Data-driven Efficient Solvers and Predictions of Conformational Transitions for Langevin Dynamics on Manifold in High Dimensions
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We work on dynamic problems with collected data {x_i} that distributed on a manifold M⊂R^p. Through the diffusion map, we first learn the reaction coordinates {y_i}⊂N where N is a manifold isometrically embedded into an Euclidean space R^ℓ for ℓ≪ p. The reaction coordinates enable us to obtain an efficient approximation for the dynamics described by a Fokker-Planck equation on the manifold N. By using the reaction coordinates, we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of N. Furthermore, we provide a weighted L^2 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 N. 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 N 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.
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