Gauge Equivariant Networks (Gauge CNNs) are convolutional neural networks designed for data defined on non-Euclidean manifolds (curved surfaces, meshes, sphere) that guarantee their output is independent of the arbitrary local coordinate system (gauge) chosen at each point on the surface — solving the fundamental problem that curved surfaces lack a globally consistent "north-east" reference frame, making standard convolution undefined without an arbitrary and physically meaningless gauge choice.

What Are Gauge Equivariant Networks?

• Definition: On a flat 2D image, convolution is well-defined because there is a global, consistent coordinate system — "right" and "up" mean the same thing everywhere. On a curved surface (sphere, protein surface, brain cortex), there is no globally consistent coordinate system — at each point, the local tangent plane has an arbitrary orientation (the "gauge"). A gauge equivariant network guarantees that its output does not depend on this arbitrary orientation choice.
• The Gauge Problem: On a sphere, the equirectangular projection defines local coordinates but introduces singularities at the poles and severe distortion. On a 3D mesh (brain surface, molecular surface), each face or vertex has a local tangent plane with an arbitrary orientation. Applying standard convolution on these surfaces produces results that change when the local gauge is rotated — a physically meaningless artifact of the coordinate choice.
• Gauge Equivariance: A gauge equivariant network transforms its features predictably when the local gauge is changed — specifically, gauge-equivariant features transform under the structure group of the fiber bundle (typically SO(2) for surfaces). This ensures that the final invariant outputs (scalar predictions) are identical regardless of gauge choice, while intermediate equivariant features carry meaningful geometric information.

Why Gauge Equivariant Networks Matter

• Spherical Data: Global weather modeling, omnidirectional vision (360° cameras), and planetary science all operate on spherical domains where standard planar convolution introduces pole distortion. Gauge equivariant networks on the sphere produce consistent predictions at all latitudes without the artifacts of projected 2D convolution.
• Mesh Processing: 3D meshes representing protein surfaces, brain cortices, automotive body panels, and architectural structures require convolution-like operations that respect the curved geometry. Gauge equivariance ensures that the results of mesh convolution are intrinsic to the surface geometry, not dependent on the arbitrary triangulation or local frame assignment.
• Theoretical Generality: Gauge equivariance provides the most general mathematical framework for equivariant neural networks on manifolds, subsumming planar equivariant CNNs, spherical CNNs, and mesh CNNs as special cases. It is grounded in the theory of fiber bundles and gauge theory from differential geometry and theoretical physics.
• Anisotropic Features: Unlike isotropic approaches (that use only rotation-invariant features like distances and angles), gauge equivariant networks support oriented features — tangent vectors, directional derivatives, and tensor fields — that carry richer geometric information. This is essential for tasks like predicting surface flow direction, fiber orientation in materials, or protein binding site directionality.

Gauge Equivariance Domains

Gauge Equivariant Networks are surface crawlers — navigating curved geometry with convolution-like operations that produce consistent results regardless of the arbitrary local coordinate frame, enabling deep learning on spheres, meshes, and manifolds where standard flat-world convolution fails.