AI method tackles one of science’s toughest math problems

Penn engineers have developed a new way to use AI to solve inverse partial differential equations (PDEs). This is a particularly difficult class of mathematical problems that have far-reaching implications for our understanding of the natural world.

This advancement, which researchers call a “mitigating layer,” could benefit fields as diverse as genetics and weather forecasting. That’s because inverse partial differential equations help scientists work backwards from observable patterns to infer the hidden dynamics that produced them.

“Solving an inverse problem is like looking at the ripples in a pond and working backwards to figure out where the pebble landed,” he says. Vivek ShenoyEduardo D. Glandt is the Presidential Distinguished Professor of Materials Science and Engineering (MSE). Transactions related to machine learning research (TMLR)will be announced. Conference on neural information processing systems (NeurIPS 2026). “The effects are clearly visible, but the real challenge is to deduce the hidden causes.”

Rather than simply throwing more computing power at the problem, the researchers sought a better mathematical approach. “Modern AI often advances by scaling up the amount of computation,” he says. Vinayak Vinayaka doctoral candidate at MSE and co-first author of the study. “But some scientific tasks require better mathematics than just computational ability.”

Why are inverse partial differential equations important?

Differential equations are essentially mathematical tools for describing change. These help scientists model how systems evolve, such as how populations grow, how heat dissipates, and how chemical reactions occur over time.

Partial differential equations (PDEs) handle more complex systems by describing how things change in both space and time. These are used to model various phenomena such as weather systems, heat flowing through materials, etc. Shenoy Institutethe organization of DNA within cells.

Inverse partial differential equations ask even more difficult questions. Rather than using known rules to predict how a system will behave, it helps scientists work backwards from what they can observe to infer the hidden forces, parameters, or dynamics that produced it.

“For many years, we have used these equations to study how chromatin, the folded state of DNA in the nucleus, is organized in living cells,” Shenoy says. “But we kept running into the same problem. Although we could see the structure and model its formation, we couldn’t reliably infer the epigenetic processes that drive this system, the chemical changes that help control which genes are activated. The more we tried to optimize existing approaches, the more it became clear that the mathematics itself had to change.”

Rethinking how AI computes

At the heart of the problem is a seemingly simple mathematical idea: differentiation, or measuring how something changes. In everyday terms, derivatives allow scientists to tell how fast a quantity is rising or falling. Higher order derivatives go further and help describe more complex patterns of change.

For years, AI systems tackling inverse partial differential equation problems have typically calculated these derivatives through a method called recursive automatic differentiation, which repeatedly calculates how quantities change through the neural networks that are the backbone of all AI models.

However, for higher-order systems, especially when the data is noisy, the process can become unstable and require significant computational power.

As the researchers explained, recursive automatic differentiation is like repeatedly zooming in on the slope of a line. If the line is jagged, each additional step can actually magnify the noise in the data, making the final result less reliable. In other words, the team realized they needed a way to reliably smooth the signal before measuring changes.

The power of emollients

In the 1940s, Kurt Otto Friedrichsa German-American mathematician who later received the National Medal of Science. “Relaxant” explained”, a mathematical tool that “relaxes” particularly noisy or jagged functions by smoothing out their sharpest features.

By applying this technique, the team was able to avoid problems caused by recursive automatic differentiation. “We initially thought the problem was related to the architecture of neural networks,” he says. Ananyae Kumar Bartalia graduate of Penn College of Engineering scientific computing M.Sc., and the other co-lead author of this paper. “But after careful tuning of the network, we ultimately discovered that the bottleneck was the recursive autodifferentiation itself.”

Implementing a “relaxation layer” to smooth the signal before measuring it significantly reduced both noise and power scaling. “This allows us to solve these equations more reliably without the same computational burden,” Bartali says.

Elucidation of chromatin

For the Shenoy lab, one immediate application of the relaxation layer is to better understand how the small domains of chromatin, the mixture of proteins and DNA that packages chromosomes for storage inside cells, regulate access to genetic material in the nucleus. previous job It shows how epigenetic reactions and physical interactions organize chromatin structure.

“These domains are only 100 nanometers in size,” Shenoy says. “However, these domains play important roles in biology and health because accessibility determines gene expression, and gene expression governs cell identity, function, aging, and disease.”

By estimating the rate of epigenetic reactions that cause these changes, or in other words, how quickly the chemical changes that control genes occur, the relaxation layer could help researchers who study chromatin go from simply observing the structure of chromatin using powerful microscopes to modeling how chromatin changes over time and how those changes affect gene expression.

“Tracking how these reaction rates change during aging, cancer, and development opens up new therapeutic possibilities. If reaction rates control chromatin organization and cell fate, then altering these rates could guide cells toward desired states,” Vinayak added.

Future direction

Relaxant layers may also prove useful far beyond biology. Many problems in materials science, fluid mechanics, and other fields of scientific machine learning involve high-order equations and noisy data, so this framework has the potential to provide a more stable and computationally efficient way to infer hidden parameters across a wide range of systems.

The researchers hope that the same mathematical approach that helped uncover chromatin’s hidden reaction rates will help scientists tackle similar difficult inverse problems in many other fields. “The ultimate goal is to quantitatively uncover the rules that generate complex patterns by observing them,” Shenoy says. “If you understand the rules that govern the system, you may be able to change it.”

This research was conducted at the University of Pennsylvania School of Engineering and Applied Sciences and was supported by National Cancer Institute (NCI) award U54CA261694 (VBS). National Science Foundation (NSF) Center for Engineering Mechanobiology (CEMB) grant CMMI -154857 (VBS); NSF grant DMS -2347834 (VBS); National Institute of Medical Imaging and Bioengineering (NIBIB) awards R01EB017753 (VBS) and R01EB030876 (VBS) and National Institute of General Medical Sciences (NIGMS) award Awarded R01GM155943 (VBS).