Learning finite symmetry groups of dynamical systems via equivariance detection

TL;DR

The paper tackles the challenge of uncovering the full finite symmetry group of nonlinear dynamical systems from trajectory data, without requiring explicit knowledge of the governing equations. It introduces the Equivariance Seeker Model (ESM), a multi-branch neural framework that learns symmetry transformations through an equivariance loss and uses a repetition loss to ensure all $K$ elements are found in a single training run. The method is validated on simple and complex systems (Thomas attractor, Lorenz, Duffing) in both theory-informed and data-driven settings, with branch-removal and a group-metric used to confirm completeness and group properties. The approach enables automatic, data-driven symmetry discovery with practical implications for simplifying dynamical analyses and spectral decompositions, and is made available as open-source code.

Abstract

In this work, we introduce the Equivariance Seeker Model (ESM), a data-driven method for discovering the underlying finite equivariant symmetry group of an arbitrary function. ESM achieves this by optimizing a loss function that balances equivariance preservation with the penalization of redundant solutions, ensuring the complete and accurate identification of all symmetry transformations. We apply this framework specifically to dynamical systems, identifying their symmetry groups directly from observed trajectory data. To demonstrate its versatility, we test ESM on multiple systems in two distinct scenarios: (i) when the governing equations are known theoretically and (ii) when they are unknown, and the equivariance finding relies solely on observed data. The latter case highlights ESM's fully data-driven capability, as it requires no prior knowledge of the system's equations to operate.

Figures (7)

• Figure 1: Schematic representation of the ESM and its loss function. Left: illustration of ESM's processing flow: the input state $\vec{x}$ is operated by $M$ parallel branches. Each $\alpha$ branch applies three consecutive blocks: first, a trainable matrix $\hat{W}_{\alpha}$. Second, the non-trainable function $\vec{\mathcal{Y}}$ that produces the dynamical system's time derivative. Third, the matrix $\hat{W}_{\alpha}^{-1}$. The ESM outputs $M$ predictions $\vec{y}_{\hat{W}_{\alpha}} = \hat{W}_{\alpha}^{-1} \vec{\mathcal{Y}} \left( \hat{W}_{\alpha} \vec{x} \right)$. Right: schematic illustration of the single-branch equivariance loss function. It has $K$ global minima of the same magnitude (zero or $l_{\hat{I}}$), corresponding to the symmetry transformations $\hat{D}_{\alpha}$. The function may present other local minima of higher magnitude, which are not associated to symmetries. Branch matrices $\hat{W}_{\alpha}$ (for $\alpha = 1, \dots, M$) are independently optimized with the equivariance loss and, at the same time, jointly optimized with the repetition loss.
• Figure 2: Illustration of ESM algorithm for the example $\dot{x} = - x^3$. The symmetry group has $K = 2$ elements: $D_1 = 1$ (identity) and $D_2 = -1$ in this case. We present two different trainings ($T_a$ and $T_b$) with different random initialization of the trainable parameters of the model, the branch weights $(W_1 , W_2)$. (a-b) Equivariance (blue) and repetition (brown) loss curves as function of epochs for $T_a$ and $T_b$ respectively. Gray curves represent the $\sigma$ (solid) and $A$ (dashed) values as function of the epochs (both normalized to the same arbitrary scale). The dashed vertical lines indicate the epochs at which each variation interval starts: first, $\sigma$ (from $\sigma_0 = 10^4$ to $\delta\sigma = 10^{-2}$) and second, $A$ (from $A_0 = 10^3$ to $\delta A = 10^{-8}$). (c-f) Colour maps of the total loss in the $(W_1 , W_2)$ plane at epochs 400, 655, 800 and 1200 respectively [these epochs are marked with black arrows in both horizontal axes of panels (a) and (b)]. The color bars represent the value of $\mathcal{L} = \mathcal{L}_{\mathrm{equiv}} + \mathcal{L}_{\mathrm{rep}}$. In all panels, trajectories in the "space of weights" [coordinates ($W_1$,$W_2$)] in $T_a$ and $T_b$ are plotted in black and red respectively. At each panel, these trajectories are plotted from the initial training location (squares) to the current state at a given epoch (triangles). In panel (f) (final epoch), the final results are represented by circles. The gray segments represent the Gaussian width in the definition of $\mathcal{L}_{\mathrm{rep}}$, corresponding to the current value of the hyperparameter $\sigma$.
• Figure 3: ESM algorithm for the illustrative system $\dot{x} = -x^3$ for different number of branches: $M = 2$ (blue), $M = 3$ (orange) and $M = 4$ (black). In the left column, the analytical function $\vec{y}(\vec{x})$ is used in the ESM. In the right column, a pretrained oracle $\vec{y}_{\mathrm{NN}} (\vec{x})$ is used instead. The gray curves in (b) provide a representation of the dependence of the hyperparameters $\sigma$ (solid), $A$ (dashed) and $\eta$ (dotted) as function of epochs (they are all normalized to the same arbitrary scale). In all panels, the dashed vertical lines mark the epoch at which a hyperparameter starts to decrease. (a-b) Equivariance loss as function of the epochs. The horizontal dashed-dotted line indicates the value $l_{\hat{I}}$ given by the NN oracle's MSE. (c-d) Repetition loss as function of the epochs. (e-f) Group metric (Eq \ref{['eq:d_group']}) as function of the epochs.
• Figure 4: ESM algorithm for the Thomas' cyclically symmetric attractor. (a-f) Optimization of the matrices $\{ \hat{W}_{\alpha} \}$, using both the analytical function $\vec{y}(\vec{x})$ and oracle $\vec{y}_{\mathrm{NN}}(\vec{x})$, for $M = 4$ (blue) and $M = 30$ (black). Three quantities are shown as function of the epochs: equivariance loss (a-b), repetition loss (c-d) and group metric (e-f). The gray curves in (b) indicate the variation of $\sigma$, $A$ and $\eta$, and the vertical dashed lines in all panels represent the starting of each hyperparameter decrease. The dashed-dotted lines in (a-b) indicate the bound $l_{\hat{I}}$ given by the oracle's MSE. In all panels, the red and purple areas indicate the regimes at which some branches have been removed by each of the two final-training processes. (g-h) Branch-removal processes for $M = 30$ using $\vec{y}_{\mathrm{NN}} (\vec{x})$. (g) Removal of matrices in local minima. The plot shows each single-matrix equivariance loss for all the matrices before the removal. The dotted-dashed line represents $l_{\hat{I}}$, and the dotted line represents the tolerance $\varepsilon_{\mathrm{equiv}}$ given by the maximum oracle squared error. The inset displays the histogram of squared errors for all ESM training samples, with the dashed-dotted line representing $l_{\hat{I}}$ the dotted line representing $\varepsilon_{\mathrm{equiv}}$. (h) Removal of redundant matrices. The plot shows the AED between all possible matrix pairs before the removal.
• Figure 5: ESM algorithm with $M = 30$ branches for the Lorenz system. Three quantities are shown as function of the epochs: equivariance loss (a-b), repetiton loss (c-d) and group metric (e-f). Panels (g-h) illustrate the final-training processes using $\vec{y}_{\mathrm{NN}} (\vec{x})$. The remaining details are similar to those in Fig. \ref{['fig:FIG_4']}.