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Description
The Fokker-Planck equation governs the time evolution of the probability density function $p_t(x)$ of a stochastic process. In high dimensions, direct approximation of $p_t$ is difficult because of normalization constraints. A common alternative is to learn the score function $s_t(x)=\nabla_x \log p_t(x)$ instead of the density itself, which avoids these difficulties.
In this work, we compare three score-based approaches which can be represented by a neural network: Score Matching (SM), Sliced Score Matching (SSM), and Score-PINN. Since neural network training involves an optimization problem that may have multiple local minima, independently initialized networks may converge to different solutions. We use the variance across their predictions to quantify this sensitivity to random initialization of score-based methods.
We further show how uncertainty in the learned score affects the recovered density by applying all three methods to the anisotropic Ornstein-Uhlenbeck process, for which the analytical solution is known. This allows exact evaluation of score and density errors, and identification of the score-based solver whose predictions are most consistent across random initializations.