SPECIAL MATHEMATICS COLLOQUIUM
Speaker: Huan Lei
Abstract. Computational modeling of multiscale dynamic systems is centered around projecting the high-dimensional full system dynamics onto a set of resolved field variables. In this talk, I will introduce a framework based on both rigorous projection and data-learning algorithms to systematically construct such models so that the non-local correlation and fluctuations arising from smaller scale interactions are properly accounted for. In particular, we develop numerical methods to address three essential challenges associated with the near-equilibrium fluctuation and the non-equilibrium dynamics. To quantify the uncertainty propagation arising from near-equilibrium fluctuations, we develop a numerical method based on data-driven basis construction and a sparsity enhancement algorithm. The developed method explores the sparse representation of the target quantity within a high-dimensional arbitrary density random space and enables us to accurately construct a surrogate model using limited sampling data. Furthermore, to quantify the non-equilibrium dynamic process, the projected dynamics are cast into the generalized Langevin Equation with an un-parameterized free energy term and memory kernel. We further develop a numerical method based on machine-learning SVGD algorithm for multi-dimensional density estimation to accurately construct free energy landscape. Finally, we develop a data-driven algorithm to efficiently parameterize the memory kernel with consistent fluctuation-dissipation and invariant measure. The proposed framework can accurately characterize the challenging non-equilibrium properties such as reaction rate with broad applications to multiscale dynamics in physics, engineering and biological systems.