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Data-driven Discovery of Lie Symmetries and First Integrals with Latent ODEs

Matt L. Wiemman, Peter Melchior

arXiv:2610.35967v1cs.LGstat.ML

TL;DR

The paper asks how to discover a system’s symmetries from noisy observations when they are not known in advance. LatentLieODE learns a decorrelated latent chart, fits linear generators against a differentiable bracket condition, and learns first integrals. Across six systems it recovers expected symmetries and closely matching invariants, while the discussion identifies limits on representable symmetries and periodic-orbit assumptions.

  • Problem

    Given noisy observations, it remains an open inverse problem to determine which symmetries, if any, a system has.

    Source: §1 · p. 2

  • Method

    LatentLieODE learns a decorrelated latent chart and fits linear generators against a differentiable Lie-bracket condition, with first-integral networks learned alongside the dynamics.

    Source: §3 · pp. 4–5

  • Results

    Across six systems, the method recovers expected symmetries and invariants that match ground-truth conserved quantities almost perfectly, including exactly three symmetries for sphere geodesics.

    Source: §4 · pp. 6, 8

  • Takeaways & Limitations

    The approach discovers linear Lie symmetries, algebra structure, and calibrated first integrals from state timeseries and video, offering a pathway for neural surrogates of dynamical systems.

    Source: §5 · p. 9

    A symmetry is found only if a chart at the chosen latent width can linearise it; the circle and speed priors presume periodic orbits, leaving chaotic regimes out of scope.

    Source: §5 · p. 10

Abstract

from arXiv · show

We present LatentLieODE, a novel neural ODE method to discover linear Lie symmetries and first integrals of the learned dynamics from timeseries observations such as video inputs. It starts with an autoencoder, which learns a low dimensional representation of the observed system and then maps trajectories to decorrelated latent coordinates $z \in \mathbb{R}^d$. A bank of linear infinitesimal generators $\mathbf{A}_k \in \mathbb{R}^{d \times d}$ is fitted jointly against a differentiable Lie-bracket residual $\mathbf{J}_f(z)\mathbf{A}z - \mathbf{A} f(z)$, while an MLP optimises the latent ODE $f(z) = \mathrm{d}z/\mathrm{d}t$ to reconstruct the observation-space timeseries. After suitable generators and $f$ are found, we fit a third network $H_\phi$ to generate the first integral of the system. Across six benchmark systems (harmonic oscillator, simple and twin pendulums, Lotka-Volterra, pixel pendulum, sphere geodesics) we find each system's relevant symmetries, and the discovered invariants match ground-truth conserved quantities at $|\rho| \geq 0.978$. On the sphere, the pipeline recovers a non-abelian $\mathfrak{so}(3)$, certified by the negative-definite Killing form of its recovered structure constants. This work presents significant advances in data-driven discovery of symmetries and provides a new pathway for neural surrogates of dynamical systems.

1. Introduction

The paper addresses the inverse problem of discovering which symmetries a system has from noisy observations, rather than building in a known symmetry or analytic structure. It proposes a latent-ODE pipeline that learns linear generators and first integrals from trajectories.

  • Given noisy observations, the inverse problem is to determine which symmetries, if any, a system has.Source: §1 · p. 2
  • LatentLieODE learns a reduced, decorrelated coordinate chart and fits an over-complete bank of linear generators to a differentiable Lie-bracket residual.Source: §1 · p. 2
  • The method makes algebra dimension an output, discovers first integrals with reparameterisation-invariant training and physical-unit calibration, and is validated on five classical trajectories and video observations.Source: §1 · p. 2

2. Related work

Prior approaches either use conservation laws as an inductive bias or learn symmetries separately from dynamics. LatentLieODE instead tests infinitesimal symmetry directly on the learned vector field and discovers algebra dimension and first integrals alongside generators.

  • Noether networks meta-learn conserved quantities to improve forecasting, while Hamiltonian and Lagrangian neural networks impose variational structure to obtain dynamics and energy conservation.Autoencoder variants extend the latter approaches to pixel observations.Source: §2 · pp. 2–3
  • LaLiGAN learns latent linear group actions adversarially, but fixes group dimension in advance and treats dynamics and downstream structure as separate steps.Source: §2 · p. 2
  • Unlike distribution-level adversarial symmetry discovery, LatentLieODE evaluates the infinitesimal Lie bracket pointwise, makes algebra dimension an output, and recovers calibrated first integrals alongside generators.The paper describes first-integral recovery as a connection to conservation laws suggested as future work by Yang et al. (2024).Source: §2 · p. 3

3. Method

The method learns a latent chart in which selected dynamical symmetries can be represented by linear generators, then jointly fits the dynamics and generator bank using differentiable constraints. It also co-trains first-integral networks and calibrates the selected invariant to physical units.

