Safe PDE Boundary Control with Neural Operators

TL;DR

This paper introduces a neural boundary control barrier function framework to ensure safety constraints in PDE boundary control, enabling model-free controllers to satisfy boundary safety conditions across various PDE dynamics.

Contribution

It proposes a novel neural boundary control barrier function and safety filtering method that guarantees boundary safety constraints in PDE control using neural operators.

Findings

Effective safety guarantees in PDE boundary control demonstrated across multiple PDE types.

Improved boundary constraint satisfaction compared to baseline controllers.

Plug-and-play applicability validated through extensive experiments.

Abstract

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 neural boundary control barrier function (BCBF) to ensure the feasibility of the trajectory-wise constraint satisfaction of boundary output. Based on the 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 plug-and-play effectiveness of the proposed method by achieving better general performance and boundary constraint satisfaction compared to the vanilla and constrained model-free controller baselines. The code is available at https://github.com/intelligent-control-lab/safe-pde-control.