(679b) Manifold Learning Techniques and Model Reduction for Dissipative Dynamics

Many chemical systems -for instance, large chemical reaction networks- are characterized by the existence of widely different time scales. Reduction of large systems of ODEs with such separation of time scales is one of the commonly studied topics in dynamical systems theory. Such systems often contain a slow, attracting, invariant manifold to which trajectories quickly decay. If this inertial manifold is low-dimensional, one can hope to obtain an efficient reduction of the system dynamics. This can be achieved by first "learning the manifold" empirically using sampled data and projecting the dynamics onto the empirically discovered slow manifold. The POD-Galerkin method [1,2], for instance, uses Principal Component Analysis (PCA [3]) to find a linear hyperspace containing the slow manifold and uses a Galerkin projection of the dynamics on the leading principal components of the sampled data. In highly non-linear systems, however, the intrinsic manifold dimension can be much smaller compared to the lowest dimensional linear subspace containing the slow manifold.

In this work, we consider a non-linear extension to the POD-Galerkin method. We propose using a non-linear machine learning technique -diffusion maps (DMAPs [4]), in particular- to empirically obtain the non-linear slow manifold of the dynamics. The ability to obtain a reduced model now rests on the successful definition of maps from the physical coordinates to the reduced (DMAP) coordinates and vice versa. The former is accomplished using the well-known Nystrom extension. The implementation of the latter, i.e., finding a consistent set of physical coordinates on the slow manifold given the reduced (DMAP) coordinates is one of the crucial points of discussion in this work. We illustrate our approach by considering two prototypical examples: a textbook singularly perturbed reacting system, and the truncated spectral discretization of a dissipative reaction-diffusion PDE. We thus link nonlinear manifold learning techniques for data analysis with model reduction techniques for evolution equations with separation of time scales.

References

[1]  G. Berkooz, P. Holmes, and J. L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Annual Review of Fluid Mechanics, 25 (1993), pp. 539–575.

[2]  K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM Journal on Numerical Analysis, 40 (2003), pp. 492–515.

[3]  I. T. Jolliffe, Principal component analysis, Springer-Verlag, 2002.

[4]  R. Coifman, S. Lafon, A. Lee, M. Maggioni, B. Nadler, F. Warner, and S. Zucker, Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps, PNAS, 102 (2005), p. 7426.