The development of ultra-scaled semiconductor devices requires device simulators that do not rely on empirical models. The Monte Carlo (MC) simulation is among the most suitable approaches for this purpose. In conventional MC simulation, device characteristics are typically obtained by repeatedly solving the Poisson equation in discrete space and the Boltzmann transport equation in continuous space. To avoid this inefficiency, we have developed a Poisson solver using Physics-Informed Neural Networks (PINNs) to directly compute the electric field from the charge distribution and implemented it in our MC simulator. In this study, we aim to enable accurate predictions even under untrained conditions by introducing an online transfer learning using the variational form of the Poisson equation. The machine learning model consists of two components: one that represents the charge density distribution and another that maps the electron position to the output location. The former employs a Convolutional Neural Network (CNN) to extract features, while the latter represents the electron position using basis functions. By combining these two components, a mesh-free representation of the output function is achieved. This framework is based on the Deep Operator Network (DeepONet), an operator-learning method. For performance evaluation, the constructed model was applied to a Si GAA FET with a gate length of 10 nm and a channel cross section of 5 nm × 5 nm. For the trained device structure, the relative error of the I-V characteristics was 2.3%. For untrained device structure, the prediction was challenging due to non-physical growth and reduction in the number of electrons. Therefore, we introduced online learning with the variational form of the Poisson equation and achieved a relative error of 4.3%. The computational speed depends on the mesh resolution; for 13,064 mesh points, approximately 7× acceleration was achieved.