Math-CSE PhD Thesis Defense: Songchen Tan

Abstract:

Automatic differentiation occupies a central role in numerical algorithms for nonlinear equations, differential equations, and optimization. This thesis reinterprets automatic differentiation as a paradigm of program transformation enabled by modern compiler techniques, and explores how such techniques can systematically generate not only derivatives but also a broader class of algorithmic components from user-defined programs. By introducing principles of symbolic-numeric computing that leverage symbolic intermediate representations, the thesis establishes a unified framework for arbitrary-order differentiation and the automatic synthesis of high-level algorithmic components. These methods are then applied to develop advanced algorithms for solving nonlinear equations, ordinary and stochastic differential equations, and physics-informed neural networks. Through both theoretical insights and practical tool development, the thesis demonstrates that compiler techniques can significantly enhance the efficiency, flexibility, and capabilities of numerical computation.

Thesis Committee Members:

• Professor Alan Edelman, Department of Mathematics, MIT (Chair)
• Professor John Urschel, Department of Mathematics, MIT
• Professor Steven Johnson, Department of Mathematics and Department of Physics, MIT