Disclosed are devices and methods of detecting and mitigating oscillatory instabilities in systems with turbulent flow, such as thermoacoustic, aeroacoustic or aeroelastic equipment. The system includes a sensor array for measuring one or more parameters of an operating equipment S, and an analysis and prediction unit. The analysis and prediction unit is configured to estimate a state tensor to identify a state of the equipment indicating stable operation or impending oscillatory instability. The system further includes an actuator array configured to implement a control action to promote stable operation of the equipment. Methods for robust prediction of the state of stability are also disclosed. A neural ordinary differential equation (ODE) method of predicting stability or instability is disclosed, involving forming a neural network that incorporates an equation characteristic of the operational state. The invention further discloses a hybrid convolutional neural network based prediction method for stability.