ExAMPC: the Data-Driven Explainable and Approximate NMPC with Physical Insights Jean Pierre Allamaa 1, 2 and Panagiotis Patrinos 2 and Tong Duy Son 1 Abstract -- Amidst the surge in the use of Artificial Intelligence (AI) for control purposes, classical and model-based control methods maintain their popularity due to their transparency and deterministic nature. However, advanced controllers like Nonlinear Model Predictive Control (NMPC), despite proven capabilities, face adoption challenges due to their computational complexity and unpredictable closed-loop performance in complex validation systems. This paper introduces ExAMPC, a methodology bridging classical control and explainable AI by augmenting the NMPC with data-driven insights to improve the trustworthiness and reveal the optimization solution and closed-loop performance's sensitivities to physical variables and system parameters. By employing a low-order spline embedding to reduce the open-loop trajectory dimensionality by over 95%, and integrating it with SHAP and Symbolic Regression from eXplainable AI (XAI) for an approximate NMPC, we enable intuitive physical insights into the NMPC's optimization routine. The prediction accuracy of the approximate NMPC is enhanced through physics-inspired continuous-time constraints penalties, reducing the predicted continuous trajectory violations by 93%. ExAMPC enables accurate forecasting of the NMPC's computational requirements with explainable insights on worst-case scenarios. Experimental validation on automated valet parking and autonomous racing with lap-time optimization NMPC, demonstrates the methodology's practical effectiveness in real-world applications. I. INTRODUCTION Linear Model Predictive Control (MPC) stands out for its inherent explainability, allowing precise analysis of the instantaneous open-loop (OL) prediction and closed-loop (CL) system behavior. However, this clarity on stability and performance diminishes with complex systems, such as chaotic dynamics or those involving a plant model that is more complicated than the linear prediction model in the MPC.

Despite the growing availability of sensing and data in general, we remain unable to fully characterise many in-service engineering systems and structures from a purely data-driven approach. The vast data and resources available to capture human activity are unmatched in our engineered world, and, even in cases where data could be referred to as ``big,'' they will rarely hold information across operational windows or life spans. This paper pursues the combination of machine learning technology and physics-based reasoning to enhance our ability to make predictive models with limited data. By explicitly linking the physics-based view of stochastic processes with a data-based regression approach, a spectrum of possible Gaussian process models are introduced that enable the incorporation of different levels of expert knowledge of a system. Examples illustrate how these approaches can significantly reduce reliance on data collection whilst also increasing the interpretability of the model, another important consideration in this context.

In this workshop we will explore how to use physical intuition and ideas to design new classes of machine learning (ML) algorithms. Physics-inspired sampling algorithms could be used to train ML structures or sample the hyper-parameter space (e.g. Additionally, physics-based models such as Ising/Potts models or energy-based models have influenced ML inference frameworks such as Markov Random Fields and Restricted Boltzmann Machines, and we want to continue the discussion to facilitate this innovation transfer. Finally, physical insight could be used to enhance learning in the situation of scarce data by enforcing smoothness, differentiability or other physical properties relevant to a given problem. We will also explore the use of Koopmans' theorem to design learning algorithms for dynamical systems.