UMass Amherst Researchers Will Develop New Predictive Math for Problems with Many Variables and Complex Interactions

Markos Katsoulakis

AMHERST, Mass. ­– Markos Katsoulakis, professor of mathematics and statistics at the University of Massachusetts Amherst, with others, recently received a $3.1 million award from U.S. Defense Advanced Research Projects Agency (DARPA) to develop a reliable, predictive computational framework for computer-simulated design of better-performing and/or lower cost materials for catalysis, energy production and energy storage applications.

Katsoulakis explains that at present, “There is strong experimental evidence that novel material architectures can provide unprecedented performance, therefore a reliable computational framework for the prediction of such materials can be a critical ingredient for decreasing development cost and time-to-market.”

He adds that this research has great potential for contributing to a number of fields because it will lay down the mathematical and computational science foundations for uncertainty quantification and reliable predictive computation in a broad class of complex systems with hundreds to millions of uncertain parameters commonly encountered in physico-chemical and biological processes, atmosphere and ocean sciences, epidemiology and various types of complex networks. Furthermore, broadly related challenges arise in machine learning where models can also be complex, multi-scale, for example, in the so-called “deep learning,” and high-dimensional, that is with thousands of parameters to be learned.

The UMass Amherst team of Katsoulakis and Matthew Dobson will be joined in this project by partners at Brown University, the universities of Delaware, North Carolina-Chapel Hill and California, San Diego, each with its own areas of expertise.

The researchers point out that despite the ever-increasing sophistication of computational capabilities, these models are not always predictive. Mathematical models are now not only asked to account for increasing complex systems with millions or even billions of variables, but also to integrate available data from different scales, for example from the molecular level all the way to the everyday macroscopic scale, and basic physical processes at different time scales.

Handling real-life systems with unprecedented levels of complexity and multi-scale features requires not only more powerful computational capabilities, but also new mathematics, Katsoulakis explains. “The vision of this new grant is to fill in this scientific and ultimately technological gap via the interdisciplinary expertise and collaborations of the participating teams and their researchers,” he adds.

The first of three key challenges to developing a reliable, predictive modeling and computing for materials is not only the millions of parameters involved, but also the fact that predictions and uncertainty quantification need to account for the multi-scale and multi-physics nature of models.

An additional challenge is “model-form uncertainties” induced by limited data and/or simplified or incomplete physics. Failing to account for such constraints leads to wrong predictions, unnecessarily extreme high dimensionality and statistical inference of too many parameters from experimental data. In this latter direction, the researchers need to develop a new mathematical theory, Katsoulakis notes.

Finally, he says, rare events play a key and sometimes dominant role in many problems from engineering, chemistry, biology and materials to traffic, wireless or computer networks, economics, finance and atmospheric and ocean sciences and predictive models need to try to accommodate them.

For example, in engineering system performance, designers know that certain types of critical, rare events such as data loss or system breakdown may be fundamental to overall operation. Likewise in physical science problems, rare events may determine important material properties and attributes in chemical reaction networks where parts of may communicate rarely but with significant implications.

To use the mundane example concerning the potential impact of a fairly rare event, the UMass Amherst mathematician says, “think of the effect that a single vehicle breakdown during rush hour may have to the overall traffic flow, even a very large distance from the scene.” Taking all the variables and mechanisms such as vehicle speeds and sizes, road network, weather, traffic volume and more into account, represents just such a complex, multi-scale modeling, simulation and analysis problem.

The goal here is to develop mathematical methods that direct rare event simulations in the most information-rich parameter or state-space regimes, Katsoulakis says. Among other things, he adds, “We also need methods capable of rare event analysis and simulation. Overall, we need to develop a quantitative understanding of when and how a complex system’s overall behavior is ultimately determined by the occurrence, or not, of rare events.”