A local squared Wasserstein-2 method for efficient reconstruction of models with uncertainty

TL;DR

This paper introduces a local squared Wasserstein-2 method for reconstructing models with uncertain parameters, efficiently estimating output distributions without prior distribution knowledge, demonstrated across various uncertainty quantification tasks.

Contribution

The proposed method uniquely reconstructs output distributions from empirical data without requiring prior distribution assumptions, improving efficiency in inverse problems with uncertainty.

Findings

Effective in linear regression with coefficient uncertainty

Successful in training neural networks with weight uncertainty

Reconstructs ODEs with latent random variables

Abstract

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.