Geometric Mechanics and Geometric Integrators for Control, Optimization, and Sensitivity Analysis
The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.
Underlying this is the fact that mechanics carries structure beyond the differential equations themselves: a symplectic form, and momentum maps encoding the conservation laws that symmetry generates. Discretizing the variational principle, rather than the equations it produces, yields geometric integrators that inherit this structure exactly, that exhibit excellent long-time energy behavior, and that evolve intrinsically on configuration manifolds such as the rotation and Euclidean groups.
We will discuss the application of geometric structure-preserving numerical schemes to the geometric optimization control of mechanical systems, such as robots and drones. I will also describe the role of geometry and geometric structure-preservation in accelerated optimization and adjoint sensitivity analysis, and how they can be used to train neural networks derived from neural differential equations.
A recurring theme is that these structures reappear one level up: accelerated optimization methods arise as trajectories of a time-dependent Lagrangian system, and the adjoint equations of sensitivity analysis are themselves symplectic. Discretizing them in a structure-preserving way makes discretize-then-optimize and optimize-then-discretize agree.
About the speaker
Melvin Leok is professor of mathematics and director of the Computational Science, Mathematics and Engineering program at the University of California, San Diego. His research interests are in computational geometric mechanics, computational geometric control theory, discrete differential geometry, and structure-preserving numerical schemes, and particularly how these subjects relate to systems with symmetry. He received his Ph.D. in 2004 from the California Institute of Technology in Control and Dynamical Systems under the direction of Jerrold Marsden. He was a PIMS Marsden Memorial Lecturer, Simons Fellow in Mathematics, three-time NAS Kavli Frontiers of Science Fellow, and has received the DoD Newton Award for Transformative Ideas, NSF Faculty Early Career Development (CAREER) Award, SciCADE New Talent Prize, SIAM Student Paper Prize, Leslie Fox Prize (second prize) in Numerical Analysis, A*STAR International Fellowship, and Loke Cheng-Kim Foundation Scholarship. He has given plenary talks at Foundations of Computational Mathematics, NUMDIFF, and the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, and is the coauthor of a research monograph entitled, “Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds.