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

arXiv:2610.35967v1 Announce Type: new 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 R^d. A bank of linear infinitesimal generators mathbf{A}_k in R^{d imes 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) = dz/dt 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.

Source: arXiv cs.LG | 2026-10-28