(373t) Exploring Energy Landscapes Using the Directed Graph Laplacian

Authors: 
Psarellis, Y. M., Johns Hopkins University
Bello-Rivas, J., Princeton University
Wichrowski, N. J., Johns Hopkins University
Kevrekidis, I. G., Princeton University
Finding all minima of a potential energy function is of essential importance in many chemical engineering problems, including protein folding, self-assembly, and plant design. Here, we combine tools for finding all critical points of a potential with data-mining of the results of the exploration. Using a dynamical systems approach [1], we first explore all critical points and connections thereof in an energy landscape. Then, treating the discovered minima network as a directed graph, we apply the Directed Graph Laplacian (DGL) algorithm [2] in order to discover a lower-dimensional underlying manifold and create an effective embedding [3]. To illustrate this methodology, we use an example case of a two-scale potential. The DGL algorithm successfully identifies those minima that lie on an effective low-dimensional (here, one-dimensional) manifold. This combination of methods has the potential to enable dimensionality reduction in optimization problems with fine and coarse scale structure, where a surrogate lower-dimensional effective description is possible.

[1] Quapp W, Hirsch M, Imig O, Heidrich D. Searching for saddle points of potential energy surfaces by following a reduced gradient. Journal of Computational Chemistry. 1998; 19(9): 1087-1100.

[2] Chung F. Spectral Graph Theory. Providence, RI: American Mathematical Society, 1996.

[3] Perrault-Joncas D, Meilă M. Estimating Vector Fields on Manifolds and the Embedding of Directed Graphs. arXiv preprint. 2014; arXiv:1406.0013