This paper proposes MoRPI-PINN, a novel method based on a Physically Informed Neural Network (PINN), that enables accurate mobile robot navigation even in the absence of satellite navigation or cameras. To address the drift problem of navigation solutions that arise when using only inertial sensors, we employ snake-like meandering motions to increase the inertial signal-to-noise ratio and reconstruct the mobile robot's position. By incorporating physical laws and constraints into the learning process, we provide an accurate and robust navigation solution, and experimental results demonstrate an accuracy improvement of over 85% compared to existing methods. This lightweight approach allows for implementation on edge devices and can be applied to general mobile robot applications.