The Information Sciences Directorate (ISD) describes its programs in mathematical sciences as follows: Programs in the Mathematical Sciences Division underlie and enable “understanding of complex nonlinear systems, stochastic networks and systems, mechanistic models of adaptive biological systems and networks, and the vast variety of partial differential equation-based phenomena. Nonlinear structures and metrics for modeling and studying complex systems are sought, as is creating theory for the control of stochastic systems, spatial-temporal statistical inference, data classification and regression analysis, predicting and controlling biology through hierarchical and adaptive models, enabling new capabilities through bio-inspired techniques, creating high-fidelity computational principles for sharp-interface flows, solving inverse problems, deriving reduced-order methods, and developing computational linguistics.” The programs are Modeling of Complex Systems, Probability and Statistics, Biomathematics, and Computational Mathematics.1 The division’s budget of $20.2 million, including $7.5 million in core funding and $12.7 in leveraged funding, supports 156 current projects.
Scientific Quality and Degree of Innovation
The objectives of this program are to develop mathematical characterizations and computational models for (1) common themes in anomalous physics; (2) material-related issues in layered and two-dimensional geometries, energetic crystals, and porous media; and (3) high-fidelity computational electrodynamic modeling and associated inverse problems. The program manager’s strategy for achieving these objectives seemed less clear and cogent than for other programs in the Mathematical Sciences Division. The program of research is substantial, but it does not evince a consistent theme across the funded projects. However, the research funded is novel and is leading the field in several important ways. The program’s thrusts in fractional order operators and models for quantum stochastic differential equations, for example, offer a potentially transformative alternative to competing approaches in anomalous physics and material. The program manager’s willingness to trade off risk versus innovation has enabled the program to have a portfolio that is distinguishable from the bulk of the research supported by the National Science Foundation (NSF) and other research sponsors.
It appears that the program’s scientific objectives will be met. The program has found some unique areas to fund that are not duplicative of those funded by other sponsors and that represent a
meaningful niche for the Army Research Office (ARO) mission. For example, the program is proposing to fund research that investigates the use of conservation laws to assist with large data sets. This is an interesting, speculative, and high-risk idea that will have enormous benefit if successful. No other sponsor in the mathematical sciences is actively supporting this type of work.
Computational Mathematics had some significant scientific advances. For example, the work on fast solvers for fractional partial differential equations represents a significant advance in the field. Also, the funded work in support of ground vehicles is important and significant.
One of the most important transitions is the work supporting the Army’s Ground Vehicle Program. This effort is transitioning into the replacement/upgrade for the North Atlantic Treaty Organization Reference Mobility Model, called Next Generation NRMM. Funded work is also supporting the Engineered Resilient Systems Program at the Army’s Engineer Research and Development Center and was used on the Joint Light Tactical Vehicle Program. Other significant transitions have been the funded work on effective Hamiltonians for monolayers and the work on mesh adaptation.
The Biomathematics Program is relatively small. The program’s budget of $4.8 million includes $1.3 million in core funding and supports 35 current projects. Consequently, its research scope is not broad, although the quality of the projects is generally high. The vision of the program is to identify and mathematize the fundamental principles of biological structure, function, and development applying across systems and scales, to enable revolutionary advances in soldier health, performance, and materiel. Consistent with this vision, the Biomathematics Program focuses on areas such as understanding circadian rhythm to improve sleep/alertness of soldiers in the field, and molecular interventions to improve healing of wounds.
Scientific Quality and Degree of Innovation
In general, the research funded under this program is of good scientific quality and is responsive to the scientific objectives of the program. The program funds research that is of interest to the Army and to the Department of Defense (DoD) in general, yet it is unlikely to receive funding from other agencies. In this light, the funding allocated by the Biomathematics Program to research projects does not overlap what other agencies would be funding. While projects are generally of good scientific quality, the tools that researchers use to address the scientific questions appear to be limited. As an example, considering stochastic techniques in addition to deterministic techniques would seem to be appropriate when one of the scientific objectives is to understand principles behind biological heterogeneity.
The program would stand to benefit if the range of research approaches used to address problems were broadened, where appropriate. Given a specific objective, such as enhancing molecular responses to improve healing, this could be accomplished by seeking out researchers who would bring differing perspectives and competing approaches.
Supported researchers appear to be chosen under the Biomathematics Program in a principled way. The portfolio of projects in this program make up a coherent whole focused on the program’s scientific objectives. The program manager seems to be in close communication with the researchers in the field and has an opportunity to discuss potentially productive changes to the research scope. The changes in focus from decision theory to learning and from determinism to stochasticity are timely.
