Minimal Curvature Trajectories: Riemannian Geometry Concepts for Model Reduction in Chemical Kinetics

TL;DR

This paper introduces a novel trajectory-based optimization method leveraging Riemannian geometry to accurately approximate slow invariant manifolds for model reduction in chemical kinetics, enhancing understanding of multi-scale dynamics.

Contribution

It presents a new geometric optimization approach for computing slow manifolds in chemical kinetics, integrating Riemannian concepts for improved model reduction.

Findings

Accurate approximations of slow invariant manifolds achieved

Geometrically motivated criteria effectively identify slow trajectories

Method applied successfully to three test reaction mechanisms

Abstract

In dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories. Model reduction methods exploit this for simplifying chemical kinetics via a time scale separation into fast and slow modes. The aim is to approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones via computation of a slow attracting manifold. We present a novel method for computing approximations of such manifolds using trajectory-based optimization. We discuss Riemannian geometry concepts as a basis for suitable optimization criteria characterizing trajectories near slow attracting manifolds and thus provide insight into fundamental geometric properties of multiple time scale chemical kinetics. The optimization criteria correspond to a suitable mathematical formulation of "minimal relaxation" of chemical forces along reaction trajectories under given constraints. We present various geometrically motivated criteria and the results of their application to three test case reaction mechanisms serving as examples. We demonstrate that accurate numerical approximations of slow invariant manifolds can be obtained.