Researchers at the University of Pennsylvania have developed a new AI approach to solving inverse partial differential equations — a notoriously difficult class of math problems with applications ranging from DNA research to weather forecasting.
A team of engineers at the University of Pennsylvania has developed a new AI-driven technique for tackling one of science’s most stubborn mathematical challenges: inverse partial differential equations, or PDEs. The advance could reshape how researchers study everything from gene regulation to fluid dynamics.
The method, which the researchers call “Mollifier Layers,” was published in Transactions on Machine Learning Research and is slated for presentation at the Conference on Neural Information Processing Systems (NeurIPS 2026).
Working Backward From the Evidence
Differential equations are the mathematical backbone of modern science, helping researchers model how systems change over time — from population growth to heat transfer to chemical reactions. Partial differential equations handle especially complex systems, describing change across both space and time. Inverse PDEs flip the script: instead of predicting how a system will behave given a set of known rules, they work backward from observable data to uncover the hidden forces that produced it.
“Solving an inverse problem is like looking at ripples in a pond and working backward to figure out where the pebble fell. You can see the effects clearly, but the real challenge is inferring the hidden cause,” senior author Vivek Shenoy, the Eduardo D. Glandt President’s Distinguished Professor in Materials Science and Engineering at the University of Pennsylvania, said in a news release.
Shenoy’s lab has long used PDEs to study chromatin — the tightly folded form that DNA takes inside the cell nucleus. While the team could observe chromatin’s structure and model how it forms, a key piece remained elusive.
“For years, we’ve used these equations to study how chromatin, which is the folded state of DNA inside the nucleus, organizes itself inside living cells,” Shenoy added. “But we kept running into the same problem: We could see the structures and model their formation, but we could not reliably infer the epigenetic processes driving this system, namely the chemical changes that help control which genes are active. The more we tried to optimize the existing approach, the clearer it became that the mathematics itself needed to change.”
Better Math, Not Just More Computing Power
Most AI systems that tackle inverse PDE problems rely on a technique called recursive automatic differentiation — essentially measuring how quantities change by repeatedly running calculations through a neural network. For higher-order equations or noisy data, this process can become unstable and computationally expensive.
“Modern AI often advances by scaling up computation,” co-first author Vinayak Vinayak, a doctoral candidate in Materials Science and Engineering, said in the news release. “But some scientific challenges require better mathematics, not just more compute.”
To address this, the team turned to a mathematical concept introduced in the 1940s by German-American mathematician Kurt Otto Friedrichs. He described “mollifiers” as mathematical tools that smooth out particularly noisy or jagged functions by softening their sharpest features. Adapting this technique allowed the team to bypass the issues caused by recursive automatic differentiation.
Co-first author Ananyae Kumar Bhartari, a graduate of Penn Engineering’s Scientific Computing master’s program, said the team initially suspected the neural network’s architecture was the problem — but eventually traced it back to the differentiation method itself. Implementing a mollifier layer, which smoothed the signal before measuring it, radically diminished both the noisiness and the power consumption scaling, leading to more reliable results without the same computational burden.
Why It Matters for Genetics — and Beyond
For the Shenoy Lab, the most immediate application is gaining a clearer picture of how chromatin domains regulate access to genetic material inside the nucleus. These domains are tiny — just 100 nanometers in size — but their influence is enormous. As Shenoy noted, “but because accessibility determines gene expression, and gene expression governs cell identity, function, aging and disease, these domains play a critical role in biology and health.”
By using mollifier layers to infer the epigenetic reaction rates that drive chromatin changes, researchers could move beyond static observation and begin modeling how chromatin evolves over time — and how those changes affect which genes are switched on or off.
“If we can track how these reaction rates evolve during aging, cancer or development, this creates the potential for new therapies: If reaction rates control chromatin organization and cell fate, then altering those rates could redirect cells to desired states,” Vinayak added.
The implications stretch well beyond biology. Because inverse PDE problems appear throughout materials science, fluid mechanics and climate modeling, the mollifier layers framework could offer a more stable and efficient tool across a wide range of scientific disciplines.
“Ultimately, the goal is to move from observing complex patterns to quantitatively uncovering the rules that generate them,” added Shenoy. “If you understand the rules that govern a system, you now have the possibility of changing it.”
The study was conducted at the University of Pennsylvania School of Engineering and Applied Science.
Source: University of Pennsylvania School of Engineering and Applied Science