Sobolev Training for Data-efficient Approximate Nonlinear MPC

Model predictive control is a powerful advanced control technique to deal with complex nonlinear systems with constraints. Despite recent advances in computing hardware and optimization algorithms, solving the nonconvex optimization problems associated to nonlinear model predictive control in real time can still be intractable. There are several possibilities to alleviate the challenge of computational complexity. An increasingly popular approach is to use neural networks to approximate the mapping between current system state and the corresponding optimal control input, which is implicitly defined by an optimization problem. These approaches typically need a large amount of data. Although the data can be generated offline by sampling the state space and solving many model predictive control problems in advance, it can become intractable especially for increasing system dimensions. In this work, we propose to use the parametric sensitivities of the nonlinear optimization problems as additional information that is provided to the neural network during training. In this way, the neural network not only fits the solution of the optimization problems itself but also its gradients. This training method, usually called Sobolev training, can lead to a higher accuracy in the approximation for the same amount of data. We illustrate the potential of the approach with a simulation study of a nonlinear chemical reactor. Copyright (c) 2023 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)