用张量训练格式的交叉近似法解Fokker-Planck方程
Solution of the Fokker-Planck equation by cross approximation method in the tensor train format
摘要：
本文提出了一种新的求解多维Fokker-Planck方程的数值格式，该格式基于Chebyshev插值和谱微分技术以及低秩张量逼近，即张量训练分解和多维交叉逼近方法，其组合使得可以显著减少随着维度增加而保持精度所需的自由度的数量。我们证明了所提出的方法对一些多维问题的有效性，包括Ornstein-Uhlenbeck过程和哑铃模型。所开发的计算效率高的求解器可以用于广泛的实际重要的问题，包括机器学习应用中的密度估计。
英文摘要：
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.
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