A major challenge for engineers today is managing the complexity of the systems they design. Consider the task of optimizing a city’s transportation network—deciding whether to invest in a subway system, expand tramlines, subsidize ride-sharing services, or build roads that accommodate cars while ensuring safety for pedestrians and cyclists. These intricate design problems involve numerous stakeholders with diverse priorities, including technical, financial, legal, environmental, and political concerns. One tool engineers can use for tackling such complexity is category theory, a branch of mathematics that offers valuable insights for solving 21st-century engineering challenges.

MIT students explored category theory in a special fall semester subject, Applied Category Theory for Engineering Design, taught by Gioele Zardini, the Rudge (1948) and Nancy Allen Career Development Assistant Professor of Civil and Environmental Engineering.

Over the semester, students learned a compositional design optimization framework language to formulate complex problems and solve them. The class focused on autonomous robotics and mobility and offered students the opportunity to put the framework into practice through a co-design application of their own real-world problem that they presented at the end of semester.

For MIT Interdepartmental Transportation Program graduate students Xingling Li and Riccardo Fiorista, their greatest learning from the class was compositional thinking.

“Composition embeds two directions in complex system decision-making: an extensive, complicated system can be decomposed into smaller systems, and the solution to the system can be derived from the composition of smaller systems and their components,” says Li.  “At the heart of the co-design paradigm, these principles allow us to represent intricate interactions that consistently arise in real-world complex systems without the usual limitations of linearity, differentiability, or comparability,” adds Fiorista

As their final project, Fiorista and Li presented the complex challenge of bus route design that many big transit agencies experience. The design problem involved multi-dimensional decisions, including route design and fleet choice, and the objective encompasses passenger experience, capital and operational cost, and environmental concerns.

“Category theory was an ideal tool for analyzing the complex system design problem and provide results in the format of tradeoff, which we think can be very helpful for supporting the decision-making in bus route design,” says Li

MIT junior Rita Zambrano presented a systems design problem for a medical residency program where she used compositional thinking to analyze and formulate the components of care, cost, and quality of medical care received by resident doctors in training. Zambrano, originally a math major, was fascinated by the idea of using theoretical math to solve engineering problems. She recently switched majors to civil and environmental engineering after two years in Course 18 because she felt disconnected from the applied mathematics that was happening in the real world. “Even though I’m in Course 1 now, I still really love mathematics and I’m so happy that I found a place in CEE where I can do math and feel like I’m contributing to solving real problems,” says Zambrano. 

She learned from this class that sophisticated mathematical concepts have their place in engineering – “you just need to know how to use them,” says Zambrano

She also enjoyed how Prof. Zardini allowed students the freedom to explore applications that interested them and how much she learned from other students’ topics. “While my project focused on medical residency, I learned a lot about transportation, game theory, robotics, and many other topics through my peers – which is something I haven’t experienced in my other classes.” 

MIT Electrical Engineering and Computer Science graduate student, Meshal Alharbi was drawn to the class for its emphasis on a composable approach to systems design and analysis, offering a methodology that is both domain-agnostic and applicable to a wide range of abstract and applied problems. “I found it compelling that the foundation of this class, category theory, not only provides a unifying framework for many areas of mathematics but also facilitates elegant and concise solutions to complex problems,” says Alharbi.

Alharbi’s final project focused on the modeling and co-design of continual learning where he used category theory to design a machine learning framework to create prediction algorithms. The class taught him how to disentangle complex systems into more manageable components, identify common structures across those components, and apply the appropriate tools to describe the components and structures formally. “The benefits of this approach are twofold: it enhances mental tractability for analysis and provides computational efficiency for finding solutions,” he says.

He also found the ideas presented in the class were inherently scale-free and adaptable across multiple levels of abstraction. “This flexibility enables problem-solving and analysis at varying levels of granularity, making the concepts both intellectually stimulating and highly practical,” adds Alharbi.

Other class projects demonstrated how the application can be used for urban planning and policy design, how to use game theory more effectively for co-design problems, and how to find the best operational plan for running a large automatic warehouse.

Zardini developed the class based on his research in category theory and expertise in developing efficient computational tools and algorithmic approaches to formulate and solve complex, interconnected system design and autonomous decision- making problems.

“I never expected the breadth of presentations when I created this class,” says Zardini. “I was impressed by the student’s excitement, interest in the abstract topic and each other’s projects.”

Both Li and Fiorista noted how well the course was designed and introduced the content in a way that allowed them to experience “aha” moments when concepts learned linked together.

“While this course seems like a big jump initially, the content is geared towards students understanding math but being situated in the physical world of engineering and cyber-physical systems, making it a perfect subject for those seeking to apply rigorous mathematical concepts to real-world problems, says Fiorista.

(Main photo credit: NASA/Lillian Gipson)