用于反应扩散偏微分方程输出反馈控制的DeepONet-LSTM神经算子
DeepONet-LSTM Neural Operator for Output Feedback Control of Reaction Diffusion PDEs
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中文总结 AI 辅助
本文提出混合DeepONet-LSTM神经算子，近似反应扩散偏微分方程输出反馈控制的因果边界算子，结合理论分析与改进损失函数，实现系统有效镇定。
中文摘要 AI 辅助
本文提出一种基于神经算子的方法，用于反应扩散偏微分方程的输出反馈边界镇定。经典的输出反馈反步设计需要针对每个反应系数求解控制核方程和观测器核方程，为避免计算这些核函数，本文将输出反馈控制律重构为因果边界算子，该算子可将反应系数和边界测量值映射为边界控制输入。本文提出一种混合DeepONet-LSTM神经算子来近似该因果算子，其中DeepONet对空间系数进行编码，LSTM捕捉测量历史的时间依赖关系。本文分析了边界算子的Lipschitz连续性，证明了采用学习得到的控制器时闭环系统的实用稳定性，还引入了改进的损失函数以提升学习得到的边界输入的时间正则性。数值结果表明，所提出的神经算子控制器可有效镇定该系统。
英文摘要
This paper presents a neural operator-based approach for the output feedback boundary stabilization of reaction diffusion PDEs. The classical output feedback backstepping design requires solving control and observer kernel equations for each reaction coefficient. To avoid computing these kernel functions, the output feedback control law is reformulated as a causal boundary operator that maps the reaction coefficient and the boundary measurement to the boundary control input. A hybrid DeepONet-LSTM neural operator is proposed to approximate this causal operator, where DeepONet encodes the spatial coefficient and LSTM captures the temporal dependence of the measurement history. We analyze the Lipschitz continuity of the boundary operator and prove the closed-loop practical stability with the learned controller. A modified loss is also introduced to improve the temporal regularity of the learned boundary input. Numerical results illustrate that the proposed neural operator controller effectively stabilizes the system.
