Interpretable Discovery of One-parameter Subgroups: A Modular Framework for Elliptical, Hyperbolic, and Parabolic Symmetries

We propose a modular, data-driven framework for jointly learning unknown functional mappings and discovering the underlying one-parameter symmetry subgroup governing the data. Unlike conventional geometric deep learning methods that assume known symmetries, our approach identifies the relevant continuous subgroup directly from data. Our framework focuses on three primary geometric components of one-parameter subgroup actions: elliptic, hyperbolic, and parabolic regimes. For the given regime, our framework instantiates a corresponding symmetry discovery architecture with invariant and equivariant representation layers structured according to the Lie algebra of the subgroup, and learns the exact generator parameters end-to-end from data. This yields models whose invariance or equivariance is guaranteed by construction and admits formal proofs, enabling symmetry to be explicitly traced to identifiable components of the architecture. The approach is applicable to one-parameter subgroups of a wide range of matrix Lie groups, including $SO(n)$, $SL(n)$, and the Lorentz group. Experiments on synthetic and real-world systems, including moment of inertia prediction, double-pendulum dynamics, and high-energy \textit{Top Quark Tagging}, demonstrate accurate subgroup recovery and strong predictive performance across both compact and non-compact regimes.

Many AI systems perform better when they know that some changes to the input should not change the answer. For example, rotating a physical system or viewing a particle collision from a different reference frame may change how the data looks, but not the underlying answer. However, in many scientific datasets, these hidden patterns are not known in advance, and assuming the wrong one can make a model worse. We introduce $H_\gamma$-Net, a neural network that learns both the prediction task and the hidden pattern of change directly from data. The key idea is to teach the model to recognize when two inputs are really different versions of the same situation. It does this by mapping such inputs into a common view, so that the parts that truly matter are kept and the irrelevant changes are removed. This lets the model uncover the hidden symmetry in the data. Our method can handle three basic kinds of geometric change: rotations, squeezes or boosts, and shears, as well as some mixtures of these motions. Because this structure is built directly into the model, its predictions are guaranteed to respect the discovered symmetry. In experiments on pendulum dynamics, moment of inertia prediction, synthetic functions, and top-quark tagging, the method recovered the correct hidden symmetries and often predicted more accurately than previous approaches. This can help build scientific machine learning models that are more reliable, data-efficient, and easier to inspect.