7,029 research outputs found
Data-driven Efficient Solvers and Predictions of Conformational Transitions for Langevin Dynamics on Manifold in High Dimensions
We work on dynamic problems with collected data that
distributed on a manifold . Through the
diffusion map, we first learn the reaction coordinates where is a manifold isometrically embedded into an
Euclidean space for . The reaction coordinates
enable us to obtain an efficient approximation for the dynamics described by a
Fokker-Planck equation on the manifold . By using the reaction
coordinates, we propose an implementable, unconditionally stable, data-driven
upwind scheme which automatically incorporates the manifold structure of
. Furthermore, we provide a weighted convergence analysis of
the upwind scheme to the Fokker-Planck equation. The proposed upwind scheme
leads to a Markov chain with transition probability between the nearest
neighbor points. We can benefit from such property to directly conduct
manifold-related computations such as finding the optimal coarse-grained
network and the minimal energy path that represents chemical reactions or
conformational changes. To establish the Fokker-Planck equation, we need to
acquire information about the equilibrium potential of the physical system on
. Hence, we apply a Gaussian Process regression algorithm to
generate equilibrium potential for a new physical system with new parameters.
Combining with the proposed upwind scheme, we can calculate the trajectory of
the Fokker-Planck equation on based on the generated equilibrium
potential. Finally, we develop an algorithm to pullback the trajectory to the
original high dimensional space as a generative data for the new physical
system.Comment: 59 pages, 16 figure
- β¦