Exact conservation laws for neural network integrators of dynamical systems

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The solution of time dependent differential equations with neural networks has attracted a lot of attention recently. The central idea is to learn the laws that govern the evolution of the solution from data, which might be polluted with random noise. However, in contrast to other machine learning applications, usually a lot is known about the system at hand. For example, for many dynamical systems physical quantities such as energy or (angular) momentum are exactly conserved. Hence, the neural network has to learn these conservation laws from data and they will only be satisfied approximately due to finite training time and random noise. In this paper we present an alternative approach which uses Noether's Theorem to inherently incorporate conservation laws into the architecture of the neural network. We demonstrate that this leads to better predictions for three model systems: the motion of a non-relativistic particle in a three-dimensional Newtonian gravitational potential, the motion of a massive relativistic particle in the Schwarzschild metric and a system of two interacting particles in four dimensions.
Original languageEnglish
Article number112234
Number of pages24
JournalJournal of Computational Physics
Early online date19 May 2023
Publication statusPublished - 1 Sept 2023

Bibliographical note

Data availability
A link to the code that was used to generate the data is included in the paper.


  • Dynamical system
  • Lagrangian mechanics
  • Machine learning
  • Noether's theorem

ASJC Scopus subject areas

  • Computational Mathematics
  • General Physics and Astronomy
  • Applied Mathematics
  • Numerical Analysis
  • Computer Science Applications
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)


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