Senior author Vivek Shenoy, at left, and co-first author Vinayak Vinayak, at right, demonstrate some of the mathematics behind mollifier layers. Credit: Sylvia Zhang

Penn Engineers have developed a new way to use AI to solve inverse partial differential equations (PDEs), a particularly challenging class of mathematical problems with broad implications for understanding the natural world.

The advance, which the researchers call "Mollifier Layers," could benefit fields as varied as genetics and weather forecasting, because inverse PDEs help scientists work backward from observable patterns to infer the hidden dynamics that produced them.

"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.…