AI 中文总结
本研究针对半线性抛物方程Dirichlet边界控制问题，提出并分析了直接与间接两种物理信息神经网络方法，建立了间接方法的条件误差估计框架，并通过数值实验验证了其行为。
AI 中文摘要
我们研究了带有Tikhonov正则化的半线性抛物方程Dirichlet边界控制的物理信息神经网络（PINNs）。考虑了两种方法。直接PINN通过独立的网络参数化状态和控制，并最小化跟踪目标的惩罚形式。间接PINN则通过无约束网络表示状态、伴随和控制，联合训练以满足一阶最优性系统，其中状态-控制耦合以及齐次伴随边界和终端条件作为软惩罚项施加，而非通过架构强制。对于间接公式，我们开发了一个误差估计框架，将总误差分解为逼近、优化、求积以及软边界/终端约束贡献。在最优解正则性和相容性、网络逼近性、软约束残差的均匀Hölder控制以及参考最优性系统解的局部邻域的常设假设下，我们推导了一个定量线性化稳定性估计和一个条件局部非线性残差到误差估计，并构造了一个具有条件可靠性界限的可计算残差指示器。在两个制造测试问题上的数值实验——一个三次反应扩散方程和一个具有非平凡边界控制和伴随的线性方程——说明了直接和间接公式的行为。
英文摘要
We study physics-informed neural networks (PINNs) for the Dirichlet boundary control of a semilinear parabolic equation with Tikhonov regularization. Two approaches are considered. A direct PINN parameterizes the state and control by separate networks and minimizes a penalized form of the tracking objective. An indirect PINN instead represents the state, adjoint, and control by unconstrained networks trained jointly to satisfy the first-order optimality system, with the state-control coupling and the homogeneous adjoint boundary and terminal conditions imposed as soft penalty terms rather than enforced architecturally. For the indirect formulation we develop an error estimation framework that decomposes the total error into approximation, optimization, quadrature, and soft boundary/terminal-constraint contributions. Under standing assumptions on optimal-solution regularity and compatibility, network approximability, uniform Hölder control of the soft-constraint residuals, and a local neighborhood of the reference optimality-system solution, we derive a quantitative linearized stability estimate and a conditional local nonlinear residual-to-error estimate, and construct a computable residual indicator with a conditional reliability bound. Numerical experiments on two manufactured test problems - a cubic reactiondiffusion equation and a linear equation with a nontrivial boundary control and adjoint - illustrate the behavior of the direct and indirect formulations.
