We present a framework to sample from high-dimensional unnormalized densities using physics-informed neural networks (PINNs). For various computational science tasks, it is essential to draw samples from a target distribution where the density is known up to a normalizing constant. Without access to any training samples, existing methods based on normalizing flows and diffusion models rely on the simulation of (stochastic) differential equations for training and suffer from mode collapse. Our approach circumvents these issues by solving the underlying continuity and Fokker-Planck equations using PINNs. Motivated by optimal transport and Schrödinger bridges, we further incorporate regularizers based on Hamilton-Jacobi-Bellman equations. Through evaluations on several benchmarks, we demonstrate that our approach can mitigate mode collapse and significantly outperform various baselines.