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

Matt L. Wiemman; Peter Melchior

cs.LGOct 25, 2026

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

Authors: Matt L. Wiemman, Peter Melchior

Organizations: Princeton University

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 $H_φ$ 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 $|ρ| \geq 0.978$ . 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.

Figures & tables

Figure 1: An autoencoder maps observations (states or video) to a decorrelated latent chart where a latent ODE

\dot{z}=f(z)

carries the dynamics and orbits are circular. A linear generator

\mathbf{A}

is then fitted by minimising the normalised Lie-bracket residual

\mathbb{E}_{z\sim data}\|\mathbf{J}_{f}(z)\mathbf{A}z-\mathbf{A}f(z)\|^{2}/\|f(z)\|^{2}

on encoded data.

	Ours	LaLiGAN	Noether Nets	HNN/LNN
Discovers Lie generators	✓	✓	–	–
Continuous-time dynamics learned jointly	✓	–	–	✓
Invariants discovered, not assumed	✓	–	✓	–
Num. of symmetries detected, not specified	✓	–	–	–
Expressivity guarantee for the action	–	✓	–	–
Exact conservation by construction	–	–	–	✓

Table 2

Figure 2: Latent phase portrait of the harmonic oscillator, Lotka-Volterra, twin pendulum, and pixel-pendulum systems. Note that if the reduced latent dimension is $>2$ , a PCA projection is used to convert the $n$ -dimensional trajectory onto a 2D map (as in the twin pendulum).

System	obs. dim	rank	expected	discovered symmetries	$\|\rho\|$	calib. $R^{2}$
Harmonic oscillator	2	2	2	rotation $+$ dilation	0.999	0.999
Pendulum ( $\sigma{=}5\%$ )	2	1	1	phase rotation	0.998	0.997
Lotka–Volterra	2	1	1	phase rotation	0.991	0.983
Twin pendulums	4	2	2	two phase rotations	0.978	0.963
Sphere geodesics	6	3	3	$\mathfrak{so}(3)$ rotations	0.997	0.996
Pixel pendulum	$2{\times}1024^{\dagger}$	1	1	phase rotation	0.995	0.996

Table 1: Symmetry discovery across the test suite. Rank counts the survivors of the generator-norm spectrum (i.e. the number of discovered symmetries). Expected is the true number of symmetries.

|\rho|

is the absolute Spearman rank correlation between the discovered invariant network

H_{\phi}

and the ground-truth invariant, both evaluated on held-out validation windows. Calib.

R^{2}

is the variance of the ground-truth invariant explained by

g(H_{\phi})

, where

g

is a monotone map fitted on validation windows ( Figure 3 a)

Figure 3: The discovered invariant, calibrated to energy on validation windows (a) , tracks the true energy of held-out trajectories over $10\times$ the training span (b) , and yields a falsifiable conservation verdict (c) : fresh pendulum data stays within the $\pm 3\times$ noise-floor band ( $1.6\times$ the floor), while a damped control from the same initial conditions is correctly rejected at $13\times$ the noise floor.

Figure 4: Pixel-pendulum reconstruction. Ground-truth frames (top) vs. frames decoded from the LatentLieODE rollout (middle) and residuals (truth minus decoded, bottom ), continued past the end of the training data. All temporal structure lives in the 2D latent ODE with the decoder rendering each latent state independently.

Figure 5: Ground truth (gray), decoded latent-ODE rollout (blue), and pure group transport

e^{s\mathbf{A}}z_{0}

(orange, dashed; no integration,

s=t

) for four held-out orbits. The transport generator is the per-orbit projection of the flow onto the discovered

\mathfrak{so}(3)

span.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Training step time against encoder input dimension, single CPU.

Figure 7: Generator norm spectrum for the sphere-geodesic system of

K=8

deliberately over-complete generators, three survive and five collapse, leaving a

263\times

gap. The detected algebra dimension matches the expected rank of 3 for the

\mathfrak{so}(3)

group.

Figure 8: Generator norm spectra for the six systems of Table 1 , in the same form as fig:sphere_generators

	Harmonic	Pendulum	Lotka–Volterra	Twin pend.	Pixel (stage 2)	Sphere
trajectories (train/val)	1024/128	1024/128	1024/128	1024/128	1024/32	3000/128
samples $\times$ span	$50\times 8$	$60\times 12$	$70\times 14$	$60\times 12$	—	$125\times 25$
noise $\sigma$	0.01	0.05	0.01	0.02	—	0.01
obs. dim	2	2	2	4	8	6
latent dim (found)	$6\to 2$	$6\to 2$	$20\to 2$	4	$8\to 2$	6
generators $K$	3	4	5	6	4	8

Appendix

Table 11

Symmetry Discovery

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LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systems

Shida Liu, Abhishek Gupta, Sumit Sinha, L. Mahadevan

School of Engineering and Applied Sciences (SEAS), Harvard University, Cambridge, MA, USA. · Tracelink, Wilmington, MA, USA. · SEAS, Physics and OEB, Harvard University, Cambridge, MA, USA.

Stochastic Differential Equations Lie Group Methods Symmetry Discovery

Open paper AskPDF arXiv

Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems the relevant continuous symmetries are rarely known, and symmetry discovery for SDEs has remained essentially unexplored. We introduce LieStoNet, an end-to-end, template-free framework for discovering Lie-point symmetries of SDEs directly from spatiotemporal trajectories, without prespecifying symmetry groups, templates, or canonical coordinates. Building on the seminal SDE Lie-symmetry theory of Gaeta and Quintero (1999), which formalizes Lie-point SDE symmetries and their relation to Fokker-Planck symmetries, LieStoNet learns neural surrogates for drift and diffusion from increments, then learns projectable generators by enforcing the SDE determining equations, separately regularizing for closure under Lie brackets, adherence to the Lie algebra axioms (bilinearity, antisymmetry, Jacobi), and a non-redundant independent basis. The surrogate also defines an associated Fokker-Planck equation, enabling optional discovery of its Lie-point symmetries in parallel. Across multiple canonical SDEs with known analytic symmetries, LieStoNet recovers generators consistent with the ground-truth symmetry algebra, providing interpretable symmetry discovery for noisy dynamics. Code is available at this link.

Abstract

Symmetry Discovery

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