Deep multi-input and multi-output operator networks method for optimal control of PDEs

Deep operator networks is a popular machine learning approach. Some problems require multiple inputs and outputs. In this work, a multi-input and multi-output operator neural network (MIMOONet) for solving optimal control problems was proposed. To improve the accuracy of the numerical solution, a physics-informed MIMOONet was also proposed. To test the performance of the MIMOONet and the physics-informed MIMOONet, three examples, including elliptic (linear and semi-linear) and parabolic problems, were presented. The numerical results show that both methods are effective in solving these types of problems, and the physics-informed MIMOONet achieves higher accuracy due to its incorporation of physical laws.