From forward-backward sweep to backpropagation for optimal control in tumor-immune dynamics.
- 1Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA. [email protected].
- 2Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN, USA.
- 3Department of Mathematics and Statistics, York University, Toronto, ON, Canada.
Abstract
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.
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BibTeX
@article{zhang2026from,
title = {{From forward-backward sweep to backpropagation for optimal control in tumor-immune dynamics.}},
author = {Wenjing Zhang and Wandi Ding and Huaiping Zhu},
journal = {Journal of mathematical biology},
year = {2026},
month = oct,
doi = {10.1007/s00285-026-02475-2},
url = {https://europepmc.org/article/MED/42955091},
note = {CC BY},
}
RIS
TY - JOUR
TI - From forward-backward sweep to backpropagation for optimal control in tumor-immune dynamics.
AU - Wenjing Zhang
AU - Wandi Ding
AU - Huaiping Zhu
PY - 2026
DA - 2026/10/29
JO - Journal of mathematical biology
DO - 10.1007/s00285-026-02475-2
UR - https://europepmc.org/article/MED/42955091
AB - 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.
SN - 0303-6812
SN - 1432-1416
ER -