Approximation of slow attracting manifolds in chemical kinetics by trajectory-based optimization approaches

Many common kinetic model reduction approaches are explicitly based on inherent multiple time scales and often assume and directly exploit a clear time scale separation into fast and slow reaction processes. They approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones. The corresponding restrictive assumption of full relaxation of fast modes often renders the resulting approximation of slow attracting manifolds inaccurate as a representation of the reduced model and makes the numerical solution of the nonlinear "reduction equations" particularly difficult in many cases where the gap in intrinsic time scales is not large enough. We demonstrate that trajectory optimization approaches can avoid such severe restrictions by computing numerical solutions that correspond to "maximally relaxed" dynamical modes in a suitable sense. We present a framework of trajectory-based optimization for model reduction in chemical kinetics and a general class of reduction criteria characterizing the relaxation of chemical forces along reaction trajectories. These criteria can be motivated geometrically exploiting ideas from differential geometry and fundamental physics and turn out to be highly successful in example applications. Within this framework, we provide results for the computational approximation of slow attracting low-dimensional manifolds in terms of families of optimal trajectories for a six-component hydrogen combustion mechanism.