mlsanityData to 14 Oct 2026

Exact conservation laws for neural network integrators of dynamical systems

arXiv 2209.11661 ↗First posted 23 Sep 2022math.DS cs.LG gr-qccited by 15

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Abstract (arXiv)

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.

Key facts. Cited by 15 papers in the map (data through 14 Oct 2026). Builds on 21 of its 50 references that are in the map. First posted on arXiv 23 Sep 2022.

Builds on

21 of 50 references.

References · 21

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9 more
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 1; 2023: 3; 2024: 4; 2025: 5; 2026: 2.

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  1. 2022Neural Conservation Laws: A Divergence-Free Perspectivecited by 70
  2. 2023Learning Physical Models that Can Respect Conservation Lawscited by 59
  3. 2023Harnessing the Power of Neural Operators with Automatically Encoded Conservation Lawscited by 24
  4. 2023Exactly conservative physics-informed neural networks and deep operator networks for dynamical systemscited by 13
  5. 2025End-to-End Probabilistic Framework for Learning with Hard Constraintscited by 9
  6. 2025Spatiotemporal Forecasting as Planning: A Model-Based Reinforcement Learning Approach with Generative World Modelscited by 4
  7. 2024Physics-embedded Fourier Neural Network for Partial Differential Equationscited by 1
  8. 2025Adaptive Correction for Ensuring Conservation Laws in Neural Operatorscited by 1
  9. 2025Physically consistent model learning for reaction-diffusion systemscited by 1
  10. 2025The Mechanics of CNN Filtering with Rectificationcited by 1
  11. 2024Scalable physical source-to-field inference with hypernetworkscited by 0
  12. 2024A nudge to the truth: atom conservation as a hard constraint in models of atmospheric composition using an uncertainty-weighted correctioncited by 0
3 more
  1. 2024Physics-Informed Graph-Mesh Networks for PDEs: A hybrid approach for complex problemscited by 0
  2. 2026Preserving Continuous Symmetry in Discrete Spaces: Geometric-Aware Quantization for SO(3)-Equivariant GNNscited by 0
  3. 2026A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Lawscited by 0