This study presents a system of physics-informed neural networks (PINNs) for determining the bending rigidities of straight Euler-Bernoulli beams with variable rigidities. The proposed PINN system consists of two artificial neural networks that collaborate to determine the bending rigidity of a beam from known displacement responses and rigidity data at a few sampling points. Known rigidity data at a few points is crucial to prevent potential inaccuracies arising from the non-uniqueness of inverse analysis solutions. Both networks receive a position on the beam axis as input, with one network outputting the transverse displacement at that position and the other network outputting the rigidity at the same position. The two networks are jointly trained under a shared set of loss functions. The beam’s governing equation and boundary conditions are incorporated into the system via specific loss functions. Additionally, known transverse displacements and rigidity data are used to train the networks through their respective loss functions. The system’s design is tested using a fixed-end beam with variable rigidity subjected to a uniformly distributed load. The achieved accuracy in predicting rigidity variation demonstrates the capability of PINNs for inverse analysis in structural mechanics.