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

Wenjing Zhang, Wandi Ding, Huaiping Zhu

TL;DR

The paper addresses numerical instability in solving optimal control problems under dynamic constraints by reframing the problem as a universal differential equation (UDE) in which the control is a neural network NN(t,p). By training NN parameters p via backpropagation, the approach yields controls that satisfy Pontryagin’s Maximum Principle (PMP) while avoiding unstable backward adjoint integrations. A cancer immunotherapy case study demonstrates stable, effective optimization of immunotherapy and chemotherapy strategies despite complex bifurcations, illustrating the method’s robustness. The authors argue that this framework is broadly applicable to high-dimensional, nonlinear dynamical systems across biology, epidemiology, engineering, and finance, providing a scalable tool for state-constrained optimal control problems.

Abstract

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 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.

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

TL;DR

The paper addresses numerical instability in solving optimal control problems under dynamic constraints by reframing the problem as a universal differential equation (UDE) in which the control is a neural network NN(t,p). By training NN parameters p via backpropagation, the approach yields controls that satisfy Pontryagin’s Maximum Principle (PMP) while avoiding unstable backward adjoint integrations. A cancer immunotherapy case study demonstrates stable, effective optimization of immunotherapy and chemotherapy strategies despite complex bifurcations, illustrating the method’s robustness. The authors argue that this framework is broadly applicable to high-dimensional, nonlinear dynamical systems across biology, epidemiology, engineering, and finance, providing a scalable tool for state-constrained optimal control problems.

Abstract

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 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.

Paper Structure

This paper contains 16 sections, 1 theorem, 42 equations, 3 figures, 2 tables.

Table of Contents

  1. Deep Learning for Optimal Control: A Pontryagin’s Maximum Principle Perspective
  2. Background
  3. Deep learning and Deep neural network
  4. Training a neural network
  5. Universal differential equations
  6. Optimal control problem solved by deep neural network
  7. Formulating the optimal control problem as a model of UDEs
  8. Gradient decent
  9. The necessary conditions
  10. An application on cancer immunotherapy
  11. The model for combination of cancer immunotherapy
  12. Optimal treatment strategies for combination of immunotherapy
  13. Immunotherapy strategy inducing responses
  14. Immunotherapy strategy inducing no response
  15. Combination of chemotherapy with immunotherapy
  16. ...and 1 more sections

Key Result

Theorem 1

If the neural network parameter set $p^\ast$ satisfies $\mathcal{L}(p^\ast) = 0$, which gives the optimal control $u^\ast(t) = NN(t, p^\ast)$ and the corresponding optimal state $y^\ast(t)=y(t, p^\ast)$ for problem eqn_NN26, then there exists a piecewise differentiable adjoint variable $\lambda(t)$ for all controls $u(t) = NN(t, p)$ at each time $t$, where the Hamiltonian $H$ is defined as and $

Figures (3)

  • Figure 1: Optimal control strategy under a quadratic cost penalty with moderate-to-high dosing range for co-stimulation and co-suppression therapy. The control successfully reduces cancer cell and non-effector T cell populations while enhancing effector T cells throughout the treatment period. This illustrates an effective immunotherapy response under optimized conditions. Optimal control with quadratic penalty in \ref{['eqn51']}, where $a=1$, $b = 10$, $c=100$, $c_{1}=2$, $c_{2}=1$, $d_1=1$, $d_2=10$, $d_3=100$.
  • Figure 2: Suboptimal control using low-dose immunotherapy. The limited dosing range fails to control cancer growth, resulting in persistent tumor burden, elevated antigen and non-effector T cell levels, and poor effector T cell activation. This scenario reflects a non-responder case despite treatment. Optimal control with quadratic penalty in \ref{['eqn51']}, where $a=1$, $b = 10$, $c=100$, $c_{1}=2$, $c_{2}=1$, $d_1=1$, $d_2=10$, $d_3=100$.
  • Figure 3: Optimal combination of low-dose chemotherapy with immunotherapy. The control strategy effectively reduces cancer cell burden and substantially enhances the effector T cell population. Here $a=1$, $b = 1$, $c=100$, $c_{1}=2$, $c_{2}=1$, $c_{3}=1$,$e_{1}=2$, $e_{2}=1$, $d_1=1$, $d_2=1$, $d_3=100$.

Theorems & Definitions (1)

  • Theorem 1