Deep Operator Neural Network Model Predictive Control

In this paper, we consider the design of model predictive control (MPC) algorithms based on deep operator neural networks (DeepONets) (Lu et al. 2021). These neural networks are capable of accurately approximating real- and complex-valued solutions (Jiang et al. 2024) of continuous-time nonlinear systems without relying on recurrent architectures. The DeepONet architecture is made up of two feedforward neural networks: the branch network, which encodes the input function space, and the trunk network, which represents dependencies on temporal variables or initial conditions. Utilizing the original DeepONet architecture (Lu et al. 2021) as a predictor within MPC for Multi-Input Multi-Output (MIMO) systems requires multiple branch networks, to generate multi-output predictions, one for each input. Moreover, to predict multiple time steps into the future, the network has to be evaluated multiple times. Motivated by this, we introduce a multi-step DeepONet (MS-DeepONet) architecture that computes in one-shot multi-step predictions of system outputs from multi-step input sequences, which is better suited for MPC. We prove that the MS-DeepONet is a universal approximator in terms of multi-step sequence prediction. Additionally, we develop automated hyperparameter selection strategies and implement MPC frameworks using both the standard DeepONet and the proposed MS-DeepONet architectures in PyTorch. We compare MS-DeepONet, standard DeepONet, and LSTM-based controllers on learning and predictive control tasks for the Van der Pol oscillator and the quadruple tank process. The MS-DeepONet is also evaluated on a challenging cart–pendulum system, where it successfully learns swing-up and stabilization policies. Across the examples, MS-DeepONet outperforms standard DeepONet in prediction accuracy and control performance, and achieves significantly lower computation times than Long Short-Term Memory (LSTM) based MPC.