AI 中文总结
研究提出结合Tikhonov正则化的PINNs框架解决抛物型PDEs终端状态跟踪最优控制问题，该问题不适定。理论上建立一致性结果和误差估计，数值实验表明该框架能准确实现目标终端状态，产生低能量和平滑轮廓控制量。
AI 中文摘要
在本研究中，我们提出了一个结合Tikhonov正则化的物理信息神经网络（PINNs）框架，以解决受抛物型偏微分方程（PDEs）约束的终端状态跟踪最优控制问题。该问题本质上是不适定的，因为无限多个分布式控制可能将系统驱动到相同的期望状态，所以正则化引导优化器趋向最小能量控制，恢复数值稳定性并产生平滑、具有物理意义的解。理论上，我们建立了一个一致性结果，表明在残差和求积近似假设下，PINNs最小化器几乎达到连续正则化目标，以及一个新的误差估计，用PINNs训练残差和正则化参数来界定学习到的控制与最小能量解的偏差。在线性热方程和非线性Burgers方程上的数值实验表明，与未正则化的基线相比，正则化的PINNs框架能准确实现目标终端状态，同时产生能量显著更低且轮廓更平滑的控制量。
英文摘要
In this study, we propose a Physics-Informed Neural Networks (PINNs) framework that incorporates Tikhonov regularization to solve terminal-state tracking optimal control constrained by parabolic partial differential equations (PDEs). This problem is inherently ill-posed, as infinitely many distributed controls may drive the system to the same desired state, so the regularization guides the optimizer toward the minimum-energy control, restoring numerical stability and yielding a smooth, physically meaningful solution. On the theoretical side, we establish a consistency result showing that PINNs minimizers nearly attain the continuous regularized objective under residual and quadrature approximation assumptions, and a novel error estimate that bounds the deviation of the learned control from the minimum-energy solution in terms of the PINNs training residuals and the regularization parameter. Numerical experiments on the linear heat equation and the nonlinear Burgers'equation demonstrate that the regularized PINNs framework accurately achieves the target terminal state while producing controls with significantly lower energy and smoother profiles compared to unregularized baselines.
Comments18 pages, 6 figures
