Neural Ordinary Differential Equations (Neural ODEs) is a class of deep neural network models that interpret the hidden state dynamics of neural networks as an ordinary differential equation, thereby capable of capturing system dynamics in a continuous time framework. In this work, I integrate symmetry regularization into Neural ODEs. In particular, I use continuous Lie symmetry of ODEs and PDEs associated with the model to derive conservation laws and add them to the loss function, making it physics-informed. This incorporation of inherent structural properties into the loss function could significantly improve robustness and stability of the model during training. To illustrate this method, I employ a toy model that utilizes a cosine rate of change in the hidden state, showcasing the process of identifying Lie symmetries, deriving conservation laws, and constructing a new loss function.

翻译：神经常微分方程（Neural ODEs）是一类深度神经网络模型，其将神经网络的隐状态动力学解释为常微分方程，从而能在连续时间框架中捕捉系统动力学。本文工作中，我将对称正则化集成到神经常微分方程中。具体而言，我利用与模型相关的常微分方程和偏微分方程的连续李对称性推导守恒律，并将其加入损失函数中，使其具有物理信息。将这种内在结构属性纳入损失函数可显著提升模型训练过程中的鲁棒性和稳定性。为阐释该方法，我采用一个隐状态变化率为余弦函数的玩具模型，演示了识别李对称性、推导守恒律以及构建新损失函数的完整流程。