The paper presents a data-driven method for discovering intrinsic Lie point symmetries of differential equation systems directly from sampled jet-space data. By formulating symmetry discovery as an operator-theoretic problem, the approach identifies admissible generators, separates them from redundant or trivial directions, and reconstructs the full symmetry algebra, including its Lie-bracket structure, on benchmark systems. The method demonstrates that explicit continuous-symmetry generators can be recovered from data generated by both exact and approximate differential systems.
arXiv:2608. 01582v1 Announce Type: cross Abstract: 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.
arXiv:2512. 20043v3 Announce Type: replace Abstract: Symmetry is fundamental to understanding physical systems and can improve performance and sample efficiency in machine learning.
The paper introduces a diagnostic tool that measures how neural emulators of partial differential equations capture physical symmetries by evaluating the overlap of loss gradients along symmetry-related states. This metric probes the local geometry of the learned loss landscape and goes beyond traditional equivariance tests by directly assessing learning dynamics. Applied to autoregressive fluid flow emulators, the study shows that orbit-wise gradient coherence enables generalization over symmetry transformations and reveals when training selects a symmetry-compatible basin.
The paper introduces LatentLieODE, a neural ODE framework that learns linear Lie symmetries and first integrals from time‑series data such as videos. It uses an autoencoder to obtain a low‑dimensional latent space, fits linear infinitesimal generators to a Lie‑bracket residual, and trains a latent ODE to reconstruct observations. The method successfully identifies symmetries and conserved quantities in six benchmark systems, achieving high correlation with ground‑truth invariants and recovering a non‑abelian σ(3) algebra on the sphere.
arXiv:2608. 08091v1 Announce Type: cross Abstract: Dynamical systems model trajectory data generated by fixed underlying dynamics, with applications ranging from biology to physics.
The paper investigates how a vanilla Transformer learns symmetries from finite data augmentation on point cloud datasets. It finds an ordering of learnability: non-angle-preserving symmetries are easiest, followed by angle-preserving symmetries, and finally base angle-preserving subgroups such as translation, rotation, and scale. The study also examines the Transformer's extrapolation behavior, performs a structural analysis of trained models to uncover interpretable mechanisms for invariance, and extends these findings to equivariant functions, suggesting that the identified mechanisms can serve as building blocks for learned equivariance.
arXiv:2606. 04108v1 Announce Type: cross Abstract: Single-view 3D generative models have achieved impressive visual quality, yet they are not designed to satisfy structural or functional requirements, and in practice, often fall short.
arXiv:2609.07031v1 Announce Type: new Abstract: Learning with group invariances is central to many scientific and geometric learning problems, yet its computational foundations remain poorly understo...
The paper introduces a basis‑independent derivative operator that can represent a wide range of geometric differential operations, such as rotations, volume forms, and curvatures, without requiring explicit coordinate arrays. This operator is built solely from existing automatic differentiation primitives and can be composed with JAX’s batching, compilation, and hardware acceleration features. The authors demonstrate its reusability across manifold operators, its integration into a spacetime inverse‑problem pipeline, and its application to optimization over rotor subgroups.
arXiv:2609.38623v1 Announce Type: new Abstract: Learning partial differential equation (PDE) dynamics across varying domains is central to predictive modelling and data-driven discovery of governing...
arXiv:2510. 25731v2 Announce Type: replace-cross Abstract: Initial-boundary value problems (IBVPs) provide the essential framework for modelling a wide range of phenomena in physics and engineering.
arXiv:2611.07410v1 Announce Type: new Abstract: Symmetry governs the organization of physical structures, while symmetry breaking reveals their transformations. In geometric deep learning, symmetry i...