We explore an application of the Physics-Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series to find optimal solutions to a few reference biharmonic problems of elasticity and elastic plate theory. Biharmonic relations are fourth-order partial differential equations (PDEs) that are challenging to solve using classical numerical methods and have not been addressed using PINNs. Our work highlights a novel application of classical analytical methods to guide the construction of efficient neural networks with a minimal number of parameters that are very accurate and fast to evaluate. In particular, we find that enriching the feature space using Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs.