Adaptive normalizing flows for solving Fokker-Planck equation.

Area of Science:

• Computational mathematics
• Stochastic processes
• Machine learning

Background:

• Fokker-Planck (FP) equations model diffusion processes from stochastic differential equations (SDEs).
• Current methods like Gaussian mixture models and deep learning solvers have limitations in interpretability and sample efficiency.
• Deep learning methods require large datasets, while mixture models rely on empirical sampling.

Purpose of the Study:

• To develop a novel framework, the adaptive normalizing flow framework for solving FP equations (ANFFP), to address the limitations of existing methods.
• To enhance the interpretability and efficiency of solving FP equations, particularly under small sample conditions.
• To provide a scalable and theoretically grounded method for high-dimensional FP equations.

Main Methods:

• Utilizing normalizing flows, a class of generative models, to approximate complex target distributions.
• Developing an adaptive framework (ANFFP) that inherently preserves probabilistic interpretability.
• Implementing efficient exact sampling strategies within the ANFFP architecture.

Main Results:

• The ANFFP method demonstrates effectiveness in solving one-, two-, and four-dimensional SDEs.
• The framework offers enhanced applicability for probabilistic response modeling with limited data.
• The computational complexity of ANFFP is analyzed, showing practical scalability.

Conclusions:

• ANFFP presents a new paradigm for solving high-dimensional FP equations.
• The method combines theoretical guarantees with practical scalability and interpretability.
• ANFFP offers a significant advancement for diffusion process modeling.