Non-asymptotic error bounds for probability flow ODEs under weak log-concavity

Score-based generative modeling, implemented through probability flow ODEs, has shown impressive results in numerous practical settings. However, most convergence guarantees rely on restrictive regularity assumptions on the target distribution—such as strong log-concavity or bounded support—or focus on the specific case of constant drift and diffusion coefficients. This work establishes non-asymptotic convergence bounds in 2-Wasserstein distance for probability flow ODEs when the target is only weakly log-concave, allowing multimodal distributions such as Gaussian mixtures and time-dependent noise schedules. Our bounds account for initialization, discretization, and score approximation errors. We show that, for fixed regularity parameters of the target distribution, weakening strong to weak log-concavity leaves the asymptotic dependence on the time horizon, step size, and score matching error. Yet, the relaxation incurs a potentially large non-asymptotic price: the bounds contain an exponential dependence on the deviation from log-concavity. We complement the upper bound with a lower bound construction, showing that this exponential dependence is unavoidable under weak log-concavity alone.