(278g) Scientific Computation in the Latent Space through Manifold Learning

Dimensionality reduction techniques, such as Diffusion Maps, have been used for discovering the reduced embedding coordinates and decreasing the stiffness of differential equations. Diffusion Maps can be used to reduce the embedding dimensions of a (finite) dataset sampled on a manifold, and to extend functions which are defined on the manifold/data set by building a functional basis termed Geometric Harmonics [1]. We can thus discover the essential latent variables for a dynamical system, and build reduced data-driven models based on those variables [2]. Such models expressed in terms of the (latent) Diffusion Maps coordinates, can help circumvent the stiffness due to the high dimensional ambient space, and thus decrease the computational cost.

We develop and (successfully) test different methodologies, either with pre-tabulation in the low-dimensional space and integration on the fly, or by going "back and forth" between the ambient space and the low dimensional space, allowing us to perform scientific computations only on the essential latent coordinates. Our examples come from combustion modeling and from the Chafee-Infante reaction-diffusion partial differential equation, for which a two-dimensional inertial manifold exists [4][5]. Sonday [4] showed that a Diffusion Maps parametrization can reveal this two-dimensional manifold embedded in a higher dimensional space. For this dynamical system, reduced models in the diffusion map coordinates are developed and integrated. The resulting "data assisted" integration results, based on three different approaches in the diffusion map embedding coordinates, are compared with the restricted (with Nyström [5]) solution coming from accurate integration of the original equations. The computational requirements and the relative values of the different schemes are compared.

[1] Lafon, S. Diffusion Maps and Geometric Harmonics. Ph.D. Thesis, Yale University, New Haven, CT, USA 2004.

[2] E. Chiavazzo, C.W. Gear, C.J. Dsilva, N. Rabin, I. G. Kevrekidis, Reduced Models in Chemical Kinetics via Nonlinear Data mining, Process 2, 112-140, (2014).

[3] Sonday, B. E. Systematic Model Reduction for Complex Systems through Data Mining and Dimensionality Reduction. PhD. Thesis, Princeton University, Princeton, NJ, USA, 2011.

[4] Gear C. W., I.G. Kevrekidis, and B.E. Sonday, Slow Manifold Integration on a Diffusion Maps Parametrization, AIP Conference Proceedings 1389, 13 (2011)

[5] Nyström, E.J. Über die praktische Auflösung von linearen Integralgleichungen mit Anwendungen
auf Randwertaufgaben der Potentialtheorie. Commentationes Physico-Mathematicae 1928, 4, 1–52 (in German).