<p>Multiscale structural optimization considers the simultaneous design of macroscale (observable) and microscale (material) features. The multiscale design space demands complex models to capture physical effects between scales, while the extended design space introduces additional numerical complexity. In this work, surrogate models based on standard finite element approaches are presented as an attractive alternative to traditional models. Fast and efficient neural network (NN) surrogate models are trained to emulate the numerical homogenization of a hyperelastic material undergoing finite deformation within a multiscale structure. Computational homogenization techniques incorporating the first-order deformation gradient are applied to resolve the effective behavior of the nonlinear microstructural material, and several NN architectures and training strategies are compared to equivalent finite element models. The NN training strategies, incorporating both forward and backward passes, introduce a Sobolev loss function of relative error norms that improves upon the state of the art. Once trained, the NN surrogates are applied to multiscale design optimization for the minimization of nonlinear compliance and the minimization of complementary work. The efficiency of NN-based surrogate models enables the comparison of traditional topology optimization approaches with their multiscale alternatives in nonlinear design.</p>