# Exact conservation laws for neural network integrators of dynamical systems

arXiv: https://arxiv.org/abs/2209.11661 · HTML: https://mlsanity.com/paper/2209.11661/

## 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 (most cited first, 21 of 21)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks](https://mlsanity.com/paper/1506.01497/) (2015)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [Machine learning structure preserving brackets for forecasting irreversible processes](https://mlsanity.com/paper/2106.12619/) (2021)
- [GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems](https://mlsanity.com/paper/2109.00092/) (2021)
- [Physical Symmetries Embedded in Neural Networks](https://mlsanity.com/paper/1904.08991/) (2019)
- [Solving Newton's Equations of Motion with Large Timesteps using Recurrent Neural Networks based Operators](https://mlsanity.com/paper/2004.06493/) (2020)
- [A simple equivariant machine learning method for dynamics based on scalars](https://mlsanity.com/paper/2110.03761/) (2021)

## Built on by (most cited first, 15 of 15)

- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Learning Physical Models that Can Respect Conservation Laws](https://mlsanity.com/paper/2302.11002/) (2023)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Exactly conservative physics-informed neural networks and deep operator networks for dynamical systems](https://mlsanity.com/paper/2311.14131/) (2023)
- [End-to-End Probabilistic Framework for Learning with Hard Constraints](https://mlsanity.com/paper/2506.07003/) (2025)
- [Spatiotemporal Forecasting as Planning: A Model-Based Reinforcement Learning Approach with Generative World Models](https://mlsanity.com/paper/2510.04020/) (2025)
- [Physics-embedded Fourier Neural Network for Partial Differential Equations](https://mlsanity.com/paper/2407.11158/) (2024)
- [Adaptive Correction for Ensuring Conservation Laws in Neural Operators](https://mlsanity.com/paper/2505.24579/) (2025)
- [Physically consistent model learning for reaction-diffusion systems](https://mlsanity.com/paper/2512.14240/) (2025)
- [The Mechanics of CNN Filtering with Rectification](https://mlsanity.com/paper/2512.24338/) (2025)
- [Scalable physical source-to-field inference with hypernetworks](https://mlsanity.com/paper/2405.05981/) (2024)
- [A nudge to the truth: atom conservation as a hard constraint in models of atmospheric composition using an uncertainty-weighted correction](https://mlsanity.com/paper/2408.16109/) (2024)
- [Physics-Informed Graph-Mesh Networks for PDEs: A hybrid approach for complex problems](https://mlsanity.com/paper/2410.02819/) (2024)
- [Preserving Continuous Symmetry in Discrete Spaces: Geometric-Aware Quantization for SO(3)-Equivariant GNNs](https://mlsanity.com/paper/2603.05343/) (2026)
- [A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Laws](https://mlsanity.com/paper/2604.15617/) (2026)

## 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.