Latent dynamics model; Rotational dynamics

  • For video, reconstruction uses temporal-variance-weighted pixel error; circular-radius and uniform-speed penalties encourage periodic trajectories to acquire action-angle coordinates.Source: §3 · pp. 3–4

Coordinate construction

  • Batchwise coordinate construction removes cross-correlations between latent dimensions while preserving their scales, helping make dynamical symmetries linearly representable.Source: §3 · pp. 3–4

Setting and assumed form

  • The search assumes one-parameter transformations commute with the learned flow and restricts generators to be linear in a learned chart, excluding symmetries without a linear representation at the chosen latent width.Source: §3 · p. 4

Symmetry discovery; Independent generators

  • An over-complete generator bank is fitted to the bracket residual; action decorrelation and a norm-spectrum gap identify surviving generators and algebra dimension.Pairwise brackets are projected onto the surviving span to check closure; stop-gradient orthogonalisation helps prevent candidate generators from collapsing onto the same symmetry.Source: §3 · pp. 4–5

Joint objective

  • A norm budget avoids the all-zero generator solution while allowing redundant candidates to collapse, and the symmetry objective is jointly optimised with reconstruction and latent consistency.Symmetry weights ramp in below fit weights because equal weighting can degrade reconstruction and make the discovered symmetry one of model error.Source: §3 · pp. 5–6

Discovering conservation laws

  • First-integral networks are trained to remain constant along the flow, while compatible generators preserve their level sets; decorrelated heads, refitting, and monotone calibration support multiple invariants and physical-unit outputs.Source: §3 · p. 6

4. Results

Across six systems, LatentLieODE recovers the expected symmetries and closely matching invariants; results also show latent phase portraits, held-out invariant tests, pixel-pendulum reconstruction, and recovery of a non-abelian sphere algebra.

Circular dynamics

  • Latent trajectories are almost entirely circular, including for nonlinear twin pendulums and pixel-only pendulum observations.The phase portraits include the harmonic oscillator, Lotka–Volterra, twin pendulum, and pixel pendulum; dimensions above two are PCA-projected for display.Source: §4 · p. 6, Figure 2 · p. 7

Symmetry discovery

  • Across all six systems, LatentLieODE recovers the expected symmetries and invariants that match ground-truth conserved quantities almost perfectly.The reported invariant comparison uses absolute Spearman rank correlation; latent dimension discovery reaches the lowest true dimensionality where enabled.Source: §4 · pp. 6–7

Transferring the discovered invariant

  • A calibrated invariant tracks true energy on unseen pendulum trajectories with |ρ| = 0.998 and remains conserved for over ten times the training period.The same invariant shows a consistent drop on damped-pendulum trajectories, providing a conservation test on unseen data.Source: §4 · p. 7, Figure 3 · p. 8

Pixels to physics

  • From 32 × 32 pixel frames without state supervision, the model reduces the latent space from 8 dimensions to 2 and finds one symmetry.The discovered invariant tracks pendulum energy at |ρ| = 0.995; the decoded rollout agrees strongly with true dynamics far into extrapolation.Source: §4 · pp. 7–8, Figure 4 · p. 8

Non-abelian symmetry groups

  • For sphere geodesics, the method recovers exactly three symmetries, and the negative-definite Killing form identifies the recovered algebra as so(3).The Killing-form eigenvalues are (−6.25, −5.98, −5.66); at least 97.8% of each commutator lies in the surviving generator span.Source: §4 · pp. 8–9

Integration-free dynamics

  • On held-out sphere orbits, pure group transport closely agrees with both ground truth and trajectories obtained by integrating the learned ODE.The transport applies the discovered generator through the exponential map without ODE integration.Source: §4 · p. 9, Figure 5 · p. 9

5. Discussion

The method discovers symmetries, their algebra, and calibrated first integrals from state time series and video frames. Its scope depends on whether a symmetry can be linearised in a learned chart at the chosen latent width.

