Solution of the Fokker-Planck Equation by Cross Approximation Method in the Tensor Train Format

Area of Science:

• Computational Physics
• Numerical Analysis
• Machine Learning

Background:

• The Fokker-Planck equation is crucial for modeling systems with many degrees of freedom.
• Solving high-dimensional Fokker-Planck equations numerically is computationally intensive.
• Existing methods often struggle with scalability as dimensionality increases.

Purpose of the Study:

• To propose a novel, computationally efficient numerical scheme for multidimensional Fokker-Planck equations.
• To reduce the number of degrees of freedom required for accurate solutions.
• To demonstrate the scheme's effectiveness on relevant test cases.

Main Methods:

• Chebyshev interpolation and spectral differentiation techniques.
• Low-rank tensor approximations, including tensor train decomposition and multidimensional cross approximation.
• Combining these methods to achieve dimensionality reduction.

Main Results:

• The proposed scheme drastically reduces the number of degrees of freedom needed for accuracy.
• Effectiveness demonstrated on multidimensional problems like the Ornstein-Uhlenbeck process and dumbbell model.
• The solver is computationally efficient, offering significant speedups.

Conclusions:

• The novel numerical scheme provides an efficient solution for multidimensional Fokker-Planck equations.
• The method's ability to handle high dimensionality makes it suitable for complex scientific problems.
• Applications include density estimation in machine learning and other fields.