Symmetry-regularized neural ordinary differential equations

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.