5. Discussion

  • The method discovers linear Lie symmetries, algebra dimension, structure constants, and calibrated first integrals from state time series and video frames.Latent-trajectory rollouts also agree strongly between direct ODE integration and sweeping the exponential-map action parameter.Source: §5 · p. 9
  • A symmetry is found only if some chart at the chosen latent width linearises it.The circle and speed priors presume (quasi-)periodic orbits, putting chaotic regimes out of scope; results are single runs checked by scripted acceptance gates, not seed ensembles.Source: §5 · p. 10

Appendix A. Cost and scaling

The symmetry machinery scales with latent width rather than observation dimension, and measured training time is mostly flat across input dimensions 2–6.

Appendix A. Cost and scaling

  • Training step time is mostly flat across D = 2 →6, while the pixel-pendulum at D = 2048 costs roughly 4 times more than the other systems.The symmetry machinery scales with latent width d, not observation dimension D; Figure 6 reports single-CPU training step time against encoder input dimension.Source: Appendix A · p. 12, Figure 6 · p. 12

Appendix B. Sphere geodesic Killing form

The appendix checks the recovered sphere-geodesic algebra using the Killing form and estimates the scale of its generator basis from the measured eigenvalues.

Appendix B. Sphere geodesic Killing form

  • The Killing form K depends only on the structure constants.Source: Appendix B · p. 12
  • For an orthonormal so(3) basis, the structure constants use ε_ijk, with ε_123 = ε_231 = ε_312 = +1 and the reversed permutations equal to −1.All other components are zero; substituting these constants into the Killing-form expression gives the orthonormal-basis check.Source: Appendix B · p. 12
  • The recovered generator basis may have an unknown uniform scale λ, and bilinearity of the commutator makes the scaled structure constants λ ε_ijk.Because the Killing form is quadratic in the structure constants, its scale can be used to estimate λ.Source: Appendix B · p. 13
  • The three measured Killing-form eigenvalues are κ = (−6.25, −5.98, −5.66), and each yields an estimate of the recovered basis scale.The Killing form is the symmetric 3 × 3 matrix built from the structure constants.Source: Appendix B · pp. 12–13

Appendix C. Generator norm ranking

After training, generators are ranked by norm; the surviving generators indicate the number of independent symmetries discovered.

Appendix C. Generator norm ranking

  • For the sphere-geodesic test, generator-norm ranking shows 3 surviving generators, corresponding to 3 independent symmetries discovered.The paper also shows generator plots for all test cases.Source: Appendix C · p. 13

Appendix D. Experimental details

The experiments use a shared differentiable latent-ODE architecture across six systems, with system-specific data and training settings. Generator spectra show three surviving generators for sphere geodesics, matching the expected rank of so(3).

Appendix D. Experimental details

  • The shared architecture uses a reverse-time GRU encoder, differentiable latent vector field, pointwise MLP decoder, and staged joint training with ramped symmetry weights.Pixel observations use convolutional encoder and decoder stacks; the same architecture is used across datasets.Source: Appendix D · p. 13
  • Three of eight sphere-geodesic generators survive while five collapse, leaving a 263× gap and recovering the expected so(3) rank of 3.Generator norm spectra are also reported for all six systems.Source: Figure 7 · p. 14, Figure 8 · p. 14
  • The harmonic oscillator and simple pendulum use two-dimensional discovered latents, while the twin-pendulum system retains a four-dimensional latent state.Dimension discovery prunes 6 → 2 for the oscillator and simple pendulum; the twin pendulums are observed jointly as a four-dimensional state.Source: Appendix D · p. 14
  • Lotka–Volterra trajectories are sampled within a filtered initial-condition box, and dimension discovery reduces the latent space from 20 to 2.The filter retains invariant values in [2.15, 3.0].Source: Appendix D · p. 15
  • The pixel pendulum uses 32 × 32 noisy frames with a temporal-difference channel and a staged perception-then-symmetry training procedure.The frozen encoder's latent width is pruned from 8 to 2 during the symmetry stage.Source: Appendix D · p. 15
  • Sphere geodesics use a six-dimensional latent embedding despite a four-dimensional data manifold, and a field-only polish phase sharpens the frequency profile.The linear so(3) realization requires embedding slack; the polish phase leaves the chart and generators fixed.Source: Appendix D · p. 15

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