Conservation Laws from Data Symmetry in Neural Networks

We explore whether intrinsic symmetries of the training data lead to conserved quantities during gradient-flow training of neural networks. Under the assumption that the loss function is analytic and non-polynomial, we prove that data symmetries generically do not induce any additional integrals of motion. For mean squared error (MSE) loss, on the other hand, there are situations in which data augmentation yields extra conserved quantities. We build a framework, utilizing tensorizable networks to describe this phenomenon. Tensorizable networks are a family of architectures whose dependence on parameters and inputs can be separated using an intermediate representation. They include linear and Figure 1: A display of how data symmetry can give polynomial networks, as well as Lightning At-rise to conservation laws. The top row shows the tention.