Mathematics rightly viewed possesses not only truth but supreme beauty. 
— Bertrand Russell

Dr. Katarina Jegdic’s love for theoretical math (sometimes called “pure” math) is grounded in the beauty she finds there. “I love how things are formulated. Everything is so beautiful and natural,” she said. 

A career mathematician and Professor of Mathematics and Statistics in the College of Sciences & Technology, Jegdic tries to help her students see math as more than just numbers and equations. “I want them to know that math is a way of thinking, a language in which we communicate how to solve problems. Math describes virtually everything in our world.”

Jegdic spent more than two decades conducting research in partial differential equations (PDEs)—a topic so complex, even Wikipedia struggles to simplify it. According to Jegdic, “PDEs are equations that express relationships between various partial derivatives of unknown multivariable functions.” 

For the non-PDE-inclined reader, examples of how PDEs are practically applied may help illuminate that definition. In Jegdic’s case, she used PDEs to study the flow of gases around a moving object (like a plane’s wing) in the aerospace engineering arena. She also applied PDEs to study the flow of gases and oil from the injection to the production wells in the oil reservoirs. “You can find applications in many areas, including in the field of cardiology, where PDEs are used to explain the blood flow,” she said.

With the advent of computers, solving PDEs entered a new era. “The classical methods that have been in use for 70 years or so to solve PDEs give good approximate solutions, but newer methods have been developed using neural networks, which is a type of machine learning,” explained Jegdic. “I became very interested in machine learning a couple of years ago, when the chair of the Mathematics & Statistics Department asked me to participate in the development of our Data Science degree.”

Machine learning, a subfield of artificial intelligence, is concerned with programming computers to learn from data. “I became interested in studying more about particular algorithms and, together with a former Data Science student Marina Zafiris, found that artificial neural networks were being used to solve PDEs,” Jegdic said. “So this is where my main research focus is now.”

There are many approaches to using neural networks to solve differential equations, Jegdic noted, mostly motivated by the so-called Universal Approximation Theorems (another Wikipedia fail). One approach she investigated with her students involved “a trial solution consisting of two parts: the first part being satisfying the initial/boundary conditions, with no adjustable parameters, and the second part being a neural network that is trained to satisfy the differential equation by minimizing the squared loss.” 

The results of her investigation? “We found that the neural network approximation methods typically yield more accurate and faster results than classical approximation methods by considering a number of examples of simple differential equations,” she said. Currently, Jegdic is working on extending these ideas to solve more complex PDEs and on incorporating dynamic neural networks. She is writing a paper on her findings, a collaboration with students and several colleagues, with a goal of publishing it this year.

Jegdic’s favorite way to engage with data science and mathematics, however, is with her students. “I really enjoy mentoring and working with our students at UHD,” she said. Along with Jegdic’s class lectures and workshops, she offers students the opportunity to engage with research.

“I serve as a faculty mentor for a group of around 10 students in the Scholars Academy each semester,” she said. She recalled one project concerning traffic flow where differential equations and analysis were tapped to assist in the practical aspects of highway and city planning. “With my colleagues, I work mostly on the theoretical aspects of mathematics,” she noted. “But for students, I focus on projects that show where mathematics is applied in real life. They enjoy that a lot.” 

In the last five years, working with Dr. Mary Jo Parker, Executive Director of the Scholars Academy, and several other faculty members from the college, Jegdic received a grant to work on nuclear science, using PDEs to describe the flow of liquids in a nuclear reactor to help prevent accidents. With her fellow researchers, she has also garnered grants from the National Science Foundation, Texas Advanced Research Program, the Department of Education, the Office of Naval Research, and the Nuclear Regulatory Commission.

“I see these grants as evidence that national organizations are interested in our work at UHD and that they see we are making important contributions,” Jegdic said. She also sits on the review boards for a dozen or so math journals. “I enjoy reading about new research areas in mathematics and in science in general—not just the area I’m working on, but others,” she noted. “You can get new ideas from reading about many different fields.”

When it comes to career success, Jegdic has found her greatest satisfaction is helping students see the beauty in numbers. “It’s been the highlight of my career,” she said. “In my early years, I was more research-oriented, pursuing research on my own. Once I saw how much students appreciate a teacher who is invested in them, that gave me a push. If I can continue to inspire and motivate students and make a difference for them the way my teachers did for me, that is the highest award for me.”