# Model Predictive Evolutionary Temperature Control via Neural-Network-Based Digital Twins
## 1 Introduction  
Model Predictive Control (MPC) represents an advanced control method that distinguishes itself by employing a mathematical system model to anticipate future system behavior and makes proactive decisions in response to expected deviations from a set point. Unlike traditional control methods that reactively rely on past and present system behavior, MPC combines the principles of feedback control and numerical optimization to achieve optimal control outcomes. By continuously optimizing the system model in real-time, MPC determines an optimal trajectory for the manipulated variable. The essential constituents of MPC encompass three fundamental elements: (i) a predictive model capturing the dynamics of the controlled system, (ii) a trajectory to be tracked and (iii) an optimal controller achieved through continuous optimization. Notably, only the initial value of the optimized output trajectory is implemented in the system, with the prediction and optimization process being repeated at each time step. This adaptive approach enables MPC to dynamically respond to changing conditions and deliver accurate control by considering future system behavior. As the MPC revolves around the iterative solution of an optimization problem, it necessitates a system model, as well as a mathematical description of the correspondingcontrol law [1]. These models are traditionally derived from first principles or obtained through system identification techniques using measured data [2]. However, an attractive alternative approach is to directly implement an MPC controller using solely measured data, without relying on prior knowledge of an accurate model [3]. The data-driven approach particularly offers practical advantages in scenarios where (i) obtaining a precise model may be challenging, time-consuming and/or expensive to evaluate; (ii) the process is ill-defined; (iii) the process is time-variant or stochastic in nature. Herein, recent advancements in machine learning facilitates the creation of input-output-based digital twin models (DT) that do not require a thorough mathematical description of the process [4], enabling the implementation of intelligent controllers that can adapt to the system dynamics and change their control policies in real time. These techniques allow for treating the system and the physical process within itself as a black box [5], while maintaining good accuracy, by approximating the mapping from the input to the output space [6]. Examples of using machine learning in MPC cover a broad range of applications. In one of the early works, [PERSON] and [PERSON] combined the linear quadratic regulator with unsupervised clustering (k-nearest neighbor) [7]. Other shallow learning applications include Gaussian process modeling for the safe exploration of dynamical systems [8], the optimal energy management in commercial building micro-grids [9], heating, ventilation and air-conditioning (HVAC) control of a hospital surgery center [10]; Bayesian regression for safe predictive learning control [11], statistical time series modeling (ARIMA) for optimal energy management [9], random forests for HVAC systems [12] and support vector machines for milling [13]. Feed-forward neural network (NN) applications within an MPC framework can also be found in various disciplines. [PERSON] et al. [14] implemented an NN to regulate set point changes in a polyethylene reactor, resulting in a 30% improvement in transition speed and a significant reduction in controlled variable fluctuations. The work of [PERSON] and [PERSON] [15] is another early example combining NNs with MPC to control the shaft speed of a gas turbine engine. Gas turbine models were created via a nonlinear autoregressive moving average model with exogenous inputs (NARMAX) and neural networks, enabling an improved control performance compared to PID controllers through various step tests. [PERSON] et al. [16] employed NNs to develop a supervisory MPC for residential heating, ventilation, and air conditioning(HVAC) systems.