The physical world dynamics are generally governed by underlying partial differential equations (PDEs) with unknown analytical forms in science and engineering problems. Neural network based data-driven approaches have been heavily studied in simulating and solving PDE problems in recent years, but it is still challenging to move forward from understanding to controlling the unknown PDE dynamics. PDE boundary control instantiates a simplified but important problem by only focusing on PDE boundary conditions as the control input and output. However, current model-free PDE controllers cannot ensure the boundary output satisfies some given user-specified safety constraint. To this end, we propose a safety filtering framework to guarantee the boundary output stays within the safe set for current model-free controllers. Specifically, we first introduce a general neural boundary control barrier function (BCBF) to ensure the feasibility of the trajectorywise constraint satisfaction of boundary output. Based on a neural operator modeling the transfer function from boundary control input to output trajectories, we show that the change in the BCBF depends linearly on the change in input boundary, so quadratic programming-based safety filtering can be done for pre-trained model-free controllers. Extensive experiments under challenging hyperbolic, parabolic and Navier-Stokes PDE dynamics environments validate the effectiveness of the proposed method in achieving better general performance and boundary constraint satisfaction compared to the model-free controller baselines.

翻译：物理世界动力学通常由具有未知解析形式的偏微分方程所支配，这在科学与工程问题中普遍存在。近年来，基于神经网络的数据驱动方法在偏微分方程模拟与求解方面得到了广泛研究，但从理解未知偏微分方程动力学推进到实现控制仍具挑战。偏微分方程边界控制通过仅关注偏微分方程边界条件作为控制输入与输出，实例化了一个简化但重要的问题。然而，当前的无模型偏微分方程控制器无法确保边界输出满足用户指定的安全约束。为此，我们提出一种安全滤波框架，以保证边界输出在当前无模型控制器下始终保持在安全集内。具体而言，我们首先引入一种通用的神经边界控制屏障函数，以确保边界输出轨迹约束满足的可行性。基于一个建模从边界控制输入到输出轨迹传递函数的神经算子，我们证明了边界控制屏障函数的变化与输入边界的变化呈线性依赖关系，从而可对预训练的无模型控制器实施基于二次规划的安全滤波。在具有挑战性的双曲型、抛物型及Navier-Stokes偏微分方程动力学环境中的大量实验表明，相较于无模型控制器基线，所提方法在实现更优综合性能及边界约束满足方面具有显著有效性。