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This paper presents a framework that integrates machine learning (ML) and optimization, accelerating inductor design workflows to achieve minimized component size and power loss for a given electrical specification. The framework is implemented using Ansys OptiSLang, which leverages ML to predict key inductor characteristics under various operating conditions. The ML models are trained as the surrogate model of core loss, copper loss, and inductance. The core loss ML is trained by experimentally measured data in a certain range of geometric conditions. Measured results of core loss are used for training to make sure the reliable data as the ground truth. Moreover, the copper loss and inductance ML is trained by the simulated results from 3D finite element analysis (FEA). These ML models are applied for inductor design optimization to rapidly and accurately predict the design values, finally find the optimized design in specified constraints and objectives in a given design space. The framework significantly increases the efficiency of design process by replacing FEA simulation, which is highly timeconsuming while considering large number of design conditions. The training results exhibit high accuracy, with scores exceeding 0.99, and demonstrate strong consistency between the predicted design surfaces and ground truths. Furthermore, optimization result produces a twodimensional Pareto front, revealing optimal solutions of minimized inductor footprint and total loss, while maintaining target inductance within the specified tolerance.