Convergence Analysis of Probability Flow ODE for Score-based Generative Models

Score-based generative models have emerged as a powerful approach for sampling high-dimensional probability distributions. Despite their effectiveness, their theoretical underpinnings remain relatively underdeveloped. In this work, we study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives. Assuming access to <inline-formula> <tex-math notation="LaTeX">$L^{2}$ </tex-math></inline-formula>-accurate estimates of the score function, we prove the total variation between the target and the generated data distributions can be bounded above by <inline-formula> <tex-math notation="LaTeX">${\mathcal {O}}(d^{3/4}\delta ^{1/2})$ </tex-math></inline-formula> in the continuous time level, where <italic>d</italic> denotes the data dimension and <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> represents the <inline-formula> <tex-math notation="LaTeX">$L^{2}$ </tex-math></inline-formula>-score matching error. For practical implementations using a <italic>p</italic>-th order Runge-Kutta integrator with step size <italic>h</italic>, we establish error bounds of <inline-formula> <tex-math notation="LaTeX">${\mathcal {O}}(d^{3/4}\delta ^{1/2} + d\cdot (dh)^{p})$ </tex-math></inline-formula> at the discrete level. Finally, we present numerical studies on problems up to 128 dimensions to verify our theory.