Partial differential equations (PDEs) underpin many existing physical phenomena, including heat conduction, fluid flow, and electromagnetic behavior. In many contexts, one may wish to optimize geometric configurations to achieve a desired physical outcome, such as improving thermal performance or minimizing electromagnetic interference.

The aim of this project is to develop a differentiable PDE solver that enables gradient-based optimization with respect to geometric parameters. Existing approaches either require remeshing and boundary tracking or rely on Monte Carlo methods with slow convergence rates. We instead propose to draw upon techniques in differentiable rendering to develop a lightweight, easy-to-implement, and fast optimizer that can handle any geometric representations (so long as a distance function exists). By achieving efficient and stable optimization for a wide range of geometric primitives, this framework makes PDE-based design more practical and accessible for real-world applications.

I hope to apply the skills I have developed through prior classes and internships during this SuperUROP project and strengthen my abilities as a researcher. I am particularly excited to continue my work in the Algorithmic Design Group and deepen my understanding of how optimization techniques can be leveraged to tangible problems in the design domain!