发表机构
University of Bath; Institute of Computational and Applied Mathematics, Foundation for Research and Technology (FORTH)（巴斯大学; 计算与应用数学研究所，研究与技术基金会）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对输运方程最优控制问题，提出残差最小二乘泛函，其极小化子等价于KKT系统解，提供连续级后验误差估计，并用神经网络参数化求解，数值验证了方法的有效性。
AI 中文摘要
我们考虑一个由二维或三维空间中的稳态线性输运方程控制的最优控制问题，其动机源于放射治疗计划中的光子输运。控制量为流入边界源，观测算子可追踪角通量或其角积分，后者作为简化剂量替代量。相应的最优性系统由具有角度依赖性的输运-伴随对组成，并通过边界最优性条件耦合。我们构造了一个残差最小二乘泛函，其极小化子与Karush--Kuhn--Tucker系统的解一致，并证明该泛函控制状态和伴随的$L^2$误差以及控制的通量加权$L^2$误差。这给出了一个无需离散化的连续级后验估计。一个额外的未加权最优性残差在目标范数下认证控制量。该估计对每个容许近似都成立，并考虑了控制惩罚和输运迹中不同的边界测度，无需$|\omega\cdot n|$的正下界。然后我们使用神经网络参数化状态、伴随和控制，并在神经网络空间上使用适当的求积法进行最小化。蒙特卡洛公式将每个残差积分替换为加权样本均值，其求积误差进入后验估计。我们在近似、求积和优化的显式假设下证明收敛性，并推导出总误差以近似、求积和优化误差表示的界。二维和三维数值示例说明了残差-误差关系以及残差框架在场跟踪控制问题中的灵活性，包括规则制造解。
英文摘要
We consider an optimal control problem governed by a stationary linear transport equation in two or three spatial dimensions, motivated by photon transport in radiotherapy treatment planning. The control is an inflow boundary source, and the observation operator may track either the angular flux or its angular integral, the latter serving as a simplified dose surrogate. The associated optimality system consists of a transport--adjoint pair with angular dependence, coupled through a boundary optimality condition. We formulate a residual least-squares functional whose minimisers coincide with solutions of the Karush--Kuhn--Tucker system and show that the functional controls the $L^2$ errors in the state and adjoint and the flux-weighted $L^2$ error in the control. This yields a continuum-level a posteriori estimate without recourse to discretisation. An additional unweighted optimality residual certifies the control in the norm of the objective. The estimate holds for every admissible approximation and accounts for the different boundary measures in the control penalty and transport traces, without a positive lower bound on $|ω\cdot n|$. We then parameterise the state, adjoint and control using neural networks and minimise over a neural network space using appropriate quadrature. The Monte Carlo formulation replaces each residual integral by a weighted sample mean, whose quadrature error enters the a posteriori estimate. We prove convergence under explicit assumptions on approximation, quadrature andoptimisation and derive bounds on the total error in terms of approximation, quadrature and optimisation error. Numerical examples in two and three dimensions illustrate the residual--error relation and the flexibility of the residual framework for field-tracking control problems, including regular manufactured solutions.
