A deep learning algorithm for the stable manifolds of the Hamilton-Jacobi equations

In this paper, we propose a deep learning method to approximate the stable manifolds of the Hamilton-Jacobi equations from nonlinear control systems, and numerically compute optimal feedback controls. Instead of discretizing the phase space, the neural network (NN) is trained on the set of randomly samples firstly and then is refined on enlarged sample set by adaptively generating samples near the points with large errors after the previous training round. Such kind of data generation may make the training more effective. Since our algorithm is meshfree basically, it has a potential to apply to various high-dimensional nonlinear systems. We illustrate the effectiveness of our method by swinging up and stabilizing the Reaction Wheel Pendulums.