Neural operator approximations of the gain kernels in PDE backstepping has emerged as a viable method for implementing controllers in real time. With such an approach, one approximates the gain kernel, which maps the plant coefficient into the solution of a PDE, with a neural operator. It is in adaptive control that the benefit of the neural operator is realized, as the kernel PDE solution needs to be computed online, for every updated estimate of the plant coefficient. We extend the neural operator methodology from adaptive control of a hyperbolic PDE to adaptive control of a benchmark parabolic PDE (a reaction-diffusion equation with a spatially-varying and unknown reaction coefficient). We prove global stability and asymptotic regulation of the plant state for a Lyapunov design of parameter adaptation. The key technical challenge of the result is handling the 2D nature of the gain kernels and proving that the target system with two distinct sources of perturbation terms, due to the parameter estimation error and due to the neural approximation error, is Lyapunov stable. To verify our theoretical result, we present simulations achieving calculation speedups up to 45x relative to the traditional finite difference solvers for every timestep in the simulation trajectory.

用神经算子来逼近偏微分方程反推中的增益核已成为实时控制器实现的一种可行方法，本文将神经算子方法从自适应控制的双曲型偏微分方程扩展到自适应控制的基准抛物型偏微分方程中，并证明了参数自适应的Lyapunov设计下植物状态的全局稳定性和渐近调节性。

通过神经算子近似增益核实现反应扩散偏微分方程的自适应控制