New AI method tackles one of science’s hardest math problems

The team's solution, called "Mollifier Layers," improves how AI handles these problems by refining the math behind the process instead of simply increasing computing power. The approach could have wide-ranging applications, from decoding genetic activity to improving weather predictions.

"Solving an inverse problem is like looking at ripples in a pond and working backward to figure out where the pebble fell," says Vivek Shenoy, Eduardo D. Glandt President's Distinguished Professor in Materials Science and Engineering (MSE) and senior author of a study published in Transactions on Machine Learning Research (TMLR), which will be presented at the Conference on Neural Information Processing Systems (NeurIPS 2026). "You can see the effects clearly, but the real challenge is inferring the hidden cause."

Instead of relying on more powerful hardware, the researchers focused on improving the underlying mathematics. "Modern AI often advances by scaling up computation," says Vinayak Vinayak, a doctoral candidate in MSE and co-first author of the study. "But some scientific challenges require better mathematics, not just more compute."

Why Inverse PDEs Matter in Science

Differential equations are the backbone of scientific modeling. They describe how systems change over time, whether it is population growth, heat flow, or chemical reactions.

Partial differential equations extend this idea further by capturing how systems evolve across both space and time. Scientists use them to study everything from weather patterns to how heat moves through materials and even how DNA is organized inside cells.

Inverse PDEs go a step further. Rather than predicting outcomes based on known rules, they allow scientists to start with observed data and work backward to uncover the hidden forces driving those observations.

"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," says Shenoy. "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."

Rethinking How AI Handles Complex Math

A key concept behind these equations is differentiation, which measures how something changes. Simple derivatives show how fast something increases or decreases, while higher-order derivatives capture more intricate patterns.

Traditionally, AI systems compute these derivatives using a process called recursive automatic differentiation. This method repeatedly calculates changes as data moves through a neural network, the foundation of modern AI.

However, this approach struggles when dealing with complex systems and noisy data. It can become unstable and demand enormous computing resources.

The researchers compare it to repeatedly zooming in on a rough, jagged line. Each step amplifies imperfections, making the final result less reliable. To overcome this, the team realized they needed a way to smooth the data before analyzing it.

Mollifier Layers Offer a Smarter Solution

The answer came from a concept introduced in the 1940s by mathematician Kurt Otto Friedrichs, who described "mollifiers," tools designed to smooth irregular or noisy functions.

By adapting this idea, the researchers created a "mollifier layer" within AI models. This layer smooths the input data before calculating changes, avoiding the instability caused by traditional methods.

"We initially assumed the issue had to do with neural network's architecture," says Ananyae Kumar Bhartari, a graduate of Penn Engineering's Scientific Computing master's program and the paper's other co-first author. "But, after carefully adjusting the network, we eventually realized the bottleneck was recursive automatic differentiation itself."

The results were striking. The new method reduced noise and significantly lowered the computational cost required to solve these equations.

Implementing a "mollifier layer," which smoothed the signal before measuring it, radically diminished both the noisiness and the power consumption scaling. "That let us solve these equations more reliably, without the same computational burden," says Bhartari.

Unlocking the Secrets of DNA Organization

One of the most promising applications of this approach lies in understanding chromatin, the complex structure of DNA and proteins inside cells.

These structures operate at an incredibly small scale, but they play a major role in determining how genes are turned on or off.

"These domains are just 100 nanometers in size," says Shenoy, "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 estimating the rates of epigenetic reactions, which control gene activity, the new AI method could help scientists move beyond simply observing chromatin to predicting how it changes over time.

"If we can track how these reaction rates evolve during aging, cancer or development," adds Vinayak, "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."

Beyond Biology: Wide-Ranging Scientific Impact

The potential uses of mollifier layers extend far beyond genetics. Many areas of science, including materials research and fluid dynamics, involve complex equations and noisy data.

This new framework could provide a more stable and efficient way to uncover hidden parameters across a wide variety of systems.

The researchers see this as a step toward a larger goal: turning observations into deeper understanding.

"Ultimately, the goal is to move from observing complex patterns to quantitatively uncovering the rules that generate them," says Shenoy. "If you understand the rules that govern a system, you now have the possibility of changing it."

This study was conducted at the University of Pennsylvania School of Engineering and Applied Science and supported by National Cancer Institute (NCI) Award U54CA261694 (V.B.S.); National Science Foundation (NSF) Center for Engineering Mechanobiology (CEMB) Grant CMMI -154857 (V.B.S.); NSF Grant DMS -2347834 (V.B.S.); National Institute of Biomedical Imaging and Bioengineering (NIBIB) Awards R01EB017753 (V.B.S) and R01EB030876 (V.B.S.) and National Institute of General Medical Sciences (NIGMS) Award R01GM155943 (V.B.S).