最小曲率轨迹：化学动力学模型简化的Riemannian Geometry概念
Minimal Curvature Trajectories: Riemannian Geometry Concepts for Model Reduction in Chemical Kinetics
摘要：
在耗散常微分方程系统中，不同的时间尺度导致各向异性相体积沿解轨线收缩。模型降阶法利用这一点，通过将时间尺度分离为快模式和慢模式来简化化学动力学。其目的是通过计算一个慢吸引流形来消除快模态，然后用一个降维的降阶模型来近似系统动力学。我们提出了一种新的方法来计算近似的流形使用基于几何的优化。我们讨论了黎曼几何概念的基础上，适当的优化标准的特点附近的轨道缓慢吸引流形，从而提供洞察多个时间尺度化学动力学的基本几何性质。的优化标准对应于一个合适的数学公式的“最小松弛”的化学力沿反应轨迹在给定的约束条件。我们提出了各种几何动机的标准和结果，他们的应用程序作为例子的三个测试用例的反应机制。我们证明了可以得到慢不变流形的精确数值逼近。
英文摘要：
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.
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