Today artificial neural networks are very useful to solve complex dynamic games of various types, i.e., to approximate optimal strategies with sufficient accuracy. Exemplarily four synthesis approaches for the solution of zero-sum, noncooperative dynamic games are outlined and discussed. Either value function, adjoint vector components or optimal strategies can be synthesized as functions of the state variables. In principle all approaches enable the solution of dynamic games. Nevertheless every approach has advantages and disadvantages which are discussed. The neural network training usually is very difficult and computationally very expensive. The coarse-grained parallelization FAUN 1.0-HPC-PVM of the advanced neurosimulator FAUN uses PVM subroutines and runs on heterogeneous and decentralized networks interconnecting general-purpose workstations, PCs and also high-performance computers. Computing times of days, weeks or months can be cut down to hours. An enhanced cornered rat game — formulated and analyzed in 1993 — serves as an example. Optimal strategies for cat and rat are synthesized. For this purpose open-loop representations of optimal strategies on an equidistant grid in the state space are used. An important end game modification is presented.