗
DOE OSTI2015

SU-E-T-466: Implementation of An Extension Module for Dose Response Models in the TOPAS Monte Carlo Toolkit

Purpose: To develop and verify an extension to TOPAS for calculation of dose response models (TCP/NTCP). TOPAS wraps and extends Geant4. Methods: The TOPAS DICOM interface was extended to include structure contours, for subsequent calculation of DVH’s and TCP/NTCP. The following dose response models were implemented: Lyman-Kutcher-Burman (LKB), critical element (CE), population based critical volume (CV), parallel-serials, a sigmoid-based model of Niemierko for NTCP and TCP, and a Poisson-based model for TCP. For verification, results for the parallel-serial and Poisson models, with 6 MV x-ray dose distributions calculated with TOPAS and Pinnacle v9.2, were compared to data from the benchmark configuration of the AAPM Task Group 166 (TG166). We provide a benchmark configuration suitable for proton therapy along with results for the implementation of the Niemierko, CV and CE models. Results: The maximum difference in DVH calculated with Pinnacle and TOPAS was 2%. Differences between TG166 data and Monte Carlo calculations of up to 4.2%±6.1% were found for the parallel-serial model and up to 1.0%±0.7% for the Poisson model (including the uncertainty due to lack of knowledge of the point spacing in TG166). For CE, CV and Niemierko models, the discrepancies between the Pinnacle and TOPAS results are 74.5%, 34.8% and 52.1% when using 29.7 cGy point spacing, the differences being highly sensitive to dose spacing. On the other hand, with our proposed benchmark configuration, the largest differences were 12.05%±0.38%, 3.74%±1.6%, 1.57%±4.9% and 1.97%±4.6% for the CE, CV, Niemierko and LKB models, respectively. Conclusion: Several dose response models were successfully implemented with the extension module. Reference data was calculated for future benchmarking. Dose response calculated for the different models varied much more widely for the TG166 benchmark than for the proposed benchmark, which had much lower sensitivity to the choice of DVH dose points. This work was supported by National Cancer Institute Grant R01CA140735.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2019

Statistically Rigorous Uncertainty Quantification for Physical Parameter Model Calibration with Functional Output

In experiments conducted on the Z-machine at Sandia National Laboratories, dynamic material properties cannot be analyzed using traditional analytic methods, necessitating solving an inverse problem. Bayesian model calibration is a statistical framework for solving an inverse problem to estimate parameters input into a computational model in the presence of multiple uncertainties. Disentangling input parameter uncertainty and model misspecification is often poorly identified problem. When using computational models for physical parameter estimation, the issue of parameter identifiability must be carefully considered to obtain accurate and precise estimates of physical parameters. Additionally, in dynamic material properties applications, the experimental output is a function, velocity over time. While we can sample an arbitrarily large number of points from the measured velocity, these curves only contain a finite amount of information about the calibration parameters. In this report, we propose modifications to the Bayesian model calibration framework to simplify and improve the estimation of physical parameters with functional outputs. Specifically, we propose scaling the likelihood function by an effective sample size rather than modeling the discrepancy function; and modularizing input nuisance parameters with weakly identified parameters. We evaluate the performance of these proposed methods using a statistical simulation study and then apply these methods to estimate parameters of the tantalum equation of state. We conclude that these proposed methods can provide simple, fast, and statistically valid alternatives to the full Bayesian model calibration procedure; and that these methods can be used to estimate parameters of the equation of state for tantalum.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Black carbon absorption at the global scale is affected by particle-scale diversity in composition

Atmospheric black carbon (BC) exerts a strong, but uncertain, warming effect on the climate. BC that is coated with non-absorbing material absorbs more strongly than the same amount of BC in an uncoated particle, but the magnitude of this absorption enhancement (E abs ) is not well constrained. Modelling studies and laboratory measurements have found stronger absorption enhancement than has been observed in the atmosphere. Here, using a particle-resolved aerosol model to simulate diverse BC populations, we show that absorption is overestimated by as much as a factor of two if diversity is neglected and population-averaged composition is assumed across all BC-containing particles. If, instead, composition diversity is resolved, we find E abs =1-1.5 at low relative humidity, consistent with ambient observations. This study offers not only an explanation for the discrepancy between modelled and observed absorption enhancement, but also demonstrates how particle-scale simulations can be used to develop relationships for global-scale models.

Science & Technology - Other Topics↗
DOE OSTI2023

Bi-fidelity modeling of uncertain and partially unknown systems using DeepONets

Recent advances in modeling large-scale, complex physical systems have shifted research focuses towards data-driven techniques. However, generating datasets by simulating complex systems can require significant computational resources. Similarly, acquiring experimental datasets can prove difficult. For these systems, often computationally inexpensive, but in general inaccurate models, known as the low-fidelity models, are available. Here in this paper, we propose a bi-fidelity modeling approach for complex physical systems, where we model the discrepancy between the true system's response and a low-fidelity response in the presence of a small training dataset from the true system's response using a deep operator network, a neural network architecture suitable for approximating nonlinear operators. We apply the approach to systems that have parametric uncertainty and are partially unknown. Three numerical examples are used to show the efficacy of the proposed approach to model uncertain and partially unknown physical systems.

MATHEMATICS AND COMPUTING↗