Latent Space Symmetry Discovery

TL;DR

LaLiGAN introduces Latent Space Symmetry Discovery, a framework that learns nonlinear group actions by decomposing them into a nonlinear encoder/decoder pair connected by a linear latent-space representation of a symmetry group. The method jointly learns the symmetry generator and the latent mappings, with a GAN-based objective and a reconstruction constraint, enabling discovery of nonlinear symmetries directly from data and producing structured latent spaces. It provides theoretical guarantees for approximating nonlinear actions under certain group-action conditions and demonstrates improved equation discovery and long-term forecasting by leveraging latent symmetries in dynamical systems such as reaction-diffusion, pendulum, and Lotka–Volterra. Empirically, LaLiGAN uncovers latent SO(2) and SO(2)×SO(2)–like symmetries, yields disentangled or regularized latent structures, and enhances downstream tasks including SINDY-based equation discovery and robust long-horizon predictions, showcasing its potential to accelerate scientific discovery from complex observations.

Abstract

Equivariant neural networks require explicit knowledge of the symmetry group. Automatic symmetry discovery methods aim to relax this constraint and learn invariance and equivariance from data. However, existing symmetry discovery methods are limited to simple linear symmetries and cannot handle the complexity of real-world data. We propose a novel generative model, Latent LieGAN (LaLiGAN), which can discover symmetries of nonlinear group actions. It learns a mapping from the data space to a latent space where the symmetries become linear and simultaneously discovers symmetries in the latent space. Theoretically, we show that our model can express nonlinear symmetries under some conditions about the group action. Experimentally, we demonstrate that our method can accurately discover the intrinsic symmetry in high-dimensional dynamical systems. LaLiGAN also results in a well-structured latent space that is useful for downstream tasks including equation discovery and long-term forecasting.

Figures (18)

• Figure 1: An example of $\mathrm{SO}(2)$ nonlinear group action $\pi'$ on $V=\mathbb R^2$ and its decomposition into an encoder $\phi$, a linear representation $\pi$ and a decoder $\psi$. Each trajectory is a group action orbit containing a random $v \in V$.
• Figure 2: Overview of the proposed LaLiGAN framework. The encoder maps the original observations to a latent space. The latent representation is transformed with the linear group action from the generator. The decoder reconstructs the inputs from original and transformed representations. The discriminator is trained to recognize the difference between the original and the transformed samples.
• Figure 3: Potential failure modes in latent space symmetry discovery. (a) Fallacious symmetry in low-dimensional subspace. (b) Absence of symmetry in a biased latent space.
• Figure 4: Symmetry discovery in reaction-diffusion system with 2D latent space. (a) Latent representations of the system at all timesteps. (b) Randomly selected samples from the dataset. (c) Samples transformed by LaLiGAN are similar to the original data. (d) Samples transformed by the baseline, linear LieGAN, are significantly different from the original data.
• Figure 5: Latent symmetry discovery in nonlinear pendulum (upper) and Lotka-Volterra equations (lower). (a) Original trajectories of the systems, where the color of each trajectory corresponds to its Hamiltonian. (b) The trajectories mapped to a symmetric latent space. (c) The trajectories transformed by LaLiGAN. (d) The trajectories transformed by linear LieGAN.

Theorems & Definitions (6)

• Theorem 4.1: Universal Approximation of Nonlinear Group Action
• Proposition 4.2
• Proposition 3.1
• proof
• Theorem 3.2: Universal Approximation of Nonlinear Group Action
• proof