In the previous article, we introduced a neural network framework based on symmetric differential equations. This novel framework exhibits complete symmetry, endowing it with perfect mathematical properties. While we have examined some of the system's mathematical characteristics, a detailed discussion of the network training methodology has not yet been presented. Drawing on the principles of the traditional backpropagation algorithm, this study proposes an alternative training approach that utilizes differential equation signal propagation instead of chain rule derivation. This approach not only preserves the effectiveness of training but also offers enhanced biological interpretability. The foundation of this methodology lies in the system's reversibility, which stems from its inherent symmetry,a key aspect of our research. However, this method alone is insufficient for effective neural network training. To address this, we further introduce a distributed Proportional-Integral-Derivative (PID) control approach, emphasizing its implementation within a closed system. By incorporating this method, we achieved both faster training speeds and improved accuracy. This approach not only offers novel insights into neural network training but also extends the scope of research into control methodologies. To validate its effectiveness, we apply this method to the MNIST dataset, demonstrating its practical utility.

翻译：在先前的研究中，我们介绍了一种基于对称微分方程的神经网络框架。这一新颖框架展现出完全的对称性，赋予了其完美的数学特性。尽管我们已经探讨了该系统的一些数学特征，但尚未对网络训练方法进行详细讨论。本研究借鉴传统反向传播算法的原理，提出了一种替代训练方法，该方法利用微分方程信号传播而非链式法则求导。该方法不仅保持了训练的有效性，还提供了更强的生物学可解释性。此方法的基础在于系统的可逆性，这源于其固有的对称性，也是我们研究的一个关键方面。然而，仅凭此方法不足以实现有效的神经网络训练。为此，我们进一步引入了分布式比例-积分-微分（PID）控制方法，并重点阐述了其在封闭系统中的实现。通过引入该方法，我们实现了更快的训练速度和更高的准确性。这一方法不仅为神经网络训练提供了新的视角，还拓展了控制方法的研究范畴。为验证其有效性，我们将该方法应用于MNIST数据集，证明了其实用价值。