AI solves a difficult inverse PDE problem – Penn’s Mollifier Layers

**AI Untangles a Tough Mathematical Puzzle in Science** Can you look at ripples in a pond and figure out where a pebble fell? In science, that’s an inverse problem: the effect is visible, the cause is hidden. To solve these inverse problems, researchers at the University of Pennsylvania have developed a new AI method. Its name: Mollifier Layers. Published on May 6, 2026, the study appeared in *Transactions on Machine Learning Research* (TMLR) and is scheduled to be presented at NeurIPS 2026. • Lead: Vivek Shenoy, Materials Science and Engineering, Penn • Team: Vinayak Vinayak, Ananyae Kumar Bhartari **What is an Inverse PDE?** Differential equations are mathematical tools that describe how a system changes—heat spreading, population growth, chemical reactions, and more. Partial differential equations (PDEs) take this a step further. They capture how a system changes across both space and time. PDEs show up everywhere: from weather and materials science to DNA organization. An inverse PDE flips the direction. Instead of using known rules to predict outcomes, it starts from observed outcomes and works backward to infer the hidden rules. That’s the hard part—and it’s also what science needs most. **Where was the problem?** Until now, AI has computed these derivatives using recursive automatic differentiation—repeatedly calculating changes as data passes through a neural network. With complex, noisy data, this becomes unstable. Like zooming in on a jagged line again and again, each step magnifies the error. The computational cost also becomes enormous. “Modern AI often advances by increasing compute. But some scientific challenges don’t need more compute—they need better mathematics,” says Vinayak Vinayak. **The solution: Mollifier Layers** The solution drew on “mollifiers,” a tool for smoothing irregular functions developed in the 1940s by mathematician Kurt Otto Friedrichs. The Penn team placed this inside the AI model as a “mollifier layer.” Before computing derivatives, it first smooths the input data. Result: less noise, significantly lower computational cost, and more stable solutions. “At first we thought the problem was the network architecture. Only later did we realize the bottleneck was recursive automatic differentiation,” says Ananyae Kumar Bhartari. **A direct application in DNA** The biggest application is in chromatin research. Chromatin is DNA in its folded state inside the cell nucleus—only about 100 nanometers in scale. Yet it determines which genes are active, influencing cell identity, aging, and disease. “We could see the structures, but we couldn’t reliably infer the epigenetic processes driving them. It became clear that the mathematics itself had to change,” says Shenoy. With the new AI method, it becomes possible to estimate epigenetic reaction rates that control gene activity. If researchers can track how these rates change during aging, cancer, or development, it could open pathways to new treatments. “If reaction rates control chromatin organization and cell fate, then by changing those rates, we can push cells toward desired states,” says Vinayak. **Beyond biology** The applications don’t stop with genetics. This framework can work in any field with noisy data and complex equations—materials research, fluid dynamics, weather prediction, and more. “The ultimate goal is to move from observing complex patterns to quantitatively extracting the rules that generate them. Once you understand the rules governing a system, you have the possibility of changing it,” says Shenoy. --- Source: University of Pennsylvania School of Engineering and Applied Science. Shenoy V., Vinayak V., Bhartari A.K. “Mollifier Layers for Inverse PDEs,” *Transactions on Machine Learning Research*, May 2026. To be presented at NeurIPS 2026.