Dielectric elastomer actuators (DEAs) are widely used as soft-robotic artificial muscles, generating muscle-like motions by coupling finite deformations with electrical excitation. Nevertheless, traditional constitutive modeling remains challenging because the mechanical response, electrostatic loading, and their mutual coupling interact strongly and nonlinearly over wide operating ranges. Building on constitutive artificial neural networks (CANNs), we propose an automated, physics-informed model discovery strategy that learns an electromechanically coupled Helmholtz free-energy potential directly from data. Nominal stress and electric displacement are obtained by differentiation, enforcing thermodynamic consistency by construction. Material frame indifference is embedded via invariant-based inputs spanning purely mechanical, purely electrical, and coupled electromechanical measures. Instead of prescribing a functional form and solely calibrating coefficients, a sparsity-promoting identification mechanism selects a minimal set of relevant invariants' functions and their coefficients, so that the complete constitutive model is discovered from data while remaining interpretable. The proposed approach integrates the strengths of machine learning with the rigor of continuum physics. The trained model accurately fits DEA data across multiple loading conditions. To evaluate robustness and practical deployability, we introduce controlled noise perturbations (representing measurement uncertainty) and quantify the stability of the discovered energy form and the derived predictions under increasing noise levels. The resulting constitutive law is benchmarked against established electroelastic formulations to highlight the algorithm's generalizability. Moreover, many DEAs are realized as slender, stacked or layered actuators, motivating the derivation of geometrically exact Cosserat beam formulations that include the electric potential as an additional degree of freedom from continuum electromechanics. Transforming the discovered Helmholtz free-energy potential to a beam description allows to model electrically driven modes like contraction, bending, and torsion under different voltage boundary conditions.