From forward-backward sweep to backpropagation for optimal control in tumor-immune dynamics

Optimal control problems in mathematical biology are commonly solved by the forward-backward sweep on the Pontryagin's Maximum Principle necessary conditions. The sweep has known stability issues at constraint-saturated optima. We frame the same problem within the universal differential equation framework. A neural network represents the control. The cost functional serves as the training loss. We prove that any stationary point of the training loss satisfies the continuous first-order necessary conditions to a discretisation-controlled accuracy. On a grid of bifurcation-parameter and initial-condition scenarios drawn from a tumor-immune dynamics model, the standard sweep fails on most scenarios in the nominal regime. Neural network training reaches a stable optimum on every scenario, with a lower final cost than the converged damped sweep iterate. The pipeline accommodates cost functional variants by changing only the loss function. A Sobol sensitivity analysis confirms robustness of the trained schedule to perturbations of the cost weights.