Chertkov Andrei, Oseledets Ivan
Skolkovo Institute of Science and Technology, Moscow, Russia.
Front Artif Intell. 2021 Aug 2;4:668215. doi: 10.3389/frai.2021.668215. eCollection 2021.
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.
我们提出了一种用于求解多维福克-普朗克方程的新型数值格式，该格式基于切比雪夫插值、谱微分技术以及低秩张量近似，即张量列分解和多维交叉近似方法，这些方法相结合使得随着维度增加，能够大幅减少保持精度所需的自由度数量。我们在包括奥恩斯坦-乌伦贝克过程和哑铃模型在内的多个多维问题上证明了所提方法的有效性。所开发的计算高效的求解器可用于广泛的实际重要问题，包括机器学习应用中的密度估计。
