In science, researchers often focus their entire careers on the pursuit of one primary field. 

Some, however, find that bridging fields opens new realms of possibility. Henry Kvinge, who trained as a mathematician but now works primarily as an AI researcher, is one such example.

Kvinge has found not only dynamic connections between math and AI but also avenues where the two can benefit each other and foster mutual progress.

The nexus where math and AI come together is important both for AI researchers who are building general-purpose models and for mathematicians who need insight, not just answers, to drive theoretical progress. 

Kvinge and his team at Pacific Northwest National Laboratory (PNNL) have designed and released datasets that represent real, research-level mathematics and developed tools to accelerate that math using AI (as in this study). Their work has mostly focused on the field of math known as combinatorics, which studies discrete (as in distinct and countable) structures like graphs and permutations. Combinatorics is an ideal access point for using AI in math because it is more concrete, is rich in data, and underlies much of modern mathematics.  

They’ve also identified mathematical symmetries within AI models that help researchers better understand and explain those models. Kvinge’s work is an example of PNNL’s broader AI expertise—the Lab is a key partner in the Genesis Mission, the Department of Energy’s national initiative to build the world’s most powerful scientific platform to accelerate discovery science, strengthen national security, and drive energy innovation.

“Our team’s guiding principle is that tools and methods shouldn’t just provide the answer, they should provide new insight,” Kvinge said. “We don’t just want to solve a problem, we want to solve the problem in such a way that a mathematician or scientist can walk away saying ‘I never thought about it in that way before—this suggests new research directions.’”

How a mathematical world model leads to “thought”

Kvinge’s team is digging into whether large language models (LLMs; the type of AI model that powers chatbots and coding agents) use the same general rules that humans do when performing mathematics, and they presented findings at a workshop held during the 2025 International Conference on Machine Learning. “We analyzed the internal computations of LLMs to see how much they rely on general rules and mathematical context, what we call ‘mathematical world models,’ when performing mathematics,” said Kvinge.

World models are systems though which humans, animals, and AI simulate the world. “I have a world model for basic physics. This is what allows me to predict that a ball at the top of a hill will roll down even before it has happened,” says Kvinge. “Because LLMs only experience a digital world, math is a good surrogate for more complex, real-world systems, helping us better understand the basis by which these systems digest and solve problems.” 

The degree to which the mathematical world models of LLMs mirror those of humans is a primary focus of Kvinge’s work, with the goal of understanding their mechanics and how they “think.” 

“For example, I visualize integers as sitting on the number line, ordered from smaller to bigger. We looked to see if LLMs think about numbers in the same way,” said Kvinge. “As another example, I know that if 7 + 3 = 10, then 3 + 7 = 10, but does an LLM understand this? Perhaps the LLM knows this fact, but it doesn’t use it to answer a user query.” He adds, “we are looking to not just clarify what it means for an LLM to have a ‘mathematical world model,’ but also to understand differences in human-style mathematical understanding and current LLM behavior.” 

Where LLMs excel (and where they don’t)

Humans understand and perform math in a particular way. Our mental framework relies both on our direct knowledge of the topic and on our trust of pure mathematics—we use its theorems to guide and check our progress. 

“Though many students might disagree, mathematics is characterized by its simplicity,” says Kvinge. “A single set of rules like the rules of long division allow us to divide any choice of integers. This elegance is at the core of what we value in mathematics.”

“When learning, LLMs have different types of constraints than humans do, and this informs the way they do math. An LLM can potentially memorize the solutions to a vast number of math problems. They don't need to find the simple and robust solutions that our cognitive limitations require us to find.”

Because Kvinge’s team can speak the languages of both AI and math, they can help facilitate collaboration and discovery between the two fields. One tool designed to connect AI researchers with important mathematical questions is OpenConjecture, a living database that currently houses thousands of unproven mathematical conjectures for investigators to solve. 

Similarly, math is an excellent testbed for studying AI and how it “thinks” because there is little ambiguity and experiments can be done digitally. Kvinge notes, “we have been surprised to find that LLMs often capture concepts that mirror the ways humans think about math. On the other hand, we have been able to scrutinize points where these models use really strange approaches. They may also try to solve a problem in a way that follows the literal request, but not the spirit of the problem, emphasizing why it is still critical to keep human mathematicians in that loop.”

Linking research communities

To further collaboration between the math and AI communities, Kvinge and fellow PNNL senior data scientists Tim Doster and Tegan Emerson launched the Topology, Algebra, and Geometry in Data Science (TAG-DS) community in 2022. While TAG-DS regularly hosts workshops at events, their first self-led conference was held December 1–2, 2025, on the University of California San Diego campus. While unaffiliated, the inaugural TAG-DS event aligned with the timing and location of NeurIPS 2025, a premier international conference on machine learning. Co-locating TAG-DS with NeurIPS expanded the opportunities for researchers in the field to connect and network at the same time and in the same place.

Looking to the future, Kvinge and colleagues will be holding TAG-DS 2026 at Northeastern University in August. They also are developing a PyTorch library called TAGTorch. PyTorch is the primary programming library for deep learning research, and TAGTorch is aimed at making it easy to apply math-inspired approaches to problems in AI. The team’s work on TAGTorch is being conducted under DOE’s Scientific Discovery through Advanced Computing program, or SciDAC.

This research has been funded by PNNL through its Generative AI for Science, Energy, and Security Science and Technology investment and other Laboratory Directed Research and Development programs/projects.