On the Theoretical and Practical Foundation of Kernel Methods and Gaussian Processes for the Solution of Partial Differential Equations

• Applicant: Dr. Aras Bacho
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 549075133

This research proposal is dedicated to the development and mathematical foundation of efficient machine learning methods, based on kernel methods and Gaussian processes, for the numerical solution of partial differential equations. It addresses the limitations of conventional approaches, which become particularly evident in dealing with high-dimensional and nonlinear problems, as well as the challenges of deep learning, which are primarily rooted in a lack of mathematical foundation and transparency. A central focus is on developing methods based on kernel and Gaussian processes for the numerical solution of set-valued and singular partial differential equations. Another important aspect is the exploration of mapping methods for transforming nonlinear partial differential equations into linear ones, similar to the Cole-Hopf transformation. The methodological approach includes the theoretical derivation of convergence guarantees and rates, the validation of developed algorithms through numerical experiments, and comparative analyses with existing methods. The innovation and significance of applying machine learning within mathematical frameworks to solve complex partial differential equations are highlighted.

• DFG Programme: WBP Fellowship
• International Connection: USA
• Hosts: Professor Dr. Houman Owhadi; Professor Dr. Andrew Stuart