The neural networks (NN) remain as black boxes, albeit their quite successful stories everywhere. It is mainly because they provide only the complex structure of the underlying network with a huge validation data set whenever their serendipities reveal themselves. In this paper, we propose the statistical NN learning model related to the concept of universal Turing computer for regression predictive model. Based on this model, we define ’statistically successful NN (SSNN) learning.’ This is mainly done by calculating the well-known Cramer–Rao lower bound for the averaged square error (ASE) of NN learning. Using such formal definition, we propose an ASE-based NN learning (ANL) algorithm. The ANL algorithm not only implements the Cramer–Rao lower bound successfully but also presents an effective way to figure out a complicated geometry of ASE over hyper-parameter space for NN. This enables the ANL to be free of huge validation data set. Simple numerical simulation and real data analysis are done to evaluate performance of the ANL and present how to implement it.

神经网络（NN）仍然是黑匣子，尽管它们到处都是相当成功的故事。这主要是因为它们的偶然性显示出来时，它们只为基础网络的复杂结构提供了巨大的验证数据集。在本文中，我们提出了与通用图灵计算机概念相关的统计神经网络学习模型，用于回归预测模型。基于此模型，我们定义了“统计上成功的NN（SSNN）学习”。这主要是通过为NN学习的平均平方误差（ASE）计算众所周知的Cramer-Rao下界来完成的。使用这种形式的定义，我们提出了一种基于ASE的NN学习（ANL）算法。ANL算法不仅成功实现了Cramer-Rao下界，而且提出了一种有效的方法来解决NN的超参数空间上ASE的复杂几何形状。这使ANL摆脱了庞大的验证数据集。完成了简单的数值模拟和真实数据分析，以评估ANL的性能并介绍如何实施。