Solver in the low-rank tensor-train format with cross approximation approach for solution of the multidimensional Fokker-Planck equation

# fpcross

## Description

This python package, named fpcross (Fokker Planck cross-approximation), provides a solver in the low-rank tensor train format with cross approximation approach for solution of the multidimensional Fokker-Planck equation (FPE) of the form

d r(x, t) / d t = D delta( r(x, t) ) - div( f(x, t) r(x, t) ),
where r(x, 0) = r0(x).


The function f(x, t), its diagonal partial derivatives d f_i (x, t) / d x_i, initial condition r0(x) and scalar diffusion coefficient D should be known. The equation is solved from the initial moment (t = 0) to the user-specified moment (t), while the solutions obtained at each time step can be used if necessary. The resulting solution r(x, t) represents both the TT-tensor on the multidimensional Chebyshev grid and the Chebyshev interpolation coefficients in the TT-format, and therefore it can be quickly calculated at an arbitrary spatial point.

## Installation

The package can be installed via pip: pip install fpcross (it requires the Python programming language of the version >= 3.8). It can be also downloaded from the repository fpcross and installed by python setup.py install command from the root folder of the project.

It is highly recommended to create a virtual environment before installing (conda create --name fpcross python=3.8 and then conda activate fpcross). The setup file contains the versions of the packages that were used when testing the software product, but it may work correct on newer versions.

## Usage

A compact example of using the solver for a user-defined FPE is provided in the script demo/demo.py (run it as python demo/demo.py from the root of the project).

The software product also implements classes for the model FPEs:

1. multidimensional simple diffusion problem (see fpcross/equation_demo/equation_dif.py);
2. multidimensional Ornstein-Uhlenbeck process (see fpcross/equation_demo/equation_oup.py);
3. 3-dimensional dumbbell model (see fpcross/equation_demo/equation_dum.py).

A demonstration of their solution is given in the script demo/check.py (run it as python demo/check.py from the root of the project).

## Citation

If you find this approach and/or code useful in your research, please consider citing:

@article{chertkov2021solution,
author    = {Chertkov, Andrei and Oseledets, Ivan},
year      = {2021},
title     = {Solution of the Fokker--Planck equation by cross approximation method in the tensor train format},
journal   = {Frontiers in Artificial Intelligence},
volume    = {4},
issn      = {2624-8212},
doi       = {10.3389/frai.2021.668215},
url       = {https://www.frontiersin.org/article/10.3389/frai.2021.668215}
}


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