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The use of computational simulation is by now so pervasive in society that it is no exaggeration to say that continued U.S. and international prosperity, security, and health depend in part on continued improvements in simulation capabilities. What if we could predict weather two weeks out, guide the design of new drugs for new viral diseases, or manage new manufacturing processes that cut production costs and times by an order of magnitude? What if we could predict collective human behavior, for example, response to an evacuation request during a natural disaster, or labor response to fiscal stimulus? (See also the companion CCC Quad Paper on Pandemic Informatics, which discusses features that would be essential to solving large-scale problems like preparation for, and response to, the inevitable next pandemic.) The past decade has brought remarkable advances in complementary areas: in sensors, which can now capture enormous amounts of data about the world, and in AI methods capable of learning to extract predictive patterns from those data. These advances may lead to a new era in computational simulation, in which sensors of many kinds are used to produce vast quantities of data, AI methods identify patterns in those data, and new AI-driven simulators combine machine-learned and mathematical rules to make accurate and actionable predictions. At the same time, there are new challenges -- computers in some important regards are no longer getting faster, and in some areas we are reaching the limits of mathematical understanding, or at least of our ability to translate mathematical understanding into efficient simulation. In this paper, we lay out some themes that we envision forming part of a cohesive, multi-disciplinary, and application-inspired research agenda on AI-driven simulators.