Continuous flow matching framework for scalable density estimation in stochastic dynamical systems

The probability density function (PDF) plays a fundamental role in characterizing the evolutionary behavior of stochastic dynamical systems. However, the inherent high dimensionality, strong nonlinearity, and multiscale nature of most stochastic dynamical systems pose formidable challenges for both theoretical analysis and computational modeling. To address these issues, we propose a flow matching density estimation framework (FMDEF) for continuous modeling of complex distributions in stochastic systems. FMDEF leverages continuous normalizing flows (CNFs) with time-dependent vector fields to capture the temporal evolution of probability densities. The framework introduces a scalable, simulation-free training mechanism that directly optimizes vector fields to match target probability paths, eliminating the need for explicit stochastic process modeling while enabling direct extraction of probabilistic structures from system measurements. Extensive numerical experiments on stochastic systems demonstrate the method’s effectiveness and robustness. Both theoretical analysis and experimental results confirm FMDEF’s potential for extension to high-dimensional problems, establishing it as a rigorous and practical solution for real-world applications in stochastic dynamical systems.