Equivariant DeepONets for Symmetry-Aware Operator Learning

Neural operators constitute a powerful class of deep learning architectures within scientific machine learning which learn mappings between function spaces, yielding natural and effective surrogates for parametric systems of partial differential equations (PDEs). A classical and potent method is the Deep Operator Network (DeepONet), which separates an input-dependent branch and output-coordinate trunk before fusing. Simultaneously, equivariant deep learning has been shown to be a powerful mechanism to imbue networks with knowledge of physical symmetries. In this work, we construct equivariant DeepONets through a representation-theoretic framework in which the observation, branch, trunk, and fusion maps obey compatible group actions. Our construction uses equivariant linear layers and symmetry-compatible branch-trunk fusion. We show that finite observations are compatible with equivariance when they recover a finite-dimensional invariant input representation or when the sensor family is closed under the spatial group and prove a universal approximation theorem for continuous equivariant operators by equivariant DeepONets. We evaluate the framework on numerous PDE systems with different symmetry groups, equivariance actions, and system sizes, including Allen–Cahn dynamics on the sphere, non-rotating spherical shallow water equations, thermoelasticity on an annulus, and a two-component Gross–Pitaevskii system. The equivariant architectures achieve order-of-magnitude reductions in prediction error and equivariance diagnostics over baseline DeepONets while often simultaneously possessing fewer learnable parameters, and the finite-sensor variants achieve comparable results. We also compare favorably to both appropriate FNO architectures and DeepONets with pre-chosen symmetry-aware trunks.