Energy management (EM) in renewable-integrated microgrids (MGs) with hybrid energy storage systems (HESS) is critical for ensuring operational reliability, reducing emissions, and minimizing costs. However, existing approaches often suffer from limited adaptability under dynamic conditions, reduced predictive accuracy, and insufficient renewable utilization. To overcome these limitations, this paper suggests a hybrid framework that integrates the Wombat Optimization Algorithm (WbOA) with an Augmented Physics-Informed Neural Network (APINN). In this framework, WbOA performs optimal scheduling of power distribution and charge–discharge operations of the HESS, while APINN enhances forecasting accuracy by embedding physical constraints such as power balance and storage dynamics into the learning process. Simulation results demonstrate that the suggested WbOA–APINN method outperforms benchmark techniques including PSO, GA, and recent hybrid models. Specifically, it achieves a system efficiency of 98.7 %, delivers 4.38 kWh of useful energy to the load over a 24-h horizon, reduces CO₂ emissions to 0.15 kg/kWh, and lowers operational cost to $124.5 per day by accounting for grid purchase, non-renewable generation, O&M, and battery degradation costs. These improvements highlight the synergistic benefits of integrating WbOA with APINN, offering a robust, scalable, and economically viable solution for intelligent EM in renewable-integrated MGs.