Self-Consistency of the Fokker-Planck Equation
This work addresses the challenge of solving the FPE, which is important for statistical physics and machine learning, by providing a novel theoretical framework and practical implementation, though it appears incremental as it builds on existing concepts like neural ODEs.
The paper tackles the problem of solving the Fokker-Planck equation (FPE) by exploiting its self-consistency property to design a potential function for hypothesis velocity fields, proving that minimizing this function leads to convergence of density trajectories to the FPE solution in the Wasserstein-2 sense, with the method being amenable to neural network parameterization for efficient training and trajectory generation.
The Fokker-Planck equation (FPE) is the partial differential equation that governs the density evolution of the Itô process and is of great importance to the literature of statistical physics and machine learning. The FPE can be regarded as a continuity equation where the change of the density is completely determined by a time varying velocity field. Importantly, this velocity field also depends on the current density function. As a result, the ground-truth velocity field can be shown to be the solution of a fixed-point equation, a property that we call self-consistency. In this paper, we exploit this concept to design a potential function of the hypothesis velocity fields, and prove that, if such a function diminishes to zero during the training procedure, the trajectory of the densities generated by the hypothesis velocity fields converges to the solution of the FPE in the Wasserstein-2 sense. The proposed potential function is amenable to neural-network based parameterization as the stochastic gradient with respect to the parameter can be efficiently computed. Once a parameterized model, such as Neural Ordinary Differential Equation is trained, we can generate the entire trajectory to the FPE.