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 $z\in {\mathbb{R}}^d$. 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 $H_{\varphi }$ 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 $|\rho |\ge 0.978$. 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.

arXiv commentAccepted into 2026 Neural Representations Proceedings