In this paper, we propose a local squared Wasserstein-2 (W_2) method to solve
the inverse problem of reconstructing models with uncertain latent variables or
parameters. A key advantage of our approach is that it does not require prior
information on the distribution of the latent variables or parameters in the
underlying models. Instead, our method can efficiently reconstruct the
distributions of the output associated with different inputs based on empirical
distributions of observation data. We demonstrate the effectiveness of our
proposed method across several uncertainty quantification (UQ) tasks, including
linear regression with coefficient uncertainty, training neural networks with
weight uncertainty, and reconstructing ordinary differential equations (ODEs)
with a latent random variable.

本文提出了一种局部平方 Wasserstein-2 方法，用于解决具有不确定潜变量或参数的模型的逆问题。我们的方法的一个关键优势是不需要关于潜变量或参数分布的先验信息，在基于经验分布的观测数据的基础上可有效重建与不同输入相关的输出的分布。我们通过几个不确定性量化任务展示了我们提出方法的有效性，包括具有系数不确定性的线性回归、具有权重不确定性的神经网络训练，以及含有潜随机变量的常微分方程重建。