Learning methods have been successfully deployed in many sciences. The program would be significantly enhanced if its scope were broadened to include researchers working on learning methods and paradigms.
Potential benefits would come from making use of short-term funding mechanisms to encourage multiple researchers to work on the same problem, ideally from divergent perspectives. There may be opportunities to increase the feedback and cooperation of managers of other programs for projects with intellectual themes that cross over into other areas of the mathematical sciences.
Most researchers appear to be accomplishing stated research goals. For example, a project with the goal of understanding group behavior found that when leaders in a network are chosen to maximize information centrality, it is possible to predict group behavior with smallest tracking errors.
Program researchers have received an impressive array of awards and recognition. That said, the program would benefit from the development and deployment of a set of metrics designed to reflect the impact that the Biomathematics Program research is having on Army programs and on science in general. The appendix of this report lists a broad set of metrics that ARO could consider for assessment of its programs.
The program is responsive to Army and DoD needs. The program manager provides ample opportunity for input to the Army at multiple phases of the research process. Multiple projects have resulted in outputs of interest to various government agencies, including the Defense Advanced Research Projects Agency’s Defense Sciences Office, the Office of Naval Research, the Army Research Laboratory’s (ARL)’s Human Research and Engineering Directorate, as well as to several organizations in the private sector (e.g., Marriott Hotels, Sustainable Bioproducts LLC). A project focused on the transfer of protein analysis techniques to the analysis of social networks has generated considerable interest in the Army for translational follow-on work.
It would be beneficial to pair researchers with different areas of expertise—that is, a basic researcher and researchers with the ability to translate research results into practice. The program has strong relevance to Army needs and would benefit from continued encouragement of active participation by Army professionals in the design of research objectives.
While much of the funding can be used to address problems that are current, the program would benefit from giving some thought to the needs of the future. This might be accomplished by engaging with a futurist who may be able to predict which problems will become relevant in 10 or 20 years.
The Biomathematics Program funds several female principal investigators (PIs) as well as researchers from minority-serving institutions. The program manager also emphasizes the value of
funding earlier-career researchers over more established researchers. This can bring to the fore new talent with new ideas, if the quality and value of the research portfolio is maintained.
Scientific Quality and Degree of Innovation
Starting from a hypothesis that direct simulation approaches (e.g., computational partial differential equations) are insufficient for effective characterization of the systems of Army/DoD interest, this program has chosen to focus its activities on the nascent fields of geometric and topological analysis of complex systems. The program oversees a relatively small component of approximately 10 percent of the overall Mathematical Sciences Division budget. Within the budgetary constraint and chosen geometric/topological focus, the program has judiciously selected topics related to data analysis and modeling, learning theory, social group dynamics, and convergence of theoretical foundations with computational aspects.
Grants have led to research of good quality that includes extension of facial recognition techniques, feature recognition from incomplete data, and combining machine learning with topological concepts (e.g., persistence diagrams). There is a mix of support of experienced researchers perfecting established methods with novel investigations. An example of the former is the research on multimodal facial recognition to extend work from the 1990s on linear algebra (Eigenfaces) or statistical approaches (Fisherfaces). Within the latter category are the works on terrain recognition from sparse observation or transformation of data into persistence diagrams.
Whereas some of the supported research (e.g., facial recognition) also receives funding from NSF, several projects seem to fit exclusively within the ARO’s niche (e.g., environment recognition and reconstruction from incomplete data), and combine innovation in mathematical approaches with relevance to Army/DoD interests.
The challenge facing the program is to balance a multitude of recently proposed techniques for geometric and topological analysis of realistic systems, of untested efficacy, with the constraints of a limited budget and need for practical results. Within these parameters, the current portfolio of funded projects does capture a good sample of the currently known or promising science in this area. The funding mechanism itself could be refined. There are a number of long-term projects (4-5 years), which allow for depth but come at the price of leaving promising areas uninvestigated. Given the nature of this program, short, exploratory investigations might be preferable unless there is cross-support from other divisions interested in a well-defined application. This approach is already present to some extent but could be further refined by analogy to the National Institutes of Health (NIH) approach of exploratory R21 projects leading to R01 projects upon buy-in from other Army entities. Morse theory, topological visualization, and statistical manifolds are examples of other promising approaches that might find space within the program if the funding model favors short-term exploratory investigations.
Funded researchers have received notable external recognitions (fellowships, including an Intel Early Career fellowship). All projects have been productive in terms of scholarly publications and the training of graduate students.
