A novel approach to expedite design optimization of nonlinear beam dynamics in storage rings is proposed and demonstrated in this study. At each iteration, a neural network surrogate model is used to suggest new trial solutions in a multi-objective optimization task. The surrogate model is then updated with the new solutions, and this process is repeated until the final optimized solution is obtained. We apply this approach to optimize the nonlinear beam dynamics of the SPEAR3 storage ring, where sextupole knobs are adjusted to simultaneously improve the dynamic aperture and the momentum aperture. The approach is shown to converge to the Pareto front considerably faster than the genetic and particle swarm algorithms.