Deep neural networks have shown many fruitful applications in this decade. A network can get the generalized function through training with a finite dataset. The degree of generalization is a realization of the proximity scale in the data space. Specifically, the scale is not clear if the dataset is complicated. Here we consider a network for the distribution estimation of the dataset. We show the estimation is unstable and the instability depends on the data density and training duration. We derive the kernel-balanced equation, which gives a short phenomenological description of the solution. The equation tells us the reason for the instability and the mechanism of the scale. The network outputs a local average of the dataset as a prediction and the scale of averaging is determined along the equation. The scale gradually decreases along training and finally results in instability in our case.

在本文中，我们研究了深度神经网络对数据集分布进行估计的问题，并发现估计结果的不稳定性取决于数据密度和训练时长，通过推导出的核平衡方程，我们得到了解释不稳定性和尺度机制的现象学描述。该网络以数据集的局部平均作为预测，并根据方程确定平均的尺度，该尺度在训练过程中逐渐减小，导致了我们所研究的问题的不稳定性。

深度神经网络的核均衡方程