Weyl Algebra Equivariant Neural Networks

Encoding physical symmetries into neural network architectures is key to improving generalization and physical consistency in domains such as molecular dynamics and particle physics. Existing equivariant networks rely primarily on Clifford algebras, which realize orthogonal-group equivariance via the sandwich action, or on partial differential operator (PDO) networks, which implicitly exploit Weyl algebra structure without an explicit unifying framework. Both approaches are limited: Clifford algebras cannot capture symplectic or differential-operator symmetries, while PDO networks lack a common algebraic language across groups. We show that the Weyl algebra—generated by position and momentum operators under the canonical commutation relation—unifies both lines of work: its subalgebra structure recovers orthogonal-group equivariance, while its differential-operator generators extend naturally to the symplectic group. Building on this, we introduce a Weyl-equivariant neural network supporting both symmetries within a single architecture, along with degree-truncation approximations for computational tractability. Empirical results on tasks involving symplectic and orthogonal symmetries demonstrate that our method matches or outperforms existing equivariant networks while offering substantially broader symmetry coverage.