mlsanityData to 14 Oct 2026

Noether's razor: Learning Conserved Quantities

arXiv 2410.08087 ↗First posted 10 Oct 2024cs.LG stat.MLcited by 4

Not on the map. Why?

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.

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

27 of 41 references.

References · 27

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2016Group Equivariant Convolutional Networkscited by 1,165
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  12. 2020Meta-Learning Symmetries by Reparameterizationcited by 76
15 more
  1. 2021Residual Pathway Priors for Soft Equivariance Constraintscited by 75
  2. 2020Learning Invariances in Neural Networkscited by 52
  3. 2018Learning Invariances using the Marginal Likelihoodcited by 51
  4. 2022Invariance Learning in Deep Neural Networks with Differentiable Laplace Approximationscited by 42
  5. 2022Relaxing Equivariance Constraints with Non-stationary Continuous Filterscited by 41
  6. 2019Equivariant Hamiltonian Flowscited by 39
  7. 2021Data augmentation in Bayesian neural networks and the cold posterior effectcited by 36
  8. 2021Symplectic Learning for Hamiltonian Neural Networkscited by 35
  9. 2021Noether Networks: Meta-Learning Useful Conserved Quantitiescited by 26
  10. 2023Learning Layer-wise Equivariances Automatically using Gradientscited by 23
  11. 2021Last Layer Marginal Likelihood for Invariance Learningcited by 21
  12. 2022Learning Invariant Weights in Neural Networkscited by 16
  13. 2019Learning Symmetries of Classical Integrable Systemscited by 15
  14. 2022Equivariance-aware Architectural Optimization of Neural Networkscited by 8
  15. 2023Learning Energy Conserving Dynamics Efficiently with Hamiltonian Gaussian Processescited by 6

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 3.

4 papers.

  1. 2025AtlasD: Automatic Local Symmetry Discoverycited by 5
  2. 2026SOAP-Bubbles: Structured Weight Uncertainty for Neural Networkscited by 3
  3. 2026Minimum-Action Learning: Energy-Constrained Symbolic Model Selection for Physical Law Identification from Noisy Datacited by 1
  4. 2026EqOD: Symmetry-Informed Stability Selection for PDE Identificationcited by 0