Wasserstein PDEs and Applications to Optimization and Learning

The investigator advances artificial intelligence and optimization through the study of partial differential equations (PDEs) on Wasserstein space, addressing fundamental challenges in machine learning that bolster national prosperity and technological innovation. This project serves the national interest by developing mathematical tools to enhance the efficiency and robustness of artificial intelligence (AI) systems, which drive advancements in healthcare, technology, and economic competitiveness. By tackling critical problems in derivative-free optimization, stochastic filtering, and adversarial decision-making, the research strengthens the scientific foundation of AI, a priority area for NSF. Additionally, the investigator mentors graduate students and postdoctoral researchers through these problems, fostering a skilled STEM workforce and contributing to national scientific leadership through education and professional development. The investigator studies the well-posedness of second-order PDEs on Wasserstein space, applying weak propagation of chaos to achieve exponential convergence in high-dimensional optimization problems central to machine learning. The project employs rigorous theoretical analysis of master equations, extending results to particle systems on unbounded domains influenced by common and idiosyncratic noise. These methods address goals in consensus-based optimization, controlled stochastic filtering, and adversarial bandit problems, with applications to AI-driven technologies. The research contributes new theoretical insights into PDEs, delivers practical advancements for optimization and learning challenges, and supports NSF’s mission by aligning with national priorities in artificial intelligence and scientific progress. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.