# Universal Differential Equation

Optimal control of tumor-immune dynamics is usually solved by a forward-backward sweep on the Pontryagin necessary conditions, but that iteration can cycle between two saturation patterns instead of converging when the control saturates. Representing the control as a neural network and the cost functional as the training loss makes every stationary point of the loss satisfy the continuous first-order necessary conditions to a discretisation-controlled accuracy. On a 4×4 grid of scenarios from a tumor-immune model, the standard sweep failed on 14 of 16 scenarios, while network training reached a stable optimum on every scenario with a final cost 2 to 10 times lower than the converged damped sweep.

Optimal control of tumor-immune dynamics is usually solved by a forward-backward sweep on the Pontryagin necessary conditions, but that iteration can cycle between two saturation patterns instead of converging when the control saturates. Representing the control as a neural network and the cost functional as the training loss makes every stationary point of the loss satisfy the continuous first-order necessary conditions to a discretisation-controlled accuracy. On a 4×4 grid of scenarios from a tumor-immune model, the standard sweep failed on 14 of 16 scenarios, while network training reached a stable optimum on every scenario with a final cost 2 to 10 times lower than the converged damped sweep.