标题： 前馈神经网络模型中的不确定性传播 显示英文标题

标题： Uncertainty propagation in feed-forward neural network models

摘要： 我们为具有泄漏ReLU激活函数的前馈神经网络架构开发了新的不确定性传播方法，这些方法在输入向量受到随机扰动的情况下适用。 特别是，我们推导了神经网络输出的概率密度函数（PDF）及其统计矩的解析表达式，作为输入不确定性以及网络参数（即权重和偏置）的函数。 一个关键发现是，对泄漏ReLU激活函数进行适当的线性化即使在输入向量有较大扰动时也能得到准确的统计结果。 这可以归因于信息在网络中传播的方式。 我们还提出了新的解析可处理的高斯copula代理模型，以近似神经网络输出的完整联合PDF。 为了验证我们的理论结果，我们在一个表示两个多项式函数空间之间非线性积分微分算子的多层神经网络上进行了蒙特卡洛模拟和详细的误差分析。 我们的研究结果表明，理论预测与蒙特卡洛模拟之间有很好的一致性。 显示英文摘要

摘要： We develop new uncertainty propagation methods for feed-forward neural network architectures with leaky ReLU activation functions subject to random perturbations in the input vectors. In particular, we derive analytical expressions for the probability density function (PDF) of the neural network output and its statistical moments as a function of the input uncertainty and the parameters of the network, i.e., weights and biases. A key finding is that an appropriate linearization of the leaky ReLU activation function yields accurate statistical results even for large perturbations in the input vectors. This can be attributed to the way information propagates through the network. We also propose new analytically tractable Gaussian copula surrogate models to approximate the full joint PDF of the neural network output. To validate our theoretical results, we conduct Monte Carlo simulations and a thorough error analysis on a multi-layer neural network representing a nonlinear integro-differential operator between two polynomial function spaces. Our findings demonstrate excellent agreement between the theoretical predictions and Monte Carlo simulations.