[Submitted on 25 Oct 2026]
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 $H_phi$ 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.
Submission history
From: Matt Wiemann [view email]
[v1] Sun, 25 Oct 2026 18:21:35 UTC (803 KB)
