Data-driven Discovery of Lie Symmetries and First Integrals with Latent ODEs
Abstract
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 ∈ ℝd. A bank of linear infinitesimal generators Ak ∈ ℝd × d is fitted jointly against a differentiable Lie-bracket residual Jf(z)Az − A f(z), while an MLP optimises the latent ODE f(z) = dz/dt to reconstruct the observation-space timeseries. After suitable generators and f are found, we fit a third network Hϕ 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 |ρ| ≥ 0.978. On the sphere, the pipeline recovers a non-abelian 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.
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BibTeX
@misc{wiemman2026datadriven,
title = {{Data-driven Discovery of Lie Symmetries and First Integrals with Latent ODEs}},
author = {Matt L. Wiemman and Peter Melchior},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35967},
eprint = {2610.35967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35967},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Data-driven Discovery of Lie Symmetries and First Integrals with Latent ODEs
AU - Matt L. Wiemman
AU - Peter Melchior
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.35967
UR - https://arxiv.org/abs/2610.35967
AB - 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 ∈ ℝ^d. A bank of linear infinitesimal generators A_k ∈ ℝ^d × d is fitted jointly against a differentiable Lie-bracket residual J_f(z)Az − A f(z), while an MLP optimises the latent ODE f(z) = dz/dt to reconstruct the observation-space timeseries. After suitable generators and f are found, we fit a third network H_ϕ 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 |ρ| ≥ 0.978. On the sphere, the pipeline recovers a non-abelian 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.
ER -