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The developments presented in this paper address the challenge of determining the optimal element positions in nonuniformly spaced broadband phased-array antennas in order to best meet desired performance criteria. Specifically, this is accomplished by introducing a new nature-based design technique that couples a robust genetic-algorithm (GA) optimizer with rapid neural-network (NN) estimation procedures. These provide performance criteria as functions of the element positions over the entire scanning range and bandwidth of operation. The objective of this GA-NN technique is to determine the optimal element positions for a broadband aperiodic linear phased-array antenna in order to minimize element VSWRs and sidelobe levels. The NN estimation procedures circumvent the need for computationally intensive full-wave numerical simulations during the optimization process, which would ordinarily render such an optimization task impractical. The effectiveness of the new GA-NN design synthesis technique is demonstrated by considering an example where a nonuniformly spaced linear phased array of ten stacked patch antennas is optimized for operation within a given bandwidth and scanning range.