Continuous symmetries are fundamental to many scientific and learning problems, yet they are often unknown a priori. Existing symmetry discovery approaches typically search directly in the space of transformation generators or rely on learned augmentation schemes. We propose a fundamentally different perspective based on spectral structure. We introduce a framework for discovering continuous one-parameter subgroups using the Generalized Fourier Transform (GFT). Our central observation is that invariance to a subgroup induces structured sparsity in the spectral decomposition of a function across irreducible representations. Instead of optimizing over generators, we detect symmetries by identifying this induced sparsity pattern in the spectral domain. We develop symmetry detection procedures on maximal tori, where the GFT reduces to multi-dimensional Fourier analysis through their irreducible representations. Across structured tasks, including the double pendulum and top quark tagging, we demonstrate that spectral sparsity reliably reveals one-parameter symmetries. These results position spectral analysis as a principled and interpretable alternative to generator-based symmetry discovery.
Figures & tables
Figure 1: Symmetry discovery and learning framework. The input is first aligned via a learnable orthogonal transformation Q∈SO(n) and decomposed into two-dimensional blocks, which are converted to polar coordinates to obtain radii and torus angles. Primitive torus Fourier features U(x) , together with radial features R(x) , are fed to the predictor ϕw . Training minimizes the prediction loss along with a spectral resonance regularizer that promotes support only on frequency directions satisfying ⟨m,λ⟩=0 , thereby enabling recovery of the latent one-parameter symmetry generator.
Method
Joint Framework
Interpretable Predictor
Scope
Baseline
Ours: Spectral Discovery
Yes
Yes
1 D Lie groups
–
Augerino
Yes
No
Generic Lie groups
✓
LieGAN
No (discovery only)
No
Generic Lie groups
–
LieGG
No (post-hoc discovery)
No
Generic Lie groups
–
InfGen
No (post-hoc discovery)
No
Generic Lie groups
–
BeyondAffine
No (post-hoc discovery)
No
1 D Lie groups (incl. non-affine)
–
Table 1: Comparison of symmetry discovery frameworks. We compare Spectral Discovery (ours) with Augerino Benton et al. [2020] , LieGAN Yang et al. [2023b] , LieGG Moskalev et al. [2022] , InfGen Ko et al. [2024] , BeyondAffine Shaw et al. [2024] , and Bispectral Neural Networks (BNNs) Sanborn et al. [2022] . Methods are evaluated along four axes: (i) whether symmetry discovery and prediction are learned in a joint training framework , (ii) whether the resulting predictor is explicitly invariant in an interpretable manner, (iii) the class of symmetry groups supported, and (iv) whether the method is used as a baseline in this work.
Figure 2: Recovered vs. True Rotational Generators. Comparison of the learned (left) and ground-truth (right) generators for (a) the Double Pendulum system (diagonal Δ(SO(2)) generator) and (b) the Top Tagging task. Both learned generators demonstrate near-perfect alignment with the physical ground truth (cosine similarity =0.9999 ).
Method
Test MSE ↓
Inv. Error ↓
Cosine Similarity ↑
Spectral Discovery
0.00298 ± 0.00024
0.00070 ± 0.00027
0.9999 ± 0.00001
Augerino
0.01053±0.00040
0.00232±0.00025
0.9233±0.0708
Table 2: Double Pendulum (6D), 32K samples. Comparison of predictive accuracy and symmetry recovery. Results are reported as mean ± standard deviation. Lower is better for Test MSE and Invariance Error; higher is better for Cosine Similarity.
Method
Accuracy (%) ↑
Cosine Similarity ↑
Spectral Discovery
84.87 ± 0.39
0.9999 ± 0.00002
Augerino
74.9±2.4
0.994±0.003
Table 3: Top Quark Tagging (Elliptic Regime). Classification accuracy and generator recovery. Results are reported as mean ± standard deviation. Higher is better for both metrics.
Symbol
Meaning
D={(xi,yi)}i=1T
Training dataset of T input–target pairs.
T
Number of training samples.
xi∈Rn
Input feature vector for sample i .
yi∈Rm
Target/output vector for sample i .
n,m
Input and output dimensions, respectively.
X⊂Rn
Input domain (assumed closed under the action of SO(n) ).
Table 4: Notation used in the method.
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Component
Configuration
Double Pendulum samples
32,000
Top Tagging samples
64,000
Epochs
40
Optimizer
Adam
Learning rate
2×10−3
Predictor ϕw
MLP (3 hidden layers, ReLU)
Appendix
Table 5: Summary of experimental settings used across all tasks.
Figure 3: Robustness to noise (noise sweep). Performance comparison between Spectral Discovery and Augerino on the 6D double pendulum task across increasing additive noise levels. Left: Mean cosine similarity between the learned and ground-truth symmetry generators. Right: Mean test loss. Spectral Discovery maintains near-perfect generator recovery under low and moderate noise and degrades gracefully at high noise, while Augerino exhibits lower alignment and higher variability.
