On fundamental unifying concepts for trajectory-based slow invariant attracting manifold computation in multiscale models of chemical kinetics

TL;DR

This paper explores geometric and analytical principles underlying various trajectory-based methods for computing slow invariant manifolds in chemical kinetics, aiming to unify and improve model reduction techniques.

Contribution

It introduces a unifying geometric and analytical framework, including a Hamiltonian formulation, for trajectory-based slow manifold approximation methods in chemical kinetics.

Findings

Relates different reduction approaches through a variational boundary value perspective.

Proposes a Hamiltonian formulation linking invariance and conservation laws.

Suggests fundamental principles underlying diverse dimension reduction techniques.

Abstract

Chemical kinetic models in terms of ordinary differential equations correspond to finite dimensional dissipative dynamical systems involving a multiple time scale structure. Most dimension reduction approaches aimed at a slow mode-description of the full system compute approximations of low-dimensional attracting slow invariant manifolds and parameterize these manifolds in terms of a subset of chosen chemical species, the reaction progress variables. The invariance property suggests a slow invariant manifold to be constructed as (a bundle of) solution trajectories of suitable ordinary differential equation initial or boundary value problems. The focus of this work is on a discussion of fundamental and unifying geometric and analytical issues of various approaches to trajectory-based numerical approximation techniques of slow invariant manifolds that are in practical use for model reduction in chemical kinetics. Two basic concepts are pointed out reducing various model reduction approaches to a common denominator. In particular, we discuss our recent trajectory optimization approach in the light of these two concepts. We relate both of them in a variational boundary value viewpoint, propose a Hamiltonian formulation and conjecture its relation to conservation laws, (partial) integrability and symmetry issues as underlying fundamental principles and potentially unifying elements of diverse dimension reduction approaches.