In this paper, we introduce a new topology of fuzzy polynomial neural networks (FPNNs) that is based on a genetically optimized multilayer perceptron with fuzzy polynomial neurons (FPNs). The study offers a comprehensive design methodology involving mechanisms of genetic optimization, especially those exploiting genetic algorithms (GAs). Let us recall that the design of the "conventional" FPNNs uses an extended group method of data handling (GMDH) and uses a fixed scheme of fuzzy inference (such as simplified, linear, and regression polynomial fuzzy inference) in each FPN of the network. It also considers a fixed number of input nodes (as being selected in advance by a network designer) at FPNs (or nodes) located in each layer. However such design process does not guarantee that the resulting FPNs will always result in an optimal networks architecture. Here, the development of the FPNN gives rise to a structurally optimized topology and comes with a substantial level of flexibility which becomes apparent when contrasted with the one we encounter in the conventional FPNNs. The design of each layer of the FPNN deals with its structural optimization involving a selection of preferred nodes (or FPNs) with specific local characteristics (such as the number of input variables, the order of the polynomial forming a consequent part of fuzzy rules and a collection of the specific subset of input variables) and addresses detailed aspects of parametric optimization. Along this line, two general optimization mechanisms are explored. The structural optimization is realized via GAs. In case of the parametric optimization we proceed with a standard least square method-based learning. Through the consecutive process of such structural and parametric optimization, an optimized and flexible fuzzy neural network becomes generated in a dynamic fashion. To evaluate the performance of the genetically optimized FPNN (gFPNN), we experimented with two time series data (gas furnace and chaotic time series) as well as some synthetic data. A comparative analysis reveals that the proposed FPNN exhibits higher accuracy and superb predictive capability in comparison to some previous models available in the literature.