用随机神经网络高效重建异质多维随机场，缓解维度灾难。
Efficient reconstruction of multidimensional random field models with heterogeneous data using stochastic neural networks
- 基于Wasserstein距离训练随机神经网络，提升多维场重建效率。
- 噪声异质时，泛化误差收敛率与维度无关，突破维度瓶颈。
- 适用于有限数据下的多维不确定性量化，鲁棒性强。
本文分析了近期基于Wasserstein距离的随机神经网络（SNN）在重建多维随机场模型中的可扩展性。我们证明了在有限训练数据条件下，使用SNN重建多维随机场模型的泛化误差上界。结果表明，当各维度噪声异质时，泛化误差的收敛速率不显式依赖于模型维度，部分缓解了从有限数据点学习多维随机场时的“维度灾难”。此外，我们改进了先前的Wasserstein距离SNN训练方法，并展示了SNN的鲁棒性。通过在多个多维不确定性量化任务上的数值实验，验证了该方法能成功训练SNN以学习多维不确定性模型。
原文摘要 · Abstract (English)
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.
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