Reduced models in chemical kinetics via nonlinear data-mining

TL;DR

This work tackles the challenge of reducing detailed chemical-kinetics models by extracting and parameterizing slow invariant manifolds with Diffusion Maps (DMAP). The authors propose a three-step workflow: (i) sample the slow manifold from detailed kinetics via brief simulations, (ii) obtain a global, low-dimensional embedding using DMAP, and (iii) construct a reduced dynamical system in the learned slow variables with accurate lifting and restriction to couple back to the full ambient space. They evaluate multiple interpolation/extension schemes (Nyström, radial basis functions, Kriging, Laplacian Pyramids, Geometric Harmonics) and find that a local lifting strategy combined with Nyström restriction yields the best accuracy for a hydrogen–air combustion example, achieving meaningful speedups (around 4×) while preserving trajectory fidelity. The work demonstrates a viable path to integrating reduced-chemistry models into flow solvers and lays out practical considerations for scalability, dimensionality variation across the phase space, and the need for non-tabulated closures in future research.

Abstract

The adoption of detailed mechanisms for chemical kinetics often poses two types of severe challenges: First, the number of degrees of freedom is large; and second, the dynamics is characterized by widely disparate time scales. As a result, reactive flow solvers with detailed chemistry often become intractable even for large clusters of CPUs, especially when dealing with direct numerical simulation (DNS) of turbulent combustion problems. This has motivated the development of several techniques for reducing the complexity of such kinetics models, where eventually only a few variables are considered in the development of the simplified model. Unfortunately, no generally applicable a priori recipe for selecting suitable parameterizations of the reduced model is available, and the choice of slow variables often relies upon intuition and experience. We present an automated approach to this task, consisting of three main steps. First, the low dimensional manifold of slow motions is (approximately) sampled by brief simulations of the detailed model, starting from a rich enough ensemble of admissible initial conditions. Second, a global parameterization of the manifold is obtained through the Diffusion Map (DMAP) approach, which has recently emerged as a powerful tool in data analysis/machine learning. Finally, a simplified model is constructed and solved on the fly in terms of the above reduced (slow) variables. Clearly, closing this latter model requires nontrivial interpolation calculations, enabling restriction (mapping from the full ambient space to the reduced one) and lifting (mapping from the reduced space to the ambient one). This is a key step in our approach, and a variety of interpolation schemes are reported and compared. The scope of the proposed procedure is presented and discussed by means of an illustrative combustion example.

Figures (15)

• Figure S1: Data manifold, dimensionality and independence of DMAP eigenvectors (a) 2000 uniformly random points initially placed in a unit square are stretched and wrapped around three fourths of a cylinder. (b) The entry in the first non-trivial eigenvector of the Markov matrix $K$ versus the first cylindrical coordinate $\theta$ for each data point. (c) Entry in the second non-trivial eigenvector of $K$ versus the first one; the quasi-one-dimensionality of the plot implies strong eigenvector correlation. (d) Entry in the third non-trivial eigenvector of $K$ versus the first one. The evident two-dimensional scatter implies that a new direction on the data manifold has been detected.
• Figure S2: The effect of data sampling. (a) A 40 by 40 array of regularly spaced points are placed on a square and subsequently wrapped around a cylinder. (b) Entry in the first non-trivial eigenvector of the Markov matrix $K$ versus the first cylindrical coordinate $\theta$ for each data point. (c) Entry in the second non-trivial eigenvector of $K$ versus the first one. (d) Entry in the third non-trivial eigenvector of $K$ versus the first one.
• Figure S3: More on the effect of data sampling: noise. (a) 1600 points are initially randomly placed in each of the 40 by 40 array small squares forming the unit square and afterwards bent around a cylinder. (b) Entry in the first non-trivial eigenvector of the Markov matrix $K$ versus the first cylindrical coordinate $\theta$ for each data point. (c) Entry in the second non-trivial eigenvector of $K$ versus the first one. (d) Entry in the third non-trivial eigenvector of $K$ versus the first one
• Figure S4: The analogy between traditional diffusion on domains and diffusion on graphs from sampled data. (a) The solution to the finite element method (FEM) formulation of the PDE (partial differential equation) eigenvalue problem $\nabla^2 \phi = \lambda \phi$ with no flux boundary conditions is reported for a narrow two-dimensional rectangular stripe. The second and seventh eigenfunctions are found to be uncorrelated and suitable to parametrize the two relevant dimensions of the manifold. (b) Entries in the first non-trivial eigenfunction of the problem in figure (a) versus entries in the fourth eigenfunction (sampled at scattered locations of the computational domain) reveals a strong correlation between those two functions: the fourth eigenvector (which we know corresponds to the third harmonic, $cos(3 \bar{x})$, does not encode a new direction on the data manifold. Right-hand side: Entries in the first non-trivial eigenfunction of the problem in figure (a) versus entries in the seventh eigenfunction (at the same scattered locations) confirms that the seventh eigenvector (which we know corresponds to $cos(\bar{y})$, encodes a new, second direction. (c) Different domain, same premise: The solution to the finite element method (FEM) formulation of the PDE (partial differential equation) eigenvalue problem $\nabla^2 \phi = \lambda \phi$ with no flux boundary conditions is reported for a two-dimensional manifold with complicated boundary. The second and seventh eigenfunctions are found to be uncorrelated and suitable to parametrize the two relevant dimensions of the manifold (an "angular" and a "radial" one).
• Figure S5: On sampling initial conditions in a convex polytope in $\Re^3$ with vertices $A=(1.8,0.5,0)$, $B=(1,0,3)$, $C=(0,1,1.5)$ and $D=(0.2,0,0)$. Left-hand side: Five hundred points are generated by Eq. (\ref{['randomcomb']}) with uniformly random values $0 \le \tilde{w}_i \le 1$ and $\bar{w}_i = \tilde{w}_i /\sum\nolimits_{j = 1}^4 {\tilde{w}_j}$; notice the poor sampling close to the boundaries. Right-hand side: Five hundred points are generated by Eq. (\ref{['randomcomb']}) with uniformly random values $0 \le z_i \le 1$, $\tilde{w}_i = \left[ { - \ln \left( {z_i } \right)} \right]^{1.5}$ (i.e. $p=1.5$) and $\bar{w}_i = \tilde{w}_i /\sum\nolimits_{j = 1}^4 {\tilde{w}_j}$. The latter approach generates a more uniform sampling of the polytope interior.