描述肿瘤动力学的ODE约束优化问题建模及通过序列物理信息神经网络的数值近似
Modeling of an ODE-constrained optimization problem describing tumor dynamics, and numerical approximation via sequential physics-informed neural networks
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中文总结 AI 辅助
该研究针对氧气影响的胶质母细胞瘤生长的ODE约束最优控制问题，采用序列物理信息神经网络（PINNs）结合梯度下降法求解，验证了其数值近似效果优于经典求解器及传统PINNs。
中文摘要 AI 辅助
本文研究与氧气影响的胶质母细胞瘤生长的ODE模型相关的最优控制问题。该模型考虑了描述化疗和抗血管生成疗法的两个控制变量。成本泛函旨在减少肿瘤生长、使氧气浓度接近期望数值，并惩罚疗法的使用。我们求解该最优控制问题，证明全局最优控制的存在性，并通过庞特里亚金极小值原理推导一阶必要最优性条件。对于数值近似，我们使用物理信息神经网络（PINNs）求解状态和伴随系统，结合带Armijo线搜索的梯度下降法处理控制变量。针对PINNs的策略，我们考虑文献[15]提出的方法：将时域分解为多个子区间，每个子区间使用不同的神经网络，并在连续时间子区间之间施加连续性条件。该策略称为时域序列PINN公式，包含软约束和硬约束版本，我们将其与经典求解器及传统PINNs的对应近似结果进行比较。
英文摘要
In this paper, we study an optimal control problem related to an ODE model of glioblastoma growth influenced by the oxygen. The model considers a couple of controls describing the chemotherapy and antiangiogenic therapies. The cost functional aims to reduce the tumor growth, bring the oxygen concentration close to a desired value, and penalize the use of therapies. We solve the optimal control problem, proving the existence of a global optimal control and deriving first-order necessary optimality conditions through the Pontryagin Minimum Principle. For the numerical approximation, we use Physics-Informed Neural Networks (PINNs) to solve the state and adjoint systems, together with a gradient descent method with Armijo line search for the controls. To address the strategy of PINNs we consider the methodology proposed in [15], making a decomposition of the time domain into several subintervals, using different neural networks in each subinterval and enforcing continuity conditions between successive time subintervals. This strategy, called sequential PINN formulations in time, including soft and hard-constrained versions, is compared with the corresponding approximation results of classical solvers and traditional PINNs counterparts.
