Reduced-order modeling using the Volterra series approach has been successfully applied in the past decades to weakly nonlinear aerodynamic and aeroelastic systems. However, aspects regarding the identification of the kernels associated with the convolution integrals of Volterra series can profoundly affect the quality of the resulting reduced-order model (ROM). An alternative method for their identification based on artificial neural networks is evaluated in this work. This relation between the Volterra kernels and the internal parameters of a time-delay neural network is explored for the application in the reduced-order modeling of nonlinear unsteady aerodynamic loads. An impulse-type Volterra-based ROM is also under consideration for comparison. All aerodynamic data used for the construction of the reduced-order models are obtained from computational fluid dynamics (CFD) simulations of the NACA 0012 airfoil using the Euler equations. Prescribed inputs in pitch and in plunge degrees of freedom at different free-stream Mach numbers are used to evaluate the range of applicability of the obtained models. For weakly nonlinear test cases, the modeling performance of the neural network Volterra ROM was comparable to the impulse-type ROM. Additional accuracy and adequate modeling of stronger nonlinearities, however, could only be attained with the inclusion of the neural network kernels of higher-order in the Volterra ROM. A generic expression is derived for the kernel function of -order from the internal parameters of a time-delay neural network.