Funded projects have led to tangible benefit to the Army such as the work on environment reconstruction delivered to the ARL Survivability and Analysis Directorate or the work on improved facial recognition delivered to the ARL Sensors and Electron Devices Directorate. In addition to these direct transitions, funding has led to efficient identification of adversarial groups from topological considerations (work by a Small Business Innovation Research program researcher for the Psychological Operations Group at Fort Bragg). All these efforts are highly relevant to the DoD mission.
The program manager has cogently and judiciously started a transition from previous strategic goals of this program (e.g., computational linguistics) to topological and geometric methods suitable for nonsmooth analysis.
Scientific Quality and Degree of Innovation
This program has been managed by a temporary program manager for over 2 years. A new program manager is expected to arrive within a few weeks and to conduct a review of program strategy in the upcoming year. Such a reassessment is necessary to avoid overlap with missions of other funding agencies (e.g., NSF) and to better tailor program funding to Army needs. The program objective of development of theory and techniques in quantum, stochastic, and statistical systems is broad and tends to lead to funding of fundamental research with a large lag-time to direct application to Army/DoD systems.
The quality of the supported work is excellent. One researcher found a near-optimal random matrix concentration bound, a major improvement over bounds from many physicists and mathematicians. This is a very significant accomplishment and would be a coup for any funding agency. The work on quantum annealing seems to have played a significant role in the establishment of the new Quantum Enhanced Optimization research program at the Intelligence Advanced Research Projects Activity (IARPA). Research on novel characterizations of stochastic differential equations and on insight into properties of random matrices are additional examples of an impressive portfolio of results from the funded research. Furthermore, all funded topics exhibit significant innovation (e.g., research on application of Skorokhod-Malliavin calculus, far-from-equilibrium transitions, and nonlinear filtering).
The webpage for the program lists an interesting collection of research areas in probability and statistics that are within the program’s purview.2 The projects that have been funded, however, are virtually all in the area of probability. This is presumably because the last permanent program manager was a probabilist, more comfortable in that domain, and his temporary successor apparently maintained the nature of the program. The proposed new program manager is a statistician and is likely to correct this imbalance.
The need for probabilistic approaches to problems of Army/DoD relevance is as great as ever, and the ARO needs to play a role not only in applications but also in nurturing development of fundamental theory. That being said, the scope of the problem is so vast that some fields may best be
funded through joint mechanisms with other interested agencies. As an example, quantum communication and information processing would seem a prime candidate for a joint funding mechanism. The collaboration of NSF/NIH in the joint Division of Mathematical Sciences/National Institute of General Medical Sciences program could serve as a useful role model and allow the ARO to leverage limited funds with more impact. In particular, once areas of common interest are identified, the ARO could concentrate its own funds on subtopics that other agencies would not fund but that are of Army interest.
The research areas funded in the program were quite different from what would be routinely funded by, say, NSF. Indeed, the previous program strategy was to find good people to do very unusual things. Enormous opportunities open up with the new program manager’s being a statistician. This is not only because this allows the engagement with data science, but because there are other programs at the ARO that could naturally develop synergies with the program. One clear example is the interest in statistical methods and reinforcement learning in biology.
The accomplishments were primarily theoretical, such as the interesting development of distribution-free solutions to stochastic differential equations (the solution method did not depend on the distribution, but the answer surely does). This could be of eventual interest practically, but it is not clear that one does better by first developing a very complicated general solution method, and then specializing, as opposed to just tackling the solution for a particular distribution directly. But this is the nature of high-risk research: you never know until you try. The improved random matrix concentration bound is another primarily theoretical development, but it will lead directly to improved bounds on the convergence rates of various practical algorithms involving random matrices, and it thus gives better insights into the practical utility of those algorithms.
Funded projects have shown excellent productivity in terms of scholarly output and professional recognition. Program management has been highly effective and original in fostering collaboration between development of theory and computational applications.
This program has seemed to have reasonable transitions between different research problems, although they have been almost entirely in probability. Evidence of the quality of the funded research is furnished by the transition of results into practical tools. Notable transitions include contributions to quantum annealing (which motivated the new Quantum Enhanced Optimization research program at IARPA), contributions to health monitoring (used at Los Alamos National Laboratory), and contributions to helicopter engine monitoring (used at United Technologies Corporation).
The paucity of female statisticians and probabilists among the funded individuals needs to be corrected. This will again be helped by seeking a better balance between probability and statistics, because the percentage of women in statistics is much higher than the percentage in probability. Still, direct action is needed, such as specifically requesting submission of white papers from women, as a start toward eventual funding.
The investigators are producing high-quality research, and the subject areas seem to be appropriate and of use to the Army. The program managers are well qualified, and the ARO system allows them to maintain a close and continuous contact with PIs. This way of operating promotes two-way contact and can help to ensure that information is shared.
