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.

翻译：从概率密度中采样的任务可以视为将易处理的密度函数传输至目标分布，这被称为动态测度传输。在本工作中，我们通过一个基于偏微分方程描述的确定性或随机演化过程的原理性统一框架来解决该问题。该框架整合了先前的基于轨迹的采样方法，如扩散模型或薛定谔桥，且无需依赖时间反转的概念。此外，它使我们能够提出新的数值方法来求解传输任务，从而从复杂的目标分布中采样，且无需归一化常数或数据样本。我们采用物理信息神经网络来近似求解相应的偏微分方程，这带来了概念和计算上的双重优势。具体而言，物理信息神经网络允许进行无仿真和无离散化的优化，并且可以非常高效地训练，从而在采样任务中实现比替代方法显著更好的模态覆盖。此外，它们可以方便地通过高斯-牛顿方法进行微调，以实现高精度的采样。