论文标题
高阶准神经网络的高阶准蒙特卡洛培训
Higher-order Quasi-Monte Carlo Training of Deep Neural Networks
论文作者
论文摘要
我们提出了一种新颖的算法方法和一个错误分析，利用了Quasi-Monte Carlo点来训练工程设计中数据对观察图（DTO）图的深神经网络（DNN）替代。我们的分析揭示了具有全体形态激活功能（如Tanh）的深层和浅神经网络中训练点一致的，确定性的选择。事实证明，这些新颖的训练点有助于促进基本概括误差的高阶衰减（根据训练样本的数量），只要隐藏层中的DNN权重满足了某些可总结性条件，却没有输入数据空间中的一致性误差界限。我们提出了来自椭圆形和抛物线PDE的DTO图的数值实验，并具有不确定的投入，这些投入确认了理论分析。
We present a novel algorithmic approach and an error analysis leveraging Quasi-Monte Carlo points for training deep neural network (DNN) surrogates of Data-to-Observable (DtO) maps in engineering design. Our analysis reveals higher-order consistent, deterministic choices of training points in the input data space for deep and shallow Neural Networks with holomorphic activation functions such as tanh. These novel training points are proved to facilitate higher-order decay (in terms of the number of training samples) of the underlying generalization error, with consistency error bounds that are free from the curse of dimensionality in the input data space, provided that DNN weights in hidden layers satisfy certain summability conditions. We present numerical experiments for DtO maps from elliptic and parabolic PDEs with uncertain inputs that confirm the theoretical analysis.
