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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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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Actualizing Change after Experiencing Significant Mathematics PD: Hearing from Teachers of Color about Their Practice and Mathematical Identities

While it is well-documented that students of color and students from low-income communities have often been denied access to a challenging education in mathematics in the United States, less is known about how teachers of color have overcome their deprived educational backgrounds to become teachers of mathematics who implement inquiry-based instructional methods as a means to improve their students’ learning and mathematical identities. In this article, we use a sociopolitical theoretical framework to examine how seven elementary school teachers of color characterized their experiences as mathematics students, themselves as teachers of mathematics, and their mathematical identities after experiencing significant professional development in mathematics. All the participants in this study had experienced extensive professional development support in mathematics over a sustained period of time. We demonstrate through the teachers’ narratives that, given the deprived mathematics education that the majority of the participants experienced as PK-12 students, having opportunities to participate in significant and focused PD in mathematics over a sustained period transformed how teachers viewed themselves as teachers and positively impacted their mathematical identities.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2017

Transient dynamics of terrestrial carbon storage: Mathematical foundation and its applications

Terrestrial ecosystems have absorbed roughly 30% of anthropogenic CO 2 emissions over the past decades, but it is unclear whether this carbon (C) sink will endure into the future. Despite extensive modeling and experimental and observational studies, what fundamentally determines transient dynamics of terrestrial C storage under global change is still not very clear. Here we develop a new framework for understanding transient dynamics of terrestrial C storage through mathematical analysis and numerical experiments. Our analysis indicates that the ultimate force driving ecosystem C storage change is the C storage capacity, which is jointly determined by ecosystem C input (e.g., net primary production, NPP) and residence time. Since both C input and residence time vary with time, the C storage capacity is time-dependent and acts as a moving attractor that actual C storage chases. The rate of change in C storage is proportional to the C storage potential, which is the difference between the current storage and the storage capacity. The C storage capacity represents instantaneous responses of the land C cycle to external forcing, whereas the C storage potential represents the internal capability of the land C cycle to influence the C change trajectory in the next time step. The influence happens through redistribution of net C pool changes in a network of pools with different residence times. Moreover, this and our other studies have demonstrated that one matrix equation can replicate simulations of most land C cycle models (i.e., physical emulators). As a result, simulation outputs of those models can be placed into a three-dimensional (3-D) parameter space to measure their differences. The latter can be decomposed into traceable components to track the origins of model uncertainty. In addition, the physical emulators make data assimilation computationally feasible so that both C flux- and pool-related datasets can be used to better constrain model predictions of land C sequestration. Altogether, this new mathematical framework offers new approaches to understanding, evaluating, diagnosing, and improving land C cycle models.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Machine Learning meets Algebraic Combinatorics: A Suite of Benchmark Datasets to Accelerate AI for Mathematics Research

The use of benchmark datasets has become an important engine of progress in machine learning (ML) over the past 15 years. Recently there has been growing interest in utilizing machine learning to drive advances in research-level mathematics. However, off-the-shelf solutions often fail to deliver the types of insights required by mathematicians. This suggests the need for new ML methods specifically designed with mathematics in mind. The question then is: what benchmarks should the community use to evaluate these? On the one hand, toy problems such as learning the multiplicative structure of small finite groups have become popular in the mechanistic interpretability community whose perspective on explainability aligns well with the needs of mathematicians. While toy datasets are a useful benchmark for initial work, they lack the scale, complexity, and sophistication of many of the principal objects of study in modern mathematics. To address this, we introduce a new collection of benchmark datasets, Algebraic Combinatorics Benchmarks (ACBench), representing either classic or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. After describing the datasets, we discuss the challenges involved in constructing “good” mathematics benchmarks, describe baseline model performance, and discuss some of the insights these datasets can provide that may be of interest even to those who are not interested in mathematics research itself.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Mathematical nuances of Gaussian process-driven autonomous experimentation

Abstract The fields of machine learning (ML) and artificial intelligence (AI) have transformed almost every aspect of science and engineering. The excitement for AI/ML methods is in large part due to their perceived novelty, as compared to traditional methods of statistics, computation, and applied mathematics. But clearly, all methods in ML have their foundations in mathematical theories, such as function approximation, uncertainty quantification, and function optimization. Autonomous experimentation is no exception; it is often formulated as a chain of off-the-shelf tools, organized in a closed loop, without emphasis on the intricacies of each algorithm involved. The uncomfortable truth is that the success of any ML endeavor, and this includes autonomous experimentation, strongly depends on the sophistication of the underlying mathematical methods and software that have to allow for enough flexibility to consider functions that are in agreement with particular physical theories. We have observed that standard off-the-shelf tools, used by many in the applied ML community, often hide the underlying complexities and therefore perform poorly. In this paper, we want to give a perspective on the intricate connections between mathematics and ML, with a focus on Gaussian process-driven autonomous experimentation. Although the Gaussian process is a powerful mathematical concept, it has to be implemented and customized correctly for optimal performance. We present several simple toy problems to explore these nuances and highlight the importance of mathematical and statistical rigor in autonomous experimentation and ML. One key takeaway is that ML is not, as many had hoped, a set of agnostic plug-and-play solvers for everyday scientific problems, but instead needs expertise and mastery to be applied successfully. Graphical abstract

