This proposal will focus on rare and extreme climate events, such as heat waves and cold spells, which have major societal impacts. Rapid developments in AI are transforming scientific research, but are difficult to apply to rare events because too few of them occur in training sets. The proposed work will develop the essential mathematical tools to leverage AI methods to significantly improve the estimation of rare event statistics, both in climate and in other fields. The broadening participation aspect of this proposal is centered on making a positive impact on the lives and studies of veteran scholars of the United States military through college and graduate school admissions mentoring and research internships.<br/><br/>This interdisciplinary project relies on an essential collaboration among AI, math, and climate to make transformational advances in knowledge that build on and enhance each field. This proposal will develop AI Dynamic Galerkin Approximation (AI-DGA) to extract long return periods from large-ensemble short-duration emulations. Then, it will leverage rare-event sampling in a novel hybrid and iterative use of numerical solvers and AI emulators (AI-RES) to develop additional estimates of return periods and generate more rare event data to re-train, and thus improve the emulator. The proposed work will deliver methods to improve 1) return period estimates for rare events and 2) the training of the AI emulators themselves. The proposal will focus on heat waves and cold snaps, but the methods developed will increase the usefulness of AI emulators across climate science, and geoscience broadly, by innovating new ways to apply them even on rare events they have never seen in their training set and even if the emulators are not reliable for long simulations.<br/><br/>This award by the Division of Research, Innovation, Synergies, and Education within the Directorate for Geosciences is jointly supported by the National Discovery Cloud for Climate initiative of the Office of Advanced Cyberinfrastructure within the Directorate for Computer and Information Science and Engineering and by the Division of Mathematical Sciences within the Directorate for Mathematical and Physical Sciences.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.