Scientific Quality and Innovation
The PIs and research outcomes are of generally high quality. There are many examples of the PIs receiving prestigious academic recognition, including Presidential Early Career Awards for Scientists and Engineers and other young investigators awards. The quality of the research outcomes has been tangibly demonstrated through the translation of a number of projects to Army applied research and development activities.
The research addresses problems of importance to the Army and DoD. Programs and projects tend to address problems or use approaches that are not considered by other funding agencies such as the NSF or NIH.
Broadly speaking, the Mathematical Sciences Division’s research portfolio reflects a view of research approaches that is very much tied to the interests and expertise of the program managers. The division would benefit from enhanced consideration of alternative approaches to address research problems. Ideally, the broadening of research approaches would be accompanied by an expanded definition of performance metrics to include impact factors of products from funded research, especially research solely funded by ARO. The appendix of this report lists a broad set of metrics that ARO could consider for assessment of its programs. The division would benefit from a systematic addition of midperformance period review of projects. The feedback from these reviews would provide a basis for the refinement of future funding allocations.
Within budget constraints, funded projects concentrated on topics with high scientific payoff combined with applicability to Army needs. By and large, the program managers followed a deliberate and reasoned process behind the choice of project areas. The programs were also distinguished by their principled and thematic approach to project choices.
Program managers have much more direct and continuing contact with PIs than would be the case in some other funding agencies, such as NSF or the Air Force Office of Scientific Research. The practice of maintaining close connections between program managers and PIs has given the managers a mastery of the research details that is deeper than that seen at larger funding agencies. There is also some evidence that such close intellectual engagement increases the career appeal of managing a program. Career appeal might be further enhanced if the ARO expanded work models—such as reduced duties or incentives for publication—that encourage program managers’ personal research efforts.
There are several examples of short-term funding (e.g., 6-9 month grants) being used to explore new lines of research. The division’s program might be strengthened if such short-term funding were used as part of an entrepreneurial model with the goal of developing a diverse set of competing technical approaches to high-opportunity topics. Promising areas for the expansion of the scientific program include statistical methods and reinforcement learning in biology and learning combined with computational partial differential equations.
Over the years, the programs have resulted in many significant research outcomes and PIs’ awards for excellence. Examples of significant research outcomes include the development of world-leading algorithms for face recognition under adverse conditions and for estimation of circadian phases. Examples of awards and recognition for the division’s PIs include appointment to the National Academy of Engineering, award of the Gold Medal of the Sobolev Institute of Mathematics, and receipt of an Intel Early Career Faculty Fellowship.
Through the efforts of the program managers, there have been a number of successful transitions of research to DoD and elsewhere. On a regular basis, program managers work with Army agencies to develop ideas for new research. In some cases, program managers have structured projects creatively to enhance potential for transition to practice. For example, researchers who are strong in fundamental work have been paired with researchers with a demonstrated record in translation. Based on the success of the ad hoc pairings to date, the division might benefit from making this a regular practice.
The division follows a formal process for setting research priorities. Army feedback is solicited during multiple phases in the process, and so it appears that there is ample opportunity for the needs of the Army to influence research direction. The division would benefit from encouraging continued input from the Army on the focus of the research programs. Strategic research plans would benefit from discussions about the problems of the future and hypotheses about where the Army’s and DoD’s needs will be in 15 or 20 years.
The division’s entrepreneurial model for research is effective. The program managers exercised their autonomy in many creative and effective ways that would not be possible under the systems in place at larger funding agencies. Successful practices in this regard include matchmaking between researchers to form partnerships, such as teaming a quantum physicist with a mathematician. There is also a focus on younger researchers that facilitates consideration of new ideas.
Evidence of a close connection between the ARO and higher Army echelons was not apparent. Such connections might exist, but they were not elucidated. This matter of connection could be especially important at present, when the Army appears to be under some pressure for review and adjustment in its structure and activities, with the aim of increased readiness in a relatively short period of time. If such a reorganization program also has longer range components, it is possible that some of those could benefit from mathematical research.
With the exception of the Biomathematics Program, there appeared to be limited demographic diversity across the programs. Program managers have the ability to encourage female and minority researchers to submit white papers and follow up with complete proposals, and there is need for an analysis and tracking of demographic diversity across the ISD.
The ARO supports research by PIs and by centers of multiple researchers. In contrast with single investigator programs, the Multidisciplinary University Research Initiative programs at the ARO support centers whose efforts intersect more than one traditional research specialty, typically at $1.25 million per year for 5 years. Research topics increasingly benefit from such multidisciplinary participation, even in pairs or small sets of investigators and over shorter time periods. Including in such collaborations researchers with knowledge of transitions would be useful.