<p>The Koopman operator theory is a widely recognized approach for data-driven modeling because it provides a global linearization representation of nonlinear dynamical systems. Designing the observable function and its inverse is the key to obtaining the finite-dimensional approximation of the Koopman operator. However, most reported works exhibit potential shortcomings, such as introducing unnecessary assumptions regarding the form of the observable and the fact that its inverses can only be obtained approximately. To address these issues, we propose a novel data-driven modeling approach for nonlinear, non-autonomous systems, known as the Invertible Koopman Network (IKN). This approach utilizes the Invertible Neural Network (INN) to simultaneously find the optimal observable function and its explicit inverse, thereby facilitating more accurate finite-dimensional approximations of the Koopman operator and achieving lossless reconstruction. Furthermore, we explore integrating the IKN with Model Predictive Control (MPC) to address the optimal control problem for unknown nonlinear systems. By leveraging the linear dynamics in the observable space, the nonconvex optimization problem in classical MPC can be transformed into a simple quadratic programming problem, ensuring high computational efficiency. The method is evaluated on various nonlinear systems and demonstrates superior performance to other Koopman operator-based approaches.</p>