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Symmetry-regularized neural ordinary differential equations

Wenbo Hao

TL;DR

This work addresses stability and physical interpretability in Neural ODEs by introducing conservation laws derived from Lie symmetries of both the forward hidden-state dynamics and the backward adjoint dynamics. The authors build symmetry-regularized loss terms from these conservation laws and demonstrate the approach on a physics-inspired toy model of a charged particle in a sinusoidal electric field. The key contributions are (i) applying Lie's algorithm to obtain one-parameter symmetry groups for forward and backward Neural ODEs, (ii) deriving corresponding conservation laws, and (iii) formulating a symmetry-regularized loss that improves interpretability and potentially generalization. The method offers a principled way to encode physical invariants into data-driven models, with implications for data-driven discovery of dynamical systems and broader physics-informed learning.

Abstract

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.

Symmetry-regularized neural ordinary differential equations

TL;DR

This work addresses stability and physical interpretability in Neural ODEs by introducing conservation laws derived from Lie symmetries of both the forward hidden-state dynamics and the backward adjoint dynamics. The authors build symmetry-regularized loss terms from these conservation laws and demonstrate the approach on a physics-inspired toy model of a charged particle in a sinusoidal electric field. The key contributions are (i) applying Lie's algorithm to obtain one-parameter symmetry groups for forward and backward Neural ODEs, (ii) deriving corresponding conservation laws, and (iii) formulating a symmetry-regularized loss that improves interpretability and potentially generalization. The method offers a principled way to encode physical invariants into data-driven models, with implications for data-driven discovery of dynamical systems and broader physics-informed learning.

Abstract

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.
Paper Structure (19 sections, 57 equations, 1 figure)

This paper contains 19 sections, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Neural Ordinary Differential Equations (Neural ODEs)
  4. Regularization Techniques in Neural Networks
  5. Lie Symmetry and Conservation Laws in Machine Learning
  6. Symmetry-Preserving Models
  7. Methodology
  8. Forward and backward dynamics of Neural ODEs
  9. Symmetries of ODEs
  10. Conservation laws
  11. Symmetry-regularized loss functions
  12. Application
  13. Toy model overview
  14. Neural network architecture
  15. Lie symmetry in the forward dynamics and conservation laws
  16. ...and 4 more sections

Figures (1)

  • Figure 1: Model architecture: $\frac{d\hat{z}}{dt} = \cos\left(\theta_1 x + \theta_2\right)$