Efficiently Learning Linear System Solvers for Fast Numerical Simulation

Accelerating partial differential equation (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 challenges, 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. By implementing a lightweight online learning algorithm in the popular open source software OpenFOAM, we obtain significant reductions in fluid simulation wallclock while using minimal additional data/computation, and while inheriting the classical solvers' precision and correctness. Furthermore, we show regret-based performance guarantees for our approach under practically reasonable distributional assumptions on the linear systems' target vectors, including a setting in which we can learn instance-optimal solvers. 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.

(This talk is part of the weekly Machine Learning Lunch Meetings (MLLM), held every Tuesday from 12:15 to 1:15 p.m.  Professors from Computer Sciences, Statistics, ECE, the iSchool, and other departments will discuss their latest research in machine learning, covering both theory and applications. This is a great opportunity to network with faculty and fellow researchers, learn about cutting-edge research at our university, and foster new collaborations. For the talk schedule, please visit https://sites.google.com/view/wiscmllm/home. To receive future weekly talk announcements, please subscribe to our UW Google Group at https://groups.google.com/u/1/a/g-groups.wisc.edu/g/mllm.)