Abstract: This work introduces a method for learning low-dimensional models from data of high-dimensional black-box dynamical systems. The key contribution is a data sampling scheme that introduces a re-projection step to obtain trajectories corresponding to Markovian dynamics in low-dimensional subspaces. Models fitted to these re-projected trajectories (data) are the very same models that traditional model reduction constructs via projection of the governing equations onto reduced spaces. The recovery of reduced models from data is guaranteed for the large class of problems with polynomial nonlinear terms. Because of the recovery guarantee, a posteriori error estimators from model reduction are applicable, which allows bounding the generalization error of the learned models in a probabilistic sense. Thus, the proposed approach enables a workflow from data to reduced models to certified predictions that establishes trust in decisions made from data. Numerical experiments with convection-diffusion problems demonstrate the workflow.

Bio: Benjamin Peherstorfer is Assistant Professor at Courant Institute of Mathematical Sciences, New York University, since 2018. Before joining Courant, he was Assistant Professor at University of Wisconsin-Madison. Before that, he was Postdoctoral Associate (PostDoc) in the Aerospace Computational Design Laboratory (ACDL) at the Massachusetts Institute of Technology (MIT), working with Professor Karen Willcox. He received B.S., M.S., and Ph.D. degrees in computer science from the Technical University of Munich (Germany) in 2008, 2010, and 2013, respectively. His Ph.D. thesis was recognized with the Heinz-Schwaertzel prize, which is jointly awarded by three German universities to an outstanding Ph.D. thesis in computer science. In 2018, Benjamin was selected for a Department of Energy (DoE) Early Career Award in the Applied Mathematics Program. Benjamin's current research focuses on computational methods for data- and compute-intensive scientific computing applications, including computational statistics, mathematics of data science, model reduction, uncertainty quantification, and Bayesian inference.