  The task of sampling from a probability density can be approached as
transporting a tractable density function to the target, known as dynamical
measure transport. In this work, we tackle it through a principled unified
framework using deterministic or stochastic evolutions described by partial
differential equations (PDEs). This framework incorporates prior
trajectory-based sampling methods, such as diffusion models or Schr\"odinger
bridges, without relying on the concept of time-reversals. Moreover, it allows
us to propose novel numerical methods for solving the transport task and thus
sampling from complicated targets without the need for the normalization
constant or data samples. We employ physics-informed neural networks (PINNs) to
approximate the respective PDE solutions, implying both conceptional and
computational advantages. In particular, PINNs allow for simulation- and
discretization-free optimization and can be trained very efficiently, leading
to significantly better mode coverage in the sampling task compared to
alternative methods. Moreover, they can readily be fine-tuned with Gauss-Newton
methods to achieve high accuracy in sampling.
