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We propose, for the first time, neural space-mapping (NSM) optimization for electromagnetic based design. NSM optimization exploits our space-mapping (SM)-based neuromodeling techniques to efficiently approximate the mapping. A novel procedure that does not require troublesome parameter extraction to predict the next point is proposed. The initial mapping is established by performing upfront fine-model analyses at a reduced number of base points. Coarse-model sensitivities are exploited to select those base points. Huber optimization is used to train, without testing points, simple SM-based neuromodels at each NSM iteration. The technique is illustrated by a high-temperature superconducting quarter-wave parallel coupled-line microstrip filter and a bandstop microstrip filter with quarter-wave resonant open stubs.