We work on dynamic problems with collected data\subsetR^p based on collected sample points } \{x_i\} that distributed on a manifold \mathcal{M\subsetR^p} ^n \subset M that probe the unknown manifold M} . Through the diffusion map, we first learn the reaction coordinates \{y_i\} \subset \mathcal{N^n\subset N corresponding to \{x_i\}_{i=1}^n, } where N is a manifold isometrically embedded into an Euclidean space \mathbb{R^\ell for }  and isometrically embedded in \mathbb{R}^\ell with } \ell \ll p. The reaction coordinates enable us to obtain an efficient approximation for the dynamics described by a Fokker-Planck equation on the manifold \mathcal{N} . By using the reaction coordinates , we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of \mathcal{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_i\}_{i=1}^n with an approximated transition probability and jump rate } 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 \mathcal{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 \mathcal{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}  .
We study the Langevin dynamics of a physical system with manifold structure \mathcal{M\subsetR^p based on collected sample points } \{x_i\} \subsetR^p} _{i=1^n \subset M that probe the unknown manifold M} . Through the diffusion map, we first learn the reaction coordinates \{y_i\} _{i=1^n\subset N corresponding to \{x_i\}_{i=1}^n, } where N is a manifold ^\ell for } diffeomorphic to \mathcal{M and isometrically embedded in \mathbb{R}^\ell with } \ell \ll p. The induced Langevin dynamics on \mathcal{N approximation for the Langevin dynamics on \mathcal{N Fokker-Planck equation on the manifold \mathcal{N} in terms of the reaction coordinates \mathsf{y propose an implementable, unconditionally stable, data-driven finite volume scheme for this Fokker-Planck equation, which automatically incorporates the manifold structure of \mathcal{N}. Furthermore, we provide a weighted L^2 convergence analysis of the finite volume scheme to the Fokker-Planck equation on \mathcal{N . The proposed finite volume scheme leads to a Markov chain on \{\mathsf{y_i\}_{i=1}^n with an approximated transition probability and jump rate } between the nearest neighbor points. . 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 \mathcal{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} After an unconditionally stable explicit time discretization, the data-driven finite volume scheme gives an approximated Markov process for the Langevin dynamics on \mathcal{N .
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work on dynamic problems with collected data
study the Langevin dynamics of a physical system with manifold structure \mathcal{M
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that distributed on a manifold \mathcal{M
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isometrically embedded into an Euclidean space \mathbb{R
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diffeomorphic to \mathcal{M
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reaction coordinates enable us to obtain an efficient
induced Langevin dynamics on \mathcal{N
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dynamics described by a
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\mathsf{y
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upwind scheme
finite volume scheme for this Fokker-Planck equation,
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upwind
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on \mathcal{N
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upwind
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with transition probability
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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 \mathcal{N
A
After an unconditionally stable explicit time discretization, the data-driven finite volume scheme gives an approximated Markov process for the Langevin dynamics on \mathcal{N