Learning Dynamical Systems with Side Information
Amir Ali Ahmadi, Princeton University
We present a mathematical and computational framework for the problem of learning a dynamical system from noisy observations of a few trajectories and subject to side information. Side information is any knowledge we might have about the dynamical system we would like to learn besides trajectory data. It is typically inferred from domain-specific knowledge or basic principles of a scientific discipline. We are interested in explicitly integrating side information into the learning process in order to compensate for scarcity of trajectory observations. We identify six types of side information that arise naturally in many applications and lead to convex constraints in the learning problem. First, we show that when our model for the unknown dynamical system is parameterized as a polynomial, one can impose our side information constraints computationally via semidefinite programming. We then demonstrate the added value of side information for learning the dynamics of basic models in physics and cell biology, as well as for learning and controlling the dynamics of a model in epidemiology. Finally, we study how well polynomial dynamical systems can approximate continuously-differentiable ones while satisfying side information (either exactly or approximately). Our overall learning methodology combines ideas from convex optimization, real algebra, dynamical systems, and functional approximation theory, and can potentially lead to new synergies between these areas. Based on joint work with Bachir El Khadir.
Amir Ali Ahmadi is a Professor at the Department of Operations Research and Financial Engineering at Princeton University and an Associated Faculty member of the Program in Applied and Computational Mathematics, the Department of Computer Science, the Department of Mechanical and Aerospace Engineering, the Department of Electrical Engineering, and the Center for Statistics and Machine Learning. Amir Ali received his PhD in EECS from MIT and was a Goldstine Fellow at the IBM Watson Research Center prior to joining Princeton. His research interests are in optimization theory, computational aspects of dynamics and control, and algorithms and complexity. Amir Ali's distinctions include the Sloan Fellowship in Computer Science, the Presidential Early Career Award for Scientists and Engineers (PECASE), the NSF CAREER Award, the AFOSR Young Investigator Award, the DARPA Faculty Award, the Google Faculty Award, the MURI award of the AFOSR, the Howard B. Wentz Junior Faculty Award as well as the Innovation Award of Princeton University, the Goldstine Fellowship of IBM Research, and the Oberwolfach Fellowship of the NSF. His undergraduate course at Princeton (ORF 363, ``Computing and Optimization'') is a three-time recipient of the Teaching Award of the Princeton Engineering Council, as well as a recipient of the Excellence in Teaching of Operations Research Award of the Institute for Industrial and Systems Engineers, and the Phi Beta Kappa Award for Excellence in Undergraduate Teaching at Princeton University. Amir Ali's research has been recognized by a number of best-paper awards, including the INFORMS Optimization Society's Young Researchers Prize, the INFORMS Computing Society Prize (for best series of papers at the interface of operations research and computer science), the Best Conference Paper Award of the IEEE International Conference on Robotics and Automation, and the prize for one of two most outstanding papers published in the SIAM Journal on Control and Optimization in 2013-2015.