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.

通过确定性或随机演化的偏微分方程（PDEs）描述的原则性统一框架，本文解决了从一个可处理的密度函数电运输到目标的任务，该框架包含了基于轨迹的传统采样方法，如扩散模型、Schrödinger桥，而不依赖于时间反演的概念，并通过提出新的数值方法来解决传输任务，从而在没有标准化常数或数据样本的情况下采样复杂目标，我们使用物理信息神经网络（PINNs）来逼近相应的PDE解，这意味着在概念和计算方面的优势，PINNs不需要模拟和离散，可以有效地进行优化和训练，并且与曲牛顿方法进行调整，以实现高准确度的采样。

动力测量传输和神经PDE求解器用于采样