Symmetry-regularized neural ordinary differential equations
This work addresses the problem of enhancing model interpretability and generalizability for researchers modeling continuous-time dynamical systems, representing an incremental advancement by incorporating symmetry principles into existing Neural ODE frameworks.
The paper tackled the challenge of improving stability and physical interpretability in Neural ODEs by introducing conservation relations based on Lie symmetries into the loss function as regularization terms, demonstrating this method in a case study of charged particles in a sinusoidal electric field.
Neural ordinary differential equations (Neural ODEs) is a class of machine learning models that approximate the time derivative of hidden states using a neural network. They are powerful tools for modeling continuous-time dynamical systems, enabling the analysis and prediction of complex temporal behaviors. However, how to improve the model's stability and physical interpretability remains a challenge. This paper introduces new conservation relations in Neural ODEs using Lie symmetries in both the hidden state dynamics and the back propagation dynamics. These conservation laws are then incorporated into the loss function as additional regularization terms, potentially enhancing the physical interpretability and generalizability of the model. To illustrate this method, the paper derives Lie symmetries and conservation laws in a simple Neural ODE designed to monitor charged particles in a sinusoidal electric field. New loss functions are constructed from these conservation relations, demonstrating the applicability symmetry-regularized Neural ODE in typical modeling tasks, such as data-driven discovery of dynamical systems.