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.

翻译：本文提出了一种局部平方Wasserstein-2（W_2）方法，用于解决具有不确定隐变量或参数的模型重构反问题。该方法的关键优势在于无需预先获知底层模型中隐变量或参数的概率分布信息。相反，基于观测数据的经验分布，我们的方法能够高效地重构出不同输入所对应输出的概率分布。我们在多个不确定性量化任务中验证了所提方法的有效性，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，以及具有隐随机变量的常微分方程重构。