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

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 $\lvert \rho \rvert \geq 0.978$. 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.