Efficient reconstruction of multidimensional random field models with heterogeneous data using stochastic neural networks

TL;DR

This paper presents a scalable Wasserstein-distance method for training stochastic neural networks to reconstruct multidimensional random field models, effectively handling heterogeneous noise and reducing the curse of dimensionality.

Contribution

It provides a theoretical generalization error bound for multidimensional random field reconstruction and improves existing Wasserstein-distance training approaches.

Findings

The approach effectively reconstructs multidimensional uncertainty models.

The convergence rate may not depend on dimensionality with heterogeneous noise.

Numerical experiments demonstrate robustness and success in uncertainty quantification tasks.

Abstract

In this paper, we analyze the scalability of a recent Wasserstein-distance approach for training stochastic neural networks (SNNs) to reconstruct multidimensional random field models. We prove a generalization error bound for reconstructing multidimensional random field models on training stochastic neural networks with a limited number of training data. Our results indicate that when noise is heterogeneous across dimensions, the convergence rate of the generalization error may not depend explicitly on the model's dimensionality, partially alleviating the "curse of dimensionality" for learning multidimensional random field models from a finite number of data points. Additionally, we improve the previous Wasserstein-distance SNN training approach and showcase the robustness of the SNN. Through numerical experiments on different multidimensional uncertainty quantification tasks, we show that our Wasserstein-distance approach can successfully train stochastic neural networks to learn multidimensional uncertainty models.