A Physics-Informed Neural Networks-Based Model Predictive Control Framework for $SIR$ Epidemics

This work introduces a physics-informed neural networks (PINNs)-based model predictive control (MPC) framework for susceptible-infected-recovered (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SIR$</tex-math></inline-formula>) spreading models. Existing studies in MPC design for epidemic control often assume either 1) measurable states of the dynamics, where the parameters are learned, or 2) known parameters of the model, where the states are learned. In this work, we address the joint real-time estimation of states and parameters within the MPC framework using only noisy infected states, under the assumption that 1) only the recovery rate is known, or 2) only the basic reproduction number is known. Under the first assumption, we propose MPC-PINNs and two novel PINNs algorithms, all of which are integrated into the MPC framework. First, we introduce MPC-PINNs, which are designed for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SIR$</tex-math></inline-formula> model with control. We then propose log-scaled PINNs (MPC-LS-PINNs), which incorporate a log-scaled loss function to improve robustness against noise. Next, we present split-integral PINNs (MPC-SI-PINNs), which leverage integral operators and state coupling in the neural network training process to effectively reconstruct the complete epidemic state information. Building upon these methods, we further extend our framework for the second assumption. We establish the necessary conditions and extend our PINNs algorithms, where MPC-SI-PINNs are simplified as split-PINNs (MPC-S-PINNs). By incorporating these algorithms into the MPC framework, we simultaneously estimate the epidemic states and parameters while generating optimal control strategies. Comparative experiments against an extended Kalman filter, ideal MPC, and different neural network structures, together with validation on real COVID-19 data from Italy, demonstrate the effectiveness of the proposed methods under different settings.