Universal differential equations for optimal control problems and its application on cancer therapy

This paper highlights a parallel between the forward–backward sweeping method for optimal control and deep learning training procedures. We reformulate a classical optimal control problem, constrained by a differential equation system, into an optimization framework that uses neural networks to represent control variables. We demonstrate that this deep learning method adheres to Pontryagin’s Maximum Principle and mitigates numerical instabilities by employing backward propagation instead of a backward sweep for the adjoint equations. As a case study, we solve an optimal control problem to find the optimal combination of immunotherapy and chemotherapy. Our approach holds significant potential across various fields, including epidemiology, ecological modeling, engineering, and financial mathematics, where optimal control under complex dynamic constraints is crucial.