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Noether's Razor: Learning Conserved Quantities
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Learns Hamiltonian dynamics jointly with quadratic conserved quantities (symmetries via Noether's theorem) by optimising a variational lower bound on the marginal likelihood, so symmetries are selected by Occam's razor without regularisers.
Problem
Learning Hamiltonian dynamical systems while automatically discovering the symmetries they obey, without validation data, hand-tuned regularisers, or fixed symmetry priors.
Approach
Parameterise symmetries as learnable quadratic conserved quantities via Noether's theorem, symmetrise the Hamiltonian by Monte Carlo averaging over their flows, and optimise a variational ELBO (matrix normal posterior, closed-form prior variance) as an approximate marginal likelihood so that the evidence selects the symmetry.
Key results
- On the 3-body system, learned symmetry reaches test MSE 0.0006 vs 0.0016 for vanilla HNN, matching the fixed SE(2) oracle (0.0006), with train -ELBO/N -12.81 vs -9.52 for vanilla.
- On 3-harmonic oscillators, learned symmetry reduces test MSE to 0.00000994 from 0.00002141 for vanilla and reaches -ELBO/N -9.63 vs -6.77, close to the U(3) oracle (-9.94).
- The method recovers the correct symmetry generators: for the 3-body system 7 singular values exceed 0.05 with parallelness above 0.95, and U(n) is recovered for n-harmonic oscillators.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Learn symmetries from data using Noether's razor | HNN · Simple harmonic oscillator | Test MSE: 0.005 -> 0.002 | Helped |
| Learn symmetries from data using Noether's razor | HNN · 3 harmonic oscillators | Test MSE: 0.00002141 -> 0.00000994 | Helped |
| Learn symmetries from data using Noether's razor | HNN · 2D 3-body system | Test MSE: 0.0016 -> 0.0006 | Helped |
| Learn symmetries from data using Noether's razor | HNN · 2D 3-body system, moved test data | Test MSE: 0.0035 -> 0.0004 | Helped |
| Learn symmetries from data using Noether's razor | HNN · 2D 3-body system, wider test data | Test MSE: 0.0016 -> 0.0006 | Helped |
Surprising findings
- Maximum-likelihood-style train fit alone does not favour the symmetric solution; the marginal likelihood (ELBO) does, which is the core Occam's razor argument.
- The authors advise against down-weighting the KL term by a beta scalar, arguing it breaks the lower bound interpretation.
Limitations
- Conserved quantities are restricted to quadratic functions, which limits expressiveness (e.g. the Laplace-Runge-Lenz vector is not captured).
- Symmetrisation uses approximate Monte Carlo sampling and a relaxed (not strictly invariant) symmetry notion, and a uniform Haar measure may not exist for non-compact groups.
- Experiments are proof-of-principle on small synthetic systems with MLPs of limited size.
Experimental setup
Datasets: simulated simple harmonic oscillator trajectories simulated n-harmonic oscillator trajectories simulated 2d 3-body gravitational trajectories
Benchmarks: simple harmonic oscillator n-harmonic oscillators 2d 3-body system
Metrics: test MSE train -ELBO/N singular values and parallelness of learned generators
Baselines: vanilla HNN HNN with fixed SO(2) symmetry (oracle) HNN with fixed U(n) symmetry (oracle) HNN with fixed SE(2) symmetry (oracle)
Builds on: Hamiltonian neural networks Noether's theorem variational inference linearised Laplace symmetry learning
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