Amorphous and non-stoichiometric hafnium oxide (a-HfOx) systems are essential for advanced electronic applications due to their superior electrical properties. Simulating their atomic behaviors under electric fields ( \({E}_{{field}}\) ) is critical but challenging. Ab-initio molecular dynamics (AIMD) offer high accuracy but is computationally expensive, while classical MD lacks precision. To address this, we develop a charge equilibration integrated graph neural network (CIGNN) model that predicts atomic charge, energy, and force under \({E}_{{field}}\) conditions. Using the CIGNN model and AIMD datasets, we develop a CIGNN-based machine learning potential (CNMP) optimized for a-HfOx systems. The CNMP achieves quantum mechanical accuracy and effectively captures the atomic behaviors and dynamic properties of these systems across varying temperatures, densities, and \({E}_{{field}}\) conditions. We expect the CNMP to serve as a valuable tool for studying field-induced phenomena in complex systems and to provide a foundation for advancing innovations in electronic applications.
