Geometric Mechanics and Geometric Integrators for Control, Optimisation and Sensitivity Analysis by Prof Melvin Leok

"Geometric Mechanics and Geometric Integrators for Control, Optimisation and Sensitivity Analysis"

The geometric approach to mechanics underpins innovative control methodologies in geometric control theory. These techniques allow satellite attitude to be controlled through changes in shape rather than chemical propulsion and explain how a falling cat can land on its feet even when released in an inverted orientation.

Mechanics carries structure beyond the differential equations themselves: a symplectic form and momentum maps encoding conservation laws generated by symmetry. Discretising the variational principle, rather than the resulting equations, yields geometric integrators that inherit this structure exactly, exhibit excellent long-time energy behavior, and evolve intrinsically on configuration manifolds such as the rotation and Euclidean groups.

We will discuss geometric structure-preserving numerical schemes for the geometric optimisation control of mechanical systems, including robots and drones. I will also describe their role in accelerated optimisation and adjoint sensitivity analysis, and their use in training neural networks derived from neural differential equations.

These structures also reappear one level up: accelerated optimization methods arise as trajectories of a time-dependent Lagrangian system, while the adjoint equations of sensitivity analysis are themselves symplectic. Structure-preserving discretisation ensures that discretise-then-optimise and optimise-then-discretise 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 PhD 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 co-author of a research monograph entitled, “Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds.