Tensor neural networks for high-dimensional Fokker-Planck equations

Area of Science:

• Computational Physics
• Machine Learning
• Numerical Analysis

Background:

• High-dimensional steady-state Fokker-Planck equations are crucial in various scientific fields.
• Solving these equations analytically or numerically in high dimensions presents significant computational challenges.
• Existing methods often struggle with the curse of dimensionality.

Purpose of the Study:

• To develop an efficient and accurate method for solving high-dimensional steady-state Fokker-Planck equations.
• To leverage tensor neural networks within a physics-informed machine learning framework.
• To demonstrate the efficacy of this approach across a range of dimensions.

Main Methods:

• Application of tensor neural networks, comprising tensor products of feedforward networks or radial basis functions.
• Integration with physics-informed neural networks and stochastic gradient descent for training.
• Strategic selection of bounded domains and parameter constraints for tensor radial basis function networks.

Main Results:

• Demonstrated efficiency of tensor neural networks for steady-state Fokker-Planck equations in dimensions 2 to 10.
• Tensor feedforward networks effectively utilize auto-differentiation.
• Radial basis function networks avoid computationally expensive auto-differentiation in high dimensions, achieving high accuracy with parameter constraints.

Conclusions:

• Tensor neural networks offer a powerful and efficient computational tool for high-dimensional Fokker-Planck equations.
• The physics-informed machine learning approach provides a scalable solution to complex problems in statistical physics and beyond.
• This method shows significant promise for advancing research in fields relying on Fokker-Planck equation modeling.