Abstract
We propose the novel numerical scheme for solution of the multidimensional Fokker–Planck equation, which is based on the Chebyshev interpolation and the spectral differentiation techniques as well as low rank tensor approximations, namely, the tensor train decomposition and the multidimensional cross approximation method, which in combination makes it possible to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases. We demonstrate the effectiveness of the proposed approach on a number of multidimensional problems, including Ornstein-Uhlenbeck process and the dumbbell model. The developed computationally efficient solver can be used in a wide range of practically significant problems, including density estimation in machine learning applications.
Highlights
Fokker–Planck equation (FPE) is an important in studying properties of the dynamical systems, and has attracted a lot of attention in different fields
The equation is discretized on a tensor-product grid, such that the solution is represented as a d-dimensional tensor, and this tensor is approximated in the low-rank tensor train format (TT-format) (Oseledets, 2011)
We propose to use a TT-format and cross approximation method (CAM) to approximate the solution of the FPE which makes it possible to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases;
Summary
Fokker–Planck equation (FPE) is an important in studying properties of the dynamical systems, and has attracted a lot of attention in different fields. The equation is discretized on a tensor-product grid, such that the solution is represented as a d-dimensional tensor, and this tensor is approximated in the low-rank tensor train format (TT-format) (Oseledets, 2011) Even with such complexity reduction, the computations often take a long time. Since we can evaluate the value of ρ(x, t) at any x, we can use the cross approximation method (CAM) (Oseledets and Tyrtyshnikov, 2010; Savostyanov and Oseledets, 2011; Dolgov and Savostyanov, 2020) in the TT-format to recover a supposedly lowrank tensor from its samples In this way we do not need to have any compact representation of f, but only numerically solve the corresponding ODE. We implement FPE solver, based on the proposed approach, as a publicly available python code, and we test our approach on several examples, including multidimensional Ornstein-Uhlenbeck process and dumbbell model, which demonstrate its efficiency and robustness
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