Abstract: We propose and analyze a deterministic sampling framework using Score-Based Transport Modeling (SBTM) for sampling an unnormalized target density $\pi$. While diffusion generative modeling relies on pre-training the score function $\nabla \log f_t$ using samples from $\pi$, SBTM addresses the more challenging setting where only the unnormalized density $\pi$ is known. SBTM approximates the Wasserstein gradient flow on $\mathrm{KL}(f_t\|\pi)$ by learning the time-varying score $\nabla \log f_t$ on the fly using score matching. We prove that SBTM dissipates relative entropy at the same rate as the exact gradient flow, provided sufficient training. We further extend our framework to annealed dynamics, to handle non log-concave targets. Numerical experiments validate our theoretical findings: SBTM converges at the optimal rate, has smooth trajectories, and is easily integrated with annealed dynamics.