自回归随机神经网络模型中的不确定性传播
Uncertainty propagation in auto-regressive random neural network models
- UC Santa Cruz（加州大学圣克鲁兹分校）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究针对输入与参数均随机的神经网络，推导了其不确定性传播的解析方法，扩展至自回归动力系统，经数值实验验证了其在高维系统中传播不确定性的准确性与适用性。
中文摘要 AI 辅助
我们针对随机神经网络模型中的不确定性传播问题，开发了解析方法和基于粒子的方法，其中输入和网络参数均允许为随机量。基于Leaky ReLU激活函数的分段线性结构，我们推导了网络输出关于其输入和参数扰动的局部近似，该近似对于保持网络激活模式的扰动是精确的，且使我们能够计算网络输出的概率密度函数和特征函数的解析表达式，以及其均值和协方差的闭式近似。我们将该不确定性传播框架扩展到自治动力系统，其一步演化映射由随机神经网络表示。该映射的重复应用定义了一个自回归模型，为此我们推导了随时间传播状态和网络参数不确定性的递归方程，这些方程明确考虑了网络连续迭代下产生的状态-参数交叉协方差。在Lorenz-63系统和Kuramoto-Sivashinsky方程上的数值实验表明，该框架在可预测性范围内能实现准确的不确定性传播，且适用于高维动力系统。
英文摘要
We develop analytical and particle-based methods for uncertainty propagation in random neural network models, where both the inputs and network parameters are allowed to be random. Building on the piecewise-linear structure of the Leaky ReLU activation function, we derive a local approximation of the neural network output with respect to perturbations in both its inputs and parameters. This approximation is exact for perturbations that preserve the network activation pattern, and it allows us to compute analytical expressions for the probability density function and characteristic function of the network output, together with closed-form approximations for its mean and covariance. We extend this uncertainty propagation framework to autonomous dynamical systems whose one-step evolution map is represented by a random neural network. Repeated application of this map defines an autoregressive model, for which we derive recursive equations to propagate uncertainty in both the state and network parameters over time. These equations explicitly account for the state-parameter cross-covariance that develops under successive iterations of the network. Numerical experiments on the Lorenz-63 system and the Kuramoto-Sivashinsky equation demonstrate accurate uncertainty propagation through the predictability horizon and the applicability of the proposed framework to high-dimensional dynamical systems.
