To stabilize PDEs, feedback controllers require gain kernel functions, which are themselves governed by PDEs. Furthermore, these gain-kernel PDEs depend on the PDE plants' functional coefficients. The functional coefficients in PDE plants are often unknown. This requires an adaptive approach to PDE control, i.e., an estimation of the plant coefficients conducted concurrently with control, where a separate PDE for the gain kernel must be solved at each timestep upon the update in the plant coefficient function estimate. Solving a PDE at each timestep is computationally expensive and a barrier to the implementation of real-time adaptive control of PDEs. Recently, results in neural operator (NO) approximations of functional mappings have been introduced into PDE control, for replacing the computation of the gain kernel with a neural network that is trained, once offline, and reused in real-time for rapid solution of the PDEs. In this paper, we present the first result on applying NOs in adaptive PDE control, presented for a benchmark 1-D hyperbolic PDE with recirculation. We establish global stabilization via Lyapunov analysis, in the plant and parameter error states, and also present an alternative approach, via passive identifiers, which avoids the strong assumptions on kernel differentiability. We then present numerical simulations demonstrating stability and observe speedups up to three orders of magnitude, highlighting the real-time efficacy of neural operators in adaptive control. Our code (Github) is made publicly available for future researchers.

翻译：为稳定偏微分方程（PDE），反馈控制器需要增益核函数，而增益核函数本身由PDE支配。此外，这些增益核PDE依赖于PDE对象的泛函系数。PDE对象中的泛函系数通常是未知的，这需要采用自适应方法进行PDE控制，即在控制过程中同时估计对象系数，且每次更新对象系数函数估计值时，需在每个时间步长求解独立的增益核PDE。在每个时间步长求解PDE计算成本高昂，成为实时自适应PDE控制的实现障碍。近年来，神经算子（NO）对泛函映射的近似结果已被引入PDE控制领域，用于替代增益核的计算——通过离线训练一次、实时复用的神经网络快速求解PDE。本文首次将NO应用于自适应PDE控制，以含再循环的一维基准双曲型PDE为例展开研究。通过李雅普诺夫分析，我们在对象和参数误差状态下建立了全局稳定性，并提出了一种基于被动标识符的替代方法，避免了核可微性的严格假设。数值仿真结果验证了稳定性，并观测到高达三个数量级的加速效果，凸显了神经算子在自适应控制中的实时效能。我们的代码（Github）已公开，供后续研究者使用。