Learning solution operators for microstructural problems is of central importance in multiscale simulations, where microscale physics can be replaced by surrogate models to reduce computational cost. In this work, we extend the Equilibrium Neural Operator (EquiNO) [1] as a physics-informed PDE surrogate for predicting three-dimensional displacement and stress fields. The proposed framework is tailored to multiscale FE² computations and combines concepts from reduced-order modeling and physics-informed operator learning to achieve efficient and accurate surrogates. To this end, EquiNO represents the solution in a reduced space spanned by divergence-free basis functions obtained through proper orthogonal decomposition (POD). This construction enforces the balance of linear momentum by design and eliminates the need for penalty terms or multi-objective loss functions during training. Furthermore, the computational cost of training is significantly reduced by selecting a set of so-called magic points using the discrete empirical interpolation method (DEIM) [2], enabling the model to be trained only on a small number of spatial locations. The method is applied to quasi-static solid mechanics problems. Numerical results demonstrate that the proposed approach can be trained efficiently and delivers accurate predictions even when trained on small datasets. [1] H. Eivazi, J.-A. Tröger, S. Wittek, S. Hartmann, and A. Rausch. EquiNO: A physics-informed neural operator for multiscale simulations. Journal of Computational Physics, 114745, 2026. https://doi.org/10.1016/j.jcp.2026.114745 [2] S. Chaturantabut and D. C. Sorensen. Nonlinear model reduction via discrete empirical interpolation. SIAM Journal on Scientific Computing, 32(5):2737–2764, 2010. https://doi.org/10.1137/090766498