We present two neural network approaches that approximate the solutions of static and dynamic conditional optimal transport (COT) problems, respectively. Both approaches enable sampling and density estimation of conditional probability distributions, which are core tasks in Bayesian inference. Our methods represent the target conditional distributions as transformations of a tractable reference distribution and, therefore, fall into the framework of measure transport. COT maps are a canonical choice within this framework, with desirable properties such as uniqueness and monotonicity. However, the associated COT problems are computationally challenging, even in moderate dimensions. To improve the scalability, our numerical algorithms leverage neural networks to parameterize COT maps. Our methods exploit the structure of the static and dynamic formulations of the COT problem. PCP-Map models conditional transport maps as the gradient of a partially input convex neural network (PICNN) and uses a novel numerical implementation to increase computational efficiency compared to state-of-the-art alternatives. COT-Flow models conditional transports via the flow of a regularized neural ODE; it is slower to train but offers faster sampling. We demonstrate their effectiveness and efficiency by comparing them with state-of-the-art approaches using benchmark datasets and Bayesian inverse problems.

我们提出了两种神经网络方法来近似求解静态和动态条件优化传输问题（COT），这两种方法能够对条件概率分布进行抽样和密度估计，在贝叶斯推断中是核心任务。我们的方法将目标条件分布表示为可处理的参考分布的转换，因此属于测度传输框架。这些方法利用了COT问题的静态和动态公式的结构，通过神经网络来参数化COT映射以提高可扩展性。我们通过使用基准数据集和贝叶斯反问题将其与最先进的方法进行比较，证明了它们的有效性和效率。

基于条件最优传输的高效神经网络方法和贝叶斯推断应用