Adaptive normalizing flows for solving Fokker–Planck equation

The Fokker-Planck (FP) equation governs the probabilistic response of diffusion processes driven by stochastic differential equations (SDEs). Gaussian mixture models and deep learning solvers are two state-of-the-art methods for solving the FP equation. Although mixture models mostly depend on empirical sampling strategies and predefined Gaussian components, deep learning techniques suffer from inherent interpretability deficits and require excessively large training samples. To address these challenges, we propose an adaptive normalizing flow framework for solving FP equations (ANFFP). Normalizing flows are generative models that produce tractable distributions to approximate the complex target distributions. The ANFFP architecture inherently preserves probabilistic interpretability while enabling efficient exact sampling advantages that significantly enhance its applicability to probabilistic response modeling under small sample conditions. Numerical examples involving one-dimensional, two-dimensional, and four-dimensional SDEs demonstrate the effectiveness of the method. In addition, the computational complexity of the ANFFP method is discussed in more detail. This work provides a new paradigm for solving high-dimensional FP equations with theoretical guarantees and practical scalability.