Over the last decade, data-driven methods have surged in popularity, emerging
as valuable tools for control theory. As such, neural network approximations of
control feedback laws, system dynamics, and even Lyapunov functions have
attracted growing attention. With the ascent of learning based control, the
need for accurate, fast, and easy-to-use benchmarks has increased. In this
work, we present the first learning-based environment for boundary control of
PDEs. In our benchmark, we introduce three foundational PDE problems - a 1D
transport PDE, a 1D reaction-diffusion PDE, and a 2D Navier-Stokes PDE - whose
solvers are bundled in an user-friendly reinforcement learning gym. With this
gym, we then present the first set of model-free, reinforcement learning
algorithms for solving this series of benchmark problems, achieving stability,
although at a higher cost compared to model-based PDE backstepping. With the
set of benchmark environments and detailed examples, this work significantly
lowers the barrier to entry for learning-based PDE control - a topic largely
unexplored by the data-driven control community. The entire benchmark is
available on Github along with detailed documentation and the presented
reinforcement learning models are open sourced.

基于学习的偏微分方程（PDE）控制的学习环境和强化学习算法，通过引入三个基础性的 PDE 问题，降低了数据驱动控制领域中学习 PDE 控制的门槛，并在稳定性方面取得了进展，虽然代价较高。

PDE 控制竞技场：基于数据驱动的偏微分方程边界控制基准

PDE Control Gym: A Benchmark for Data-Driven Boundary Control of Partial  Differential Equations

The recently introduced DeepONet operator-learning framework for PDE control
is extended from the results for basic hyperbolic and parabolic PDEs to an
advanced hyperbolic class that involves delays on both the state and the system
output or input. The PDE backstepping design produces gain functions that are
outputs of a nonlinear operator, mapping functions on a spatial domain into
functions on a spatial domain, and where this gain-generating operator's inputs
are the PDE's coefficients. The operator is approximated with a DeepONet neural
network to a degree of accuracy that is provably arbitrarily tight. Once we
produce this approximation-theoretic result in infinite dimension, with it we
establish stability in closed loop under feedback that employs approximate
gains. In addition to supplying such results under full-state feedback, we also
develop DeepONet-approximated observers and output-feedback laws and prove
their own stabilizing properties under neural operator approximations. With
numerical simulations we illustrate the theoretical results and quantify the
numerical effort savings, which are of two orders of magnitude, thanks to
replacing the numerical PDE solving with the DeepONet.

通过 DeepONet 操作学习框架，我们扩展了基本双曲线和抛物线 PDE 的结果至涉及系统状态和输出或输入延迟的高级双曲线类。我们利用 DeepONet 神经网络逼近算子，并通过数值模拟验证了理论结果，并量化了通过 DeepONet 取代数值 PDE 求解所节省的计算工作量，达到两个数量级。