When Nazanin (Naz) Ahmadi first came to Brown, she knew she would be working on Physics-Informed Neural Networks (PINNs) with her advisor—Dr. George Karniadakis, Professor of Applied Mathematics and Engineering. 

She didn’t know she would be working in Quantitative Systems biology and Pharmacometrics—fields that require a background in chemistry and biology, subjects Naz hadn’t studied since her early high school years. 

Before Brown, Naz had never been to the United States. In high school, she chose her majors as physics and mathematics. “I didn’t take any biology courses at a university level,” said Naz. Now, though, Naz has become an interdisciplinarian. She’s taken courses in public health, applied mathematics, biology, engineering, and computer science. 

She’s one of the first authors on AI-Aristotle, an AI system that is helping researchers find the mathematical rules that govern physical and biological systems. 

“I knew that I would work on the mathematical part of Neural Networks,” said Naz. The idea for AI-Aristotle was prompted by the fact that, “in most cases, we don’t know the exact governing physics” behind biological and physical processes. “We know that such processes happen in the body, but we don’t know how,” said Naz.

Researchers often only have partial information about the equations that govern systems and only a limited amount of “real world” data, said Naz. “The question here is ‘how do we blend our understanding of biophysics and interactions within the system into the work of a neural network?’” said Naz. 

Neural networks are AI systems that approximate outputs or solutions to patterns and equations based on the information developers feed them. The developers of AI-Artistotle took known scientific equations and fed them to AI-Aristotle with some missing pieces. Then, they would observe if AI-Aristotle could accurately solve the equations and find the value of parameters as well as missing terms in the mathematical model. This process helped researchers ensure that the outputs generated by AI-Aristotle were accurate and consistent with known physics. 

After getting outputs from AI-Aristotle, they used symbolic regression techniques to turn the outputs into what Naz called “very beautiful mathematical equations.” Naz and the AI-Aristotle team found that AI-Aristotle accurately estimated the value of parameters with limited data and only partial physics.

The team working on AI-Aristotle came from a wide variety of backgrounds. Mario De Florio, her co-first author, studied systems and industrial engineering before becoming a postdoctoral researcher in applied mathematics. AI-Aristotle’s third and fourth authors, Khemraj Shukla and Professor Karniadakis, both have backgrounds in math. Naz herself studied electrical engineering before coming to Brown for biomedical engineering. 

Naz said this enabled them to work on different aspects of AI-Aristotle based on their areas of expertise. 

“We had three different parts in this research,” she said. First, they had to see if PINNs could find a missing part of the equations as well as the value of the parameters. Then, they had to apply the physics-informed eXtreme Theory of Functional Connections to look for its solution—which De Florio was the expert in. After that, they compared these two Physics-informed methods. When it came time for symbolic regressions, Shukla found out which algorithms worked better in each problem. 

After six months of research, the team submitted an archive paper for review. The reviewers wanted the team to try AI-Aristotle on “noisier data” that more accurately mirrored “real world” applications. In the end, they determined that the AI-Aristotle framework could be applied across a wide variety of scientific disciplines to help find the mathematical equations that govern systems.

Through the process of developing AI-Aristotle, Naz tested the framework on Pharmacokinetics and Pharmacodynamics equations and got “very good results.” She decided to focus on Quantitative Systems Biology and Pharmacology for her thesis, exploring how to use AI-Aristotle and other systems to mathematically model the exposure effects of drugs in the body. 

As for the specific questions Naz hopes to answer through her research, she has several. “What happens (to organs) when you administrate a drug in the body, what happens to the drug concentration in the body, how (do) the organs affect the drug, and how (does) the drug affect the organs? How can we predict the drug response and medical events?”

AI-Aristotle has the potential to answer those questions, alongside other important problems facing researchers in their understanding of the math behind physical and biological systems. As the team wrote in their final research paper, their “work offers guidance to researchers addressing gray-box identification challenges in complex dynamic systems, including applications in biomedicine and beyond.” 

Currently, they are working with Dr. Carl Leake, to develop an automated framework that will enable other researchers without backgrounds in neural networks or symbolic regression to use AI-Aristotle’s promising new framework.