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

Adaptive control of reaction-diffusion PDEs via neural  operator-approximated gain kernels

Deep neural networks have excelled on a wide range of problems, from vision
to language and game playing. Neural networks very gradually incorporate
information into weights as they process data, requiring very low learning
rates. If the training distribution shifts, the network is slow to adapt, and
when it does adapt, it typically performs badly on the training distribution
before the shift. Our method, Memory-based Parameter Adaptation, stores
examples in memory and then uses a context-based lookup to directly modify the
weights of a neural network. Much higher learning rates can be used for this
local adaptation, reneging the need for many iterations over similar data
before good predictions can be made. As our method is memory-based, it
alleviates several shortcomings of neural networks, such as catastrophic
forgetting, fast, stable acquisition of new knowledge, learning with an
imbalanced class labels, and fast learning during evaluation. We demonstrate
this on a range of supervised tasks: large-scale image classification and
language modelling.

通过存储案例，使用基于上下文的查找直接修改神经网络的权重，以 Memory-based Parameter Adaptation 为核心的方法，可以在监督任务中有效地改善深度神经网络的性能和训练速度。