Figure 4: Sample efficiency (sample sweep). Performance comparison between Spectral Discovery and Augerino as the number of training samples increases. Left: Mean cosine similarity between the learned and ground-truth symmetry generators. Right: Mean test mean-squared error. Spectral Discovery achieves near-perfect symmetry recovery with fewer samples and consistently lower predictive error, while Augerino improves more gradually with data.
Figure 5: LieGAN comparison: sample sweep (mean cosine similarity). Mean cosine similarity between the recovered and ground-truth symmetry generators on the 6D double pendulum task as the number of training samples increases. Spectral Discovery achieves higher generator alignment across all sample sizes, reaching near-perfect recovery at larger sample regimes.
Figure 6: LieGAN comparison: noise sweep (mean cosine similarity). Mean cosine similarity between the recovered and ground-truth symmetry generators on the 6D double pendulum task as additive noise increases. Spectral Discovery maintains near-perfect alignment for low and moderate noise and degrades gracefully at high noise, while LieGAN exhibits lower alignment throughout.
Discovering underlying symmetries from data has emerged as a crucial challenge in scientific discovery. Existing data-driven methods for symmetry discovery fail to determine the exact number and mathematical form of unknown infinitesimal generators. Recent explicit methods represent generators using a predefined function library and identify them through algebraic optimization, but they often struggle to capture complex symmetries involving high-order polynomials or transcendental functions. To address this limitation, we formulate symmetry discovery as a joint optimization problem over the function library and coefficients. We propose a novel framework that leverages an encoder-decoder architecture to dynamically generate symbolic expressions and expand the library. This generation process is optimized via reinforcement learning, which accelerates the exploration of the symbolic search space through step-wise rewards. Experiments demonstrate that LieDiscover can successfully uncover open-form infinitesimal generators involving high-order polynomials or transcendental functions, which remain intractable for existing methods. The discovered symmetries also improve performance in downstream PDE solving and discovery tasks.
Xinxin Li, Jianming Ma, Xingyu Cui +3
East China Normal University · Shanghai Jiao Tong University · Institute of Applied Physics and Computational Mathematics +2
Continuous symmetry discovery is typically formulated as a separate estimation problem for each function or dataset. We instead investigate whether the map from a function to its symmetry structure can be learned across a family of functions from finite observations. Using the infinitesimal differential characterization of continuous symmetries, we formulate this map as an inverse symmetry operator, analogous to an inverse PDE problem. For global Lie-group symmetries, we establish a representation-theoretic population characterization: the symmetry Lie algebra is the nullspace of a positive-semidefinite operator constructed from representation coefficients and the infinitesimal group action. We also show that the inverse symmetry operator is equivariant under coordinate changes through the adjoint action. This characterization separates the geometric symmetry map from finite-sample estimation, motivating a hybrid approach that amortizes statistically difficult components while retaining analytic structure. On SO(5) benchmarks, amortized equivariant estimation of harmonic coefficients improves symmetry recovery from sparse observations, whereas ordinary least squares suffices for SL(2,R). We extend this viewpoint to local symmetries, where symmetry directions form smoothly varying tangent distributions rather than a fixed Lie algebra. Under the Lie-bracket consistency condition, Frobenius' theorem guarantees local integral submanifolds along which observables remain unchanged. We recover this geometry using a hybrid estimator that learns observation weights while preserving analytic differential and geometric recovery. Across global, non-compact, local, and physically motivated synthetic benchmarks, our results support learning the statistically difficult finite-sample components while preserving the known geometric structure of symmetry recovery.
Pavan Karjol, Scindhiya Laxmi, Prathosh AP
Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India · Department of Mathematics and Computing, IIT (ISM) Dhanbad, India
Parameter space symmetries are important for understanding neural networks' loss landscape, training dynamics, and generalization. However, systematically identifying these symmetries remains a challenge. In this paper, we formalize data-dependent parameter symmetries and characterize loss invariance and the group-action axioms through infinitesimal conditions, which provide objectives for jointly learning group generators and nonlinear action maps. Our framework systematically uncovers parameter symmetries, including previously unknown ones. To study larger networks, we establish conditions under which subnetwork symmetries extend to the full model. The same construction gives an explicit family of finite-batch symmetries, providing both analytical examples and a foundation for discovery through small subnetworks. Using the infinitesimal characterization and subnetwork construction, we implement a framework for automated discovery of parameter symmetries, and successfully uncovered symmetries in various architectures, including pretrained transformer models.
Bo Zhao, Nima Dehmamy, Robin Walters +1
Harvard University · IBM Research · Northeastern University +1