# Noether's razor: Learning Conserved Quantities

arXiv: https://arxiv.org/abs/2410.08087 · HTML: https://mlsanity.com/paper/2410.08087/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 41 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Builds on (most cited first, 27 of 27)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [AutoAugment: Learning Augmentation Policies from Data](https://mlsanity.com/paper/1805.09501/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [A Kronecker-factored approximate Fisher matrix for convolution layers](https://mlsanity.com/paper/1602.01407/) (2016)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Scalable Marginal Likelihood Estimation for Model Selection in Deep Learning](https://mlsanity.com/paper/2104.04975/) (2021)
- [Structured and Efficient Variational Deep Learning with Matrix Gaussian Posteriors](https://mlsanity.com/paper/1603.04733/) (2016)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [Learning Invariances in Neural Networks](https://mlsanity.com/paper/2010.11882/) (2020)
- [Learning Invariances using the Marginal Likelihood](https://mlsanity.com/paper/1808.05563/) (2018)
- [Invariance Learning in Deep Neural Networks with Differentiable Laplace Approximations](https://mlsanity.com/paper/2202.10638/) (2022)
- [Relaxing Equivariance Constraints with Non-stationary Continuous Filters](https://mlsanity.com/paper/2204.07178/) (2022)
- [Equivariant Hamiltonian Flows](https://mlsanity.com/paper/1909.13739/) (2019)
- [Data augmentation in Bayesian neural networks and the cold posterior effect](https://mlsanity.com/paper/2106.05586/) (2021)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Noether Networks: Meta-Learning Useful Conserved Quantities](https://mlsanity.com/paper/2112.03321/) (2021)
- [Learning Layer-wise Equivariances Automatically using Gradients](https://mlsanity.com/paper/2310.06131/) (2023)
- [Last Layer Marginal Likelihood for Invariance Learning](https://mlsanity.com/paper/2106.07512/) (2021)
- [Learning Invariant Weights in Neural Networks](https://mlsanity.com/paper/2202.12439/) (2022)
- [Learning Symmetries of Classical Integrable Systems](https://mlsanity.com/paper/1906.04645/) (2019)
- [Equivariance-aware Architectural Optimization of Neural Networks](https://mlsanity.com/paper/2210.05484/) (2022)
- [Learning Energy Conserving Dynamics Efficiently with Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2303.01925/) (2023)

## Built on by (most cited first, 4 of 4)

- [AtlasD: Automatic Local Symmetry Discovery](https://mlsanity.com/paper/2504.10777/) (2025)
- [SOAP-Bubbles: Structured Weight Uncertainty for Neural Networks](https://mlsanity.com/paper/2606.23357/) (2026)
- [Minimum-Action Learning: Energy-Constrained Symbolic Model Selection for Physical Law Identification from Noisy Data](https://mlsanity.com/paper/2603.16951/) (2026)
- [EqOD: Symmetry-Informed Stability Selection for PDE Identification](https://mlsanity.com/paper/2605.11524/) (2026)

## Abstract (arXiv)

Symmetries have proven useful in machine learning models, improving generalisation and overall performance. At the same time, recent advancements in learning dynamical systems rely on modelling the underlying Hamiltonian to guarantee the conservation of energy. These approaches can be connected via a seminal result in mathematical physics: Noether's theorem, which states that symmetries in a dynamical system correspond to conserved quantities. This work uses Noether's theorem to parameterise symmetries as learnable conserved quantities. We then allow conserved quantities and associated symmetries to be learned directly from train data through approximate Bayesian model selection, jointly with the regular training procedure. As training objective, we derive a variational lower bound to the marginal likelihood. The objective automatically embodies an Occam's Razor effect that avoids collapse of conservation laws to the trivial constant, without the need to manually add and tune additional regularisers. We demonstrate a proof-of-principle on n-harmonic oscillators and n-body systems. We find that our method correctly identifies the correct conserved quantities and U(n) and SE(n) symmetry groups, improving overall performance and predictive accuracy on test data.