Frontiers in applied mathematics

Lead: Jonathan Weare

How can AI extend the reach of scientific computing while preserving rigorous guarantees on the results? We will study how AI can help design and analyze numerical methods and derive reduced models of complex systems. One question we share with the formalization group is how to certify the accuracy of a computation. For example, can an AI system propose a numerical approximation together with a verifiable bound on its error?

Applied mathematics also shapes the foundations of AI. Training a model poses large-scale optimization problems, and generative modeling depends on the ability to sample efficiently from complex distributions. We will draw on Courant’s strengths in scientific machine learning, generative modeling, and the analysis of learning algorithms to study these questions and to design training environments where models can test proposed methods against computational problems whose solutions can be checked.