Penn Engineers Develop AI Method to Reverse-Engineer Hidden Processes in Complex Systems
Researchers at the University of Pennsylvania have created a new AI technique that works backward from observable patterns to identify the underlying processes driving them. The method, called "Mollifier Layers," addresses inverse partial differential equations - mathematical problems used across genetics, materials science, and weather forecasting.
The work appears in Transactions on Machine Learning Research and will be presented at the Conference on Neural Information Processing Systems in 2026.
The Core Problem
Inverse problems reverse the typical scientific workflow. Rather than starting with known rules to predict outcomes, researchers begin with observed data and attempt to uncover the hidden dynamics responsible for it.
"Solving an inverse problem is like looking at ripples in a pond and working backward to figure out where the pebble fell," said Vivek Shenoy, senior author and Eduardo D. Glandt President's Distinguished Professor in Materials Science and Engineering. "You can see the effects clearly, but the real challenge is inferring the hidden cause."
Partial differential equations describe how systems change across both time and space. They model weather patterns, chemical reactions, material behavior, and even how DNA organizes inside cells. But calculating inverse PDEs has required enormous computational power and produces unreliable results when handling higher-order systems or noisy data.
Why Existing Methods Fail
Most AI systems solving inverse PDEs rely on recursive automatic differentiation - repeatedly calculating changes throughout a neural network to measure how things shift and transform. This approach becomes unstable with noisy data.
The team compared the problem to repeatedly zooming in on a jagged line. Each magnification amplifies imperfections, making the final calculation less reliable. They realized the data needed smoothing before measuring those changes.
The Solution: Mollifier Layers
The researchers adapted "mollifiers," mathematical tools developed in the 1940s that smooth rough or noisy functions. They created a "mollifier layer" that smooths signals before the system calculates derivatives.
"We initially assumed the issue had to do with the neural network's architecture," said Ananyae Kumar Bhartari, a graduate of Penn Engineering's Scientific Computing master's program. "But after carefully adjusting the network, we eventually realized the bottleneck was recursive automatic differentiation itself."
The new layer dramatically reduced noise and improved computational efficiency without requiring larger, more power-hungry AI systems. "That let us solve these equations more reliably, without the same computational burden," Bhartari said.
Application to Chromatin and Gene Expression
The first application targets chromatin - the tightly packed DNA and proteins inside cells that control access to genetic information. The Shenoy Lab studies chromatin domains roughly 100 nanometers across that regulate which genes activate.
These microscopic structures influence cell identity, function, aging, and disease. The new framework could help scientists infer the epigenetic reaction rates that drive chromatin changes and gene expression.
"If we can track how these reaction rates evolve during aging, cancer, or development," said Vinayak Vinayak, a doctoral candidate and co-first author, "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."
Broader Applications
Mollifier layers could extend beyond biology. Materials science, fluid mechanics, and other fields dealing with noisy data and higher-order equations could benefit from a more stable, efficient way to uncover hidden parameters in complex systems.
Shenoy said the broader goal is straightforward: "If you understand the rules that govern a system, you now have the possibility of changing it."
The work received support from the National Cancer Institute, National Science Foundation, National Institute of Biomedical Imaging and Bioengineering, and National Institute of General Medical Sciences.
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