NAPI-MPC: Neural Accelerated Physics-Informed MPC for Nonlinear PDE Systems

The control of systems governed by nonlinear partial differential equations (PDEs) can present substantial challenges for traditional linear model predictive control (MPC) approaches. Data-driven MPC has emerged as a solution for dealing with unknown nonlinear dynamics, but issues regarding safety guarantees, accuracy, and data efficiency remain a concern. We propose NAPI-MPC: a novel physics-informed scenario-based MPC approach for control of nonlinear PDE systems with partially unknown dynamics. Unlike other physics-informed learning methods that require extensive knowledge on the governing equations, our NAPI-MPC leverages distributed Port-Hamiltonian systems as a generalized, energy-based representation of the PDE dynamics, in which the Hamiltonian is modeled and learned by a Gaussian process. The Bayesian nature of the Gaussian process enables the drawing of scenario samples that are used in scenario-based predictive control to determine the optimal control action for the PDE system. To ensure applicability in time-sensitive contexts, we leverage a neural network as proxy for the MPC controller, trained offline on states and optimal control actions to enable fast inference for real-time operation.