Speaker: Professor Misha Khodak , University of Wisconsin-Madison
Abstract: Accelerating PDE solving is an important emerging AI application, but popular approaches that fully replace classical solvers using neural networks often struggle to compete due to insufficient data, optimization issues, low precision, and a lack of guarantees. We consider the alternative paradigm of integrating learning directly into solvers, focusing specifically on initial value PDEs, for which the main computational cost is often solving a sequence of linear systems. We introduce PCGBandit, a lightweight online learning algorithm that has performance guarantees under practically reasonable distributional assumptions on the linear systems' target vectors, and implement it in the popular open-source software OpenFOAM. In evaluations across six different settings, including two MHD simulations, PCGBandit yields significant wallclock reductions while inheriting the classical solvers' precision and correctness. Lastly, we highlight several future directions for analyzing scientific computing via the lens of learning theory/online algorithms and for further data-driven impact on numerical simulation.