In the context of nonlinear multi-scale problems, the inverse estimation of macroscopic distribution of some microscopic parameters based on macroscopic measurements poses significant challenges. These challenges arise from (1) the high computational cost to solve the complex forward problem, and (2) the need for derivatives of the complex multi-scale forward model, which combines macro-scale and micro-scale simulations, both of which are typically nonlinear. To address these challenges, we propose a novel approach that combines ensemble Kalman inversion for derivative-free inverse estimation and a physics-informed deep learning-based model order reduction (DL-MOR) to accelerate the micro-scale simulation. We evaluate the performance of our method using a non-linear hyper-elastic model. The results demonstrate the effectiveness of DL-MOR in significantly speeding up the micro-scale simulation and enabling relatively accurate estimation of the microscopic parameter using only macro-scale boundary measurements.