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Finite Mathematics, Finite Quantum Theory and a Conjecture on the Nature of Time

We first give a rigorous mathematical proof that classical mathematics (involving such notions as infinitely small/large, continuity etc.) is a special degenerate case of finite one in the formal limit when the characteristic p of the field or ring in finite mathematics goes to infinity. We consider a finite quantum theory (FQT) based on finite mathematics and prove that standard continuous quantum theory is a special case of FQT in the formal limit p→∞. The description of states in standard quantum theory contains a big redundancy of elements: the theory is based on real numbers while with any desired accuracy the states can be described by using only integers, i.e. rational and real numbers play only auxiliary role. Therefore, in FQT infinities cannot exist in principle, FQT is based on a more fundamental mathematics than standard quantum theory and the description of states in FQT is much more thrifty than in standard quantum theory. Space and time are purely classical notions and are not present in FQT at all. In the present paper we discuss how classical equations of motions arise as a consequence of the fact that p changes, i.e. p is the evolution parameter. It is shown that there exist scenarios when classical equations of motion for cosmological acceleration and gravity can be formulated exclusively in terms of quantum quantities without using space, time and standard semiclassical approximation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2019

DiaMonD: An Integrated Multifaceted Approach to Mathematics at the Interfaces of Data, Models, and Decisions

For many problems in computational science and engineering, the so-called forward problem—solution of the underlying mathematical model to yield output quantities of interest, given input parameters—is difficult enough for frontier complex models, which are often characterized by coupled multiphysics and possibly stochastic behavior over a wide range of length and time scales. However, we face the need to move beyond the forward problem, to address what is often the ultimate goal of CS&E: decision making. This requires us to tackle a spectrum of mathematical problems that subsume, and are thus even more difficult than, the forward problem. First, given experimental data, we wish to estimate unknown parameters characterizing a model of a natural or engineered system by solving an inverse problem. Second, we seek the optimal configuration of the system (or experiments) by solving an optimal design problem. Third, the optimal operation of the system must be determined by solving an optimal control problem. And fourth, we must quantify uncertainties as they propagate through all of the preceding problems, from data to model inference to prediction and finally to optimal design and control. Optimal mathematical methods and algorithms for such end-to-end, data-to-decisions modeling and simulation of complex problems—requiring integrated solution of forward, inverse, optimization, and UQ problems for large, multiphysics, multiscale models—entails challenges of the highest order, while also presenting great opportunities for applied mathematics research. Unfortunately, research on forward, inverse, optimization, and UQ problems has all-too-often progressed in isolation. This has led to mathematical methods that perform well when considering the forward simulation in isolation, but are prohibitive or suboptimal or unstable when combined with other methods within the framework of inversion, optimization, or UQ. The reverse is also true: general-purpose methods developed within the inversion, optimization, and UQ fields often become prohibitive when applied to complex CS&E problems since they do not exploit their structure.

58 GEOSCIENCES↗
DOE OSTI2024

Mathematics Integrated with Computer Science through Scratch: Curricular Modules for the Middle Grades

A project funded by the National Science Foundation, Computer Science Integrated with Mathematics in Middle Schools (CSIMMS) (DRL-1640039), brought middle-school mathematics teachers together with university computer science (CS) faculty and STEM education faculty to design, develop, and test curriculum modules in which CS is integrated into instruction for middle-school general-mathematics courses. The project developed integrated math/CS curriculum modules (for grades 6, 7, and 8), complete with student tasks and teacher materials to guide classroom implementation. Across the project, sixteen teachers from four middle schools representing different school contexts (both urban and suburban, serving students from a range of demographic and socioeconomic backgrounds) participated as members of the design team and trial testers of the math/CS modules. All modules underwent multiple years of testing and refinement, with extensive input from the teachers who co-designed and implemented them. Approximately 50-100 middle-school students participated in the classes of those who taught each year. The modules in this volume feature a range of models for integrating mathematics and computer science at the 6th and 7th grade levels. All modules foreground the teaching of grade-level mathematics content, with computer science functioning to motivate and/or reinforce these ideas.

Andrews larson, Christine↗
DOE OSTI2018

A mathematical discussion concerning the performance of multilayer perceptron-type artificial neural networks through use of orthogonal bipolar vectors

This work proposes the use of orthogonal bipolar vectors (OBV) as targets for multilayer perceptron (MLP) Artificial neural networks pertaining to the pattern recognition field. Implications involving this proposal are supported by a proper mathematical discussion. This study shows that the use of OBV as targets improves the class identification performance of MLP networks for pattern recognition with more than two classes when compared to the widely used conventional target bipolar vector (CBV). In addition to the overall performance improvement, it is possible to obtain high recognition rates with fewer training epochs, while keeping a feasible generalization with other data. This behavior is related to the Euclidean distance of the points in the output space, which is much higher in the case of OBVs. Therefore, with a greater distance between the points in the output space, the MLP network becomes less prone to associate trained outputs with incorrect targets. The mathematical discussion in this study is based on the backpropagation training algorithm. In addition to the mathematical discussion, results from experiments with real data are presented. These results follow those obtained by mathematical derivations. Real data sets used in the experiments are available from: (a) the Semeion Handwritten Digit of Machine Learning Repository, international repository; (b) Iris Image Database from the Chinese Academy of Sciences—CASIA; (c) Australian Sign Language, signs of Machine Learning Repository, international repository. Experiments were conducted to measure the mean Euclidean distance between the MLP network output subjected to a given input pattern and outputs related to other pattern classifications. Experimental results show that MLP networks trained with OBVs were able to ensure a Euclidean distance between the output and incorrect targets greater than MLP networks trained with CBVs (by up to 4 times for handwritten, by up to 12 times for iris images and by up to 10 times for Auslan signs language). As such, the adoption of OBVs as target vectors for MLPs reduces the computational effort for training the network.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Bioelectrochemical reactor to manage anthropogenic sulfate pollution for freshwater ecosystems: Mathematical modeling and experimental validation

Anthropogenic sulfate loading into otherwise low-sulfate freshwater systems can cause significant ecological consequences as a biogeochemical stressor. To address this challenge, in situ bioremediation technologies have been developed to leverage naturally occurring microorganisms that transform sulfate into sulfide rather than implementing resource-intensive physio-chemical processes. However, bioremediation technologies often require the supply of electron donors to facilitate biological sulfate reduction. Bioelectrochemical systems (BES) can be an alternative approach for supplying molecular hydrogen as an electron donor for sulfate-reducing bacteria through water electrolysis. Although the fundamental mechanisms behind BESs have been studied, limited research has evaluated the design and operational parameters of treatment systems when developing BESs on a scale relevant to environmental systems. Further, this study aimed to develop an application-based mathematical model to evaluate the performance of BESs across a range of reactor configurations and operational modes. The model was based on sulfate transformation by hydrogenotrophic sulfate-reducing bacteria coupled with the recovery of solid iron sulfide species formed by the oxidative dissolution of dissolved ferrous iron from a stainless steel anode. Sulfate removal closely corresponded to the rate of electrolytic hydrogen production and hydraulic residence time but was less sensitive to specific microbial rate constants. The mathematical model results were compared to experimental data from a pilot-scale BES tested with nonacidic mine drainage as a case study. The close agreement between the mathematical model and the pilot-scale BES experiment highlights the efficacy of using a mathematical model as a tool to develop a conceptual design of a scaled-up treatment system.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Report on Pure Mathematics of Machine Learning

We report on the findings conducted by the Pure Mathematics in Machine Learning (PMML) Project under the umbrella of the Mathematics in Artificial Reasoning in Sciences (MARS) Laboratory Initiative at the Pacific Northwest National Laboratory. The findings contained in this report spans three years of research from the project team. The objectives of the PMML project are two-fold: 1. To gain understanding of modern machine learners through empirical research via structured and sound experimental design, and 2. To prove mathematical theories based upon these empirical findings. We present both empirical results along with mathematical theories in this document.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Leveraging public AI tools to explore systems biology resources in mathematical modeling

Predictive mathematical modeling is an essential part of systems biology and is interconnected with information management. Systems biology information is often stored in specialized formats to facilitate data storage and analysis. These formats are not designed for easy human readability and thus require specialized software to visualize and interpret results. Therefore, comprehending modeling and underlying networks and pathways is contingent on mastering systems biology tools, which is particularly challenging for users with no or little background in data science or system biology. To address this challenge, we investigated the usage of public Artificial Intelligence (AI) tools in exploring systems biology resources in mathematical modeling. We tested public AI’s understanding of mathematics in models, related systems biology data, and the complexity of model structures. Our approach can enhance the accessibility of systems biology for non-system biologists and help them understand systems biology without a deep learning curve.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2019

Within-host mathematical models of hepatitis B virus infection: Past, present, and future

Mathematical modeling has been instrumental in enhancing our understanding of the viral dynamics of hepatitis B virus (HBV) infection. We give a primer on HBV infection in humans and a brief overview of the development of within-host mathematical models of HBV infection. In the last decade, models have advanced from considering chronic HBV infections under therapy to the pathogenesis of infection. We also summarize estimates of key viral dynamic parameters that have varied greatly among studies, and show that they impact model predictions. Here, future directions for mathematical modeling of HBV infection are proposed to better understand emerging therapies, the HBV life cycle, predicting cure, and the mechanisms involved in the immune response to HBV.

59 BASIC BIOLOGICAL SCIENCES↗