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This paper proposes Runge-Kutta neural disturbance observer to enhance the robustness of PID control of a system with general multicompartment lung mechanics. It is designed to observe the states of a particular type continous time, single-input single-output system where the states cannot be measured but can be observed through the single output and there exists parametric uncertainity or disturbance affecting the underlying system. It utilizes artificial neural network to estimate the disturbance online. Once an accurate disturbance estimation is obtained, it is incorporated in the system state equation and passed through the well-known Runge-Kutta integrator to predict the state values. Hence, the predicted states are obtained considering the disturbance and more robust state observation is achieved. The proposed observer is simple and easy to implement. Adaptation of the neural network is performed using gradient descent with an adaptive learning rate which guarantees convergence. The simulation results demonstrate that the proposed observer gains a significant success in enhancing the robustness of PID control at even high level of disturbance. Note that, multicompartment lung mechanics system is a stand-in model that can mimic the behavior of human lung. Thus, it is appropriate for hardware-in-the-loop simulation which opens a path to the real-patient-tests of mechanical respiratory systems in the future. Copyright (C) 2020 The Authors.