Neural Symmetry Learning: Integrating SE(3) Equivariant Networks, Neural ODEs, and Hyperdimensional Computing for Invariant Machine Intelligence

Introduction

We live in a world governed by symmetries — both physical and mathematical. In classical physics, symmetries often manifest in forms such as conservation laws, geometric invariances and group-theoretic transformations. In the realm of machine learning, and especially deep learning, symmetries hold an equally pivotal place. They enable models to better generalize by capturing fundamental structures in the data and in the underlying physical processes.

The notion of symmetry learning — where we train computational systems to respect, exploit, and even discover symmetries — has gained increasing traction. Researchers have been weaving together techniques from group theory, differential geometry and neural networks to build better, more robust, and more interpretable models. One particularly exciting direction is the exploitation of SE(3) symmetries — the group of rigid motions (rotations and translations) in three-dimensional space (3D). This is especially crucial for tasks involving 3D data such as computer vision, robotics, autonomous driving, and